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Root groupoid and related Lie superalgebras

We introduce a notion of a root groupoid as a replacement of the notion of Weyl group for (Kac-Moody) Lie superalgebras. The objects of the root groupoid classify certain root data, the arrows are defined by generators and relations. As an abstract groupoid the root groupoid has many connected components and we show that to some of them one can associate an interesting family of Lie superalgebras which we call root superalgebras. We classify root superalgebras satisfying some additional assumptions. To each root groupoid component we associate a graph (called skeleton) generalizing the Cayley graph of the Weyl group. We establish the Coxeter property of the skeleton generalizing in this way the fact that the Weyl group of a Kac-Moody Lie algebra is Coxeter.

1.1.1. In this paper we present an attempt to generalize the notion of Weyl group to Lie superalgebras. For a semisimple Lie algebra Weyl group parametrizes Borel subalgebras containing a fixed torus. This cannot be directly extended to Lie superalgebras since there are essentially different choices of Borel subalgebras. In order to describe all Borel subalgebras, the notion of odd (or isotropic) reflection was introduced many years ago, [Sth], [P], [DP]. An odd reflection can not be naturally extended to an automorphism of the Lie superalgebra. For many years a strong feeling persisted among the experts that one should extend Weyl group to "Weyl groupoid". One attempt was made in [S]. A somewhat reminiscent construction of groupoid was suggested by I. Heckenberger and collaborators [He], [HY], see also [AA]. More recently another notion named Weyl groupoid was introduced by Sergeev and Veselov [SV] for finite-dimensional superalgebras in order to describe the character ring. In 1.10 we comment on these definitions.
The notion of root groupoid presented in this paper is close to the notion defined in [S]. 1.1.2. The connection between semisimple Lie algebras and root systems can be described from two opposite perspectives. One can start with a Lie algebra, choose a Cartan subalgebra and study the geometry of the set of roots. On the other hand, one can start with a Cartan matrix and construct a Lie algebra by generators and relations. The second approach was vastly extended to construct a very important family of infinite-dimensional Lie algebras by Kac, Moody, Borcherds and others. Our approach follows the same pattern for construction of Lie superalgebras from combinatorial data. 1.1.3. Another idea that motivated our work is the observation that the classical Serre relation can be interpreted as reflected Chevalley relations. This led us to the notion of root algebra which, roughly speaking, respects the symmetries determined by a root groupoid. In many cases there is a unique root algebra which can be defined by Chevalley relations reflected in all possible ways. Sometimes there is a number of root algebras defined by a given root datum. The description of all root algebras is an open question -we don't know the answer even for Lie algebras. For finite dimensional and affine Lie superalgebras all Serre's relations were described in [Y]. One can see from this description that Serre's relations may involve more than two generators.
1.2. Root groupoid. In [K2], 1.1, Kac defines a realization of a Cartan matrix A = (a xy ), x, y ∈ X, as a triple (h, a, b) such that a = {a(x) ∈ h}, b = {b(x) ∈ h * } and a(x), b(x) = a xy . Adopting this definition to Lie superalgebras, we add the parity function p : X → Z 2 on the index set X and make a quadruple v = (h, a, b, p) an object of the root groupoid R -the main object of our study. Every quadruple v defines a Cartan matrix by the formula above. The pair (A, p) is called in this paper Cartan datum. There are three kinds of generators in the set of arrows in R. Two of them are quite dull: one (a homothety) rescales a(x), another (tautological arrow) is defined by an isomorphism θ : h → h ′ ; the third kind are reflexions that retain the same vector space h but change the collections a(x) and b(x) by usual (even or odd) reflection formulas. Each generator f : v → v ′ defines a linear transformation h v → h v ′ (it is identity for homotheties and reflexions, and θ for the tautological arrow defined by θ); two compositions of generators leading from v to v ′ are equivalent if they define the same linear transformation. The root groupoid R has a lot of components, some of them, most probably, useless. However, some connected components (we call them admissible) lead to interesting Lie superalgebras. It is worth mentioning that Cartan matrices A v are different even inside one component: one type of reflexions, isotropic reflexions, modify Cartan matrices in a certain way (see the formulas in 2.2.4).

Root algebras.
For each v ∈ R one defines a (huge) Lie superalgebra g(v) (we call it half-baked Lie superalgebra) basically in the same way as did our predecessors V. Kac and R. Moody, see 3.1.1 in this paper. For an arrow γ : v → v ′ in R the isomorphism h(γ) : h(v) → h(v ′ ) does not extend to a homomorphism of of the half-baked algebras. We define a root algebra supported on a component R 0 of R as a collection of quotients g(v) of g(v) such that for any γ : v → v ′ the isomorphism h(γ) extends to an isomorphism g(v) → g(v ′ ).
A component R 0 of R is called admissible if it admits a root algebra. Admissibility can be expressed in terms of weak symmetricity of the Cartan matrices at R 0 , see Theorem 3.3.1. For an admissible component R 0 there always exists an initial and a final object in the category of root algebras. The initial root algebra g U is called universal. The final root algebra g KM is called Kac-Moody. Note that g KM in the admissible case is defined as the quotient of g(v) by the maximal ideal having zero intersection with h. The universal root algebra g U is obtained by imposing on g(v) reflected Chevalley relations -so generalizing the classical Serre relations.
Note that these were two different approaches of the founding fathers of Kac-Moody Lie algebras: Victor Kac [K2] factored the half-baked algebra by the maximal ideal having zero intersection with the Cartan subalgebra, whereas Robert Moody [M] imposed on it the Serre relations. 1.4. Groups associated to the root groupoid. An only algebraic invariant of a connected groupoid is the automorphism group of its object. The group Aut R (v) is one of a plethora of groups we assign to an admissible component R 0 . It acts (up to a torus) on any root Lie algebra and on the set of its roots. For the component corresponding to a semisimple Lie algebra, Aut R (v) coincides with the Weyl group. In the case of conventional Kac-Moody Lie algebras Aut R (v) is the product of the Weyl group and a certain group of "irrelevant" automorphisms. The group of irrelevant automorphism K(v) is very easy to describe. This is a subgroup of automorphisms θ ∈ GL(h(v)) preserving all b(x) ∈ h * as well as all a(x) up to constant. It is a unipotent abelian group in the case of Kac-Moody algebras. The equality Aut R (v) = W × K does not hold already for gl(1|1), see 2.6.3.
1.4.1. Skeleton. We will now present a combinatorial description of the quotient group Aut R (v)/K(v). Let Sk ⊂ R (skeleton) be the subgroupoid whose arrows are the compositions of reflexions. Denote by Sk(v) the connected component of v ∈ Sk. This is a contractible groupoid; it makes sense to study it as a marked graph, whose edges are reflexions marked by the elements of the index set X. We denote by Sk D (v) the subset of vertices in Sk(v) having a Cartan datum D-equivalent to A v , see 2.4.6. The set Sk D (v) has a group structure and Proposition 5.4.7 (3) claims that there is an isomorphism Aut R (v)/K(v) = Sk D (v).
1.4.2. Weyl group. For a vertex v in an admissible R 0 we define Weyl group W (v) (up to isomorphism, it depends on the component only) as a certain subgroup of GL(h) generated by reflections (more precisely, by the reflections with respect to anisotropic roots, see 4. 3). The Weyl group W (v) is a normal subgroup of Aut R (v). 1.4.3. Spine. We define the spine Sp as the subgroupoid of Sk whose arrows are generated by isotropic reflexions only. For instance, if there are no isotropic reflexions (for example, if p(x) = 0 for all x) then Sp has no arrows. The connected component of v in Sp is denoted Sp(v) and the intersection Sp D (v) = Sp(v) ∩ Sk D (v) is a subgroup. Proposition 5.4.7 claims that Aut R (v)/K(v) = Sk D (v) is a semidirect product W (v) ⋊ Sp D (v). In particular, if Sp D is trivial, this gives Aut(v) = W × K.
1.5. Coxeter properties. A fundamental property of Kac-Moody Lie algebras is that its Weyl group is a Coxeter group. We generalize this result to the Weyl groups appearing in any admissible component. Similarly to the classical result, the length of an element w ∈ W can be expressed as the number of positive anisotropic roots that become negative under w, see Corollary 5.2.8.
An analog of Coxeter property holds also for the skeleton Sk(v). The length of the shortest path from v to v ′ in Sk(v) can also be expressed as the number of real positive roots that become negative, see 5. 3.9.
Coxeter property for groups provides its presentation in terms of generators and relations, with relations defined by "pairwise interaction" of the generators. It turns out that a similar presentation exists for the skeleton. In Section 6 we define the notion of Coxeter graph that generalizes that of Coxeter group, and prove that the skeleton Sk(v) satisfies this property.
1.6. Fully reflectable components. Admissible Cartan matrices are not in general required to allow reflexions r x for all x ∈ X. We call a component R 0 fully reflectable if all reflexions are allowed at all vertices of R 0 . This means that Sk(v) is a regular graph of degree |X|. In Section 7 we divide all admissible indecomposable fully reflectable components into three types: finite, affine and indefinite. This trichotomy extends the similar trichotomy for Kac-Moody Lie algebras. There is a full classification of those types that contain an isotropic root; it has been done by C. Hoyt and V. Serganova, see [H], [S]. Curiously, there are only two new indefinite series having an isotropic root; they are called Q ± (m, n, k).

1.7.
On the (lack of) uniqueness of a root Lie superalgebra. We have already mentioned that, for an admissible component R 0 there is an initial g U and a final g KM root algebra supported at R 0 . The natural map g U → g KM is surjective and all root algebras are factors lying in between. In Sections 8 and 9 we study the gap between g U and g KM in the fully reflectable case. The result of these sections can be summarized as follows.
The similar result for symmetrizable Kac-Moody Lie algebras was proven by Gabber-Kac, [GK]. Their proof was adapted to our symmetrizable case in Section 8. In the case when g KM = gl(1|1) the algebra g U has dimension (4|2) and the algebras g U and g KM are the only two root algebras in this component, see 3.2.5. Note that the explicit realization of g KM for g U = sl(n|n) (i) , (i = 1, 2, 4) and sq(n) (2) is given in [S].
We also prove that if g KM = gl(1|1) then any algebra g sandwiched between g KM and g U is a root algebra.
1.8. Examples of calculation of Aut(v). In the last Section 10 we compute the group Aut R (v) for two classes of connected components.
The first one is the case of a "star-shaped" spine. It includes the algebras sq (3) (2) , B(1|1) (1) , D(2|1, a), D(2|1, a) (1) , Q ± (m, n, k). Here one has Aut R (v) = W × K as in this case Sp D (v) is trivial. For the same reason Aut R (v) = W for all finite dimensional Lie superalgebras except for the case of gl(n|n); the latter is considered in 10.3.6. The second class is the class of components whose skeleton identifies with that of sl (1) n . This includes the root data for sl(k|l) (1) , sq(n) (2) and S(2|1, b). In these cases the Weyl group W (sl (1) n ) acts simply transitively on the skeleta Sk(v). This allows one to realize the Weyl group and Sk D (v) = Aut R (v)/K(v) as subgroups of W (sl (1) n ). 1.9. Borcherds-Kac-Moody algebras. R. Borcherds in [Bo] introduced a generalization of Kac-Moody algebras, where the Cartan matrix is real symmetric and satisfies additional conditions. The proof of Gabber-Kac is valid for this class (see [K2], 11.13) and give g KM = g U if the Cartan matrix is symmetrizable and satisfies the conditions (C1')-(C3') in [K2], 11.13). Borcherds-Kac-Moody (BKM) superalgebras were studied by M. Wakimoto in [W]. Note that any Kac-Moody algebra is a BKM algebra, but many Kac-Moody superalgebras (including gl(m|n) for m, n > 2) are not BKM superalgebras.
1.10. Comment on the groupoids studied in [HY, AA]. The objects of "Coxeter groupoid" appearing in the definition in [HY] correspond to different choices of (accessible) Borel subalgebras; thus, they correspond to the vertices of what we call a skeleton component. The arrows are generated by reflections with respect to all simple roots. Our result claiming coxeterity of the skeleton (Theorem 6.1.7) means that the groupoid defined in [HY] is contractible. For instance, it assigns to a semisimple Lie algebra the Cayley graph of its Weyl group (which is contractible when considered as a groupoid). In order to get for a semisimple Lie algebra the classifying groupoid of the Weyl group instead of the contractible Cayley graph, one has to identify the vertices having the same Cartan matrix as it is done in [AA].
In the present paper we do something similar to [AA]. However, from a categorical point of view it is better to replace factoring the set of objects of a groupoid by an equivalence relation with adding isomorphisms connecting the objects that we are willing to declare equivalent. This is precisely what we do. In our root groupoid we have generators for the arrows of three different types: apart from reflexions, we have tautological arrows and homotheties that connect vertices with D-equivalent Cartan matrices. In absence of isotropic reflexions (for instance for Kac-Moody algebras) the Weyl group coincides with the automorphism group of an object of the corresponding component of a root groupoid. However, this does not hold in general, see 5.4.7 and Section 10.
1.11. Acknowledgements. V.S. enjoyed numerous visits to Weizmann Institute whose pleasant atmosphere is gratefully acknowledged. We are grateful to J. Bernstein whose comment initiated the project. R. Moody's question triggered our Sections 5 and 6 devoted to Coxeter properies of the root groupoid. 2. Setup 2.1. Groupoid of root data. For a complex vector space h and a set X, a map a : X → h will be called injective if the induced map Span C (X) → h is an injective map of vector spaces.
Once and forever we fix a finite set X. The cardinality of X will be called the rank of root data and of Lie superalgebras connected to them.
2.1.1. We now define the root groupoid R.
The objects of R (the root data) are the quadruples (h, a : X → h, b : X → h * , p : X → Z 2 ) where h is a finite dimensional vector space over C such that a, b are injective.
We will define the arrows of R by generators and relations.
We have generating arrows of three types: and a reflectable element x ∈ X, see 2.2 for the explicit formulas; 2. a tautological arrow This collection of objects and arrows (=quiver) generates a free category denoted (temporarily) F. The groupoid R will be defined as the one with the same objects as F, and whose arrows are equivalence classes of the arrows above. The equivalence relation is defined below.
First of all, we define a functor h : F → Vect to the category of vector spaces carrying (h, a, b, p) to h, carrying the reflexions and the homotheties to the identities, and tautological arrows to the respective isomorphisms of the underlying vector spaces.
2.1.2. The equivalence relation on each Hom-set of F is defined as follows: two 2.2. Formulas for the reflexions. Any root datum (h, a, b, p) determines a Cartan matrix A(a, b) = (a xy ) x,y∈X given by the formula a xy := a(x), b(y) .

2.2.1.
Definition. An element x ∈ X is called reflectable at v = (h, a, b, p) if the following conditions hold.
(anisotropic) If a xx = 0, then p ′ := p and The pair (A = {a xy }, p) will be called Cartan datum for v.
Note that the reflectability of x ∈ X, as well as the formulas for the reflexion r x depend only on the Cartan datum.

Let us indicate what happens to a Cartan matrix under a reflexion.
Anisotropic reflexions preserve the Cartan matrix. If r x : v → v ′ is an isotropic reflexion (a xx = 0), the Cartan matrix (a ′ yz ) is given by the following formulas Proof. It is enough to verify that each generating arrow in F has invertible image in R. First of all, in our category the composition of two tautological arrows is tautological, defined by the composition of the corresponding isomorpisms h Similarly, composition of two homotheties is a homothety. This implies that tautological arrows and homotheties are invertible. Invertibility of reflexions follows from the formulas: one has r 2 x = id for all x (this is an explicit calculation).
Note the following observation.
The morphisms r x , t θ and h λ commute with each other.
The root groupoid R consists of connected components some of which will lead to interesting Lie superalgebras.
We present below properties of a component R 0 of R that will be relevant to Lie theory. This is weak symmetricity. 2.4.4. Remark. The root groupoid R is an object of "mixed" nature. It is a groupoid, but its objects and Hom sets carry an extra information (markings a, b, p, generators r x , t θ , h λ for arrows). This is why we cannot easily replace R with any equivalent groupoid (for instance, leaving only one object for each connected component). Nevertheless, we can safely assume that h is the same vector space at all objects of a given connected component R 0 , allowing however the tautological arrows t θ for automorphisms θ : h → h.
2.4.5. Remark. Tautological arrows and anisotropic reflexions (those with a xx = 0) preserve the Cartan datum. Homotheties also preserve local weak symmetricity. Isotropic reflexions usually do not satisfy this property. For this reason admissible root data with isotropic reflexions can be classified under the assumption that all elements x ∈ X are reflectable at every vertex, [H].
2.4.6. Definition. Two Cartan data, (A, p) and (A ′ , p ′ ), will be called D-equivalent if p = p ′ and there exists an invertible diagonal matrix D such that A ′ = DA.
Obviously, homotheties carry a Cartan datum to a D-equivalent one.
2.4.7. Remark. In studying a connected component R 0 of R it is often important to describe Cartan data (A(v), p) at all vertices v ∈ R 0 , up to D-equivalence. Since only isotropic reflections change the Cartan data, it is sufficient to perform only sequences of isotropic reflections, see 4.2.8.
2.5.1. We present an example of a reflexion r x : v → v ′ such that all y ∈ X are reflectable at v but some are not reflectable at v ′ .
Take the root datum v with X = {x, y}, the Cartan matrix 0 −s −s 1 , s ∈ N, p(x) = p(y) = 1. Then x and y are reflectable at v. For the reflexion r x : v → v ′ the reflected Cartan matrix is 0 s s 1 − 2s and p ′ (x) = 1, p ′ (y) = 0. Thus y is reflectable at v ′ only if 2s 2s−1 ∈ N that is for s = 0, 1.
2.6.1. Semisimple case. Let v = (h, a, b, p) represent a root system of a finite dimensional semisimple Lie algebra. This means that p(x) = 0, a : X → h is a set of simple coroots and b : X → h * is the set of simple roots. Both a and b give bases in h and h * . Let us calculate the group of automorphisms of (h, a, b, 0). Any reflexion r with the action of the Weyl group W and of Aut R (v) on h. Since the actions are faithful, this defines an injective group homomorphism We claim that it is bijective. In fact, any automorphism η : v → v in R is a composition of reflexions r x , tautological arrows and homotheties. Since reflexions, tautological arrows and homotheties commute, one can, using 2.4.1, present for a certain w ∈ W . It remains to verify that if h λ • t θ ∈ Aut R (v), then it is identity. Since t θ does not change the Cartan matrix, h λ = id. Since any automorphism of v carries a(x) and b(x) to themselves, and a(x) generate h, θ = id.
2.6.2. Kac-Moody case. In the case when (h, a, b, 0) has the Cartan matrix satisfying the conditions of [K2], 1.1, the calculation of 2.6.1 works almost as well. Let W denote the Weyl group and let W = Aut R (v). We have a group homomorphism i : W → W defined exactly as in the semisimple case. Precisely as in the semisimple case we have a decomposition (1) of an automorphism η ∈ W and deduce that h λ = id as the Cartan matrix has no zero rows. Denote Let us show K is a commutative unipotent group. Denote A ⊂ h and B ⊂ h * the subspaces spanned by the images of a and b. One has dim A = dim B = |X| and dim h = 2|X| − r where r is the rank of the Cartan matrix. This is equivalent to saying that the orthogonal complement B ⊥ ⊂ h of B lies in A. If θ is an automorphism of the triple (h, a, b), θ − 1 vanishes on A and has image in B ⊥ . This means that (θ − 1) 2 = 0. Moreover, any two such automorphisms commute. The dimension of K is (|X| − r) 2 . 2.6.3. Root datum for gl(1|1). We assume dim(h) = 2, It is easy to see that Aut(v) = Z 2 × K where Z 2 is generated by the automorphism described above and K = {θ : h → h| θ(a) ∈ C * a, θ * (b) = b}.
For more examples see 5.4.11 and Section 10.

Root Lie superalgebras
In this section we define root Lie superalgebras corresponding to certain (admissible) connected components of the groupoid R of root data.
We call g(v) the half-baked Lie superalgebra defined by the root datum v ∈ R.
3.1.2. The following properties of g := g(v) are proven in Thm. 1.2 of [K2] 2 . 1. The algebra h acts diagonally on g. We denote by g µ the weight space of weight µ, so that g = ⊕ µ∈Span Z (b) g µ , where Span Z (b) denotes the abelian subgroup of h * generated by b(x), x ∈ X. 2. There is a standard triangular decomposition where n + is freely generated byẽ x , x ∈ X and n − is freely generated bỹ f x . 3. For each x = y one has g jb(x)+b(y) = 0 for j ∈ Z ≥0 and g jb(x)+b(y) is spanned by (adẽ x ) jẽ y . The following theorem is very similar to [K2], Thm. 2.2.

Proposition.
Let v ∈ R have a symmetric Cartan matrix (a xy ). Let (·|·) be a nondegenerate symmetric form on h satisfying the condition (a(x)|h) = b(x), h for any x ∈ X, h ∈ h 3 . Then there exists a unique extension of (·|·) to an invariant symmetric bilinear form on g = g(v). This extension enjoys the following properties. 1 where ν : h * → h is the isomorphism defined by the original nondegenerate form.
3.1.4. The algebra g(v) admits a standard superinvolution θ, that is an automorphism whose square is id on the even part and −id on the odd part of g(v).
We will define the superinvolution θ by the following formulas.
If a xx = 0 and p(x) = 0, we have g = sl 2 ; if p(x) = 1, we have g = osp(1|2). 3.1.6. The space [ g jb(x)+b(y) , g −jb(x)−b(y) ] lies in h for any j ≥ 0 and is at most one-dimensional. We wish to describe, under certain assumptions, the greatest value of j for which it is nonzero. Assume axx for a xx = 0, j 0 = 1 for a xx = 0, a xy = 0, and j 0 = 0 for a xx = 0 = a xy .
(c) By (a), (b) it follows that I ∩ h = 0 if a xx = 0, a xy = 0. Therefore we may assume that a xy = a yx = 0 or a xx = 0. It is enough to verify that [ẽ z , F ] = [f z , E] = 0 for z = x, y. These formulas are similar so we will check only the formula [ẽ z , F ] = 0.
3.1.8. Proposition. Assume that x = y ∈ X and x is reflectable. We also assume that if a xx = 0 and a xy = 0 then a yx = 0.
Proof. The claim immediately reduces to the case X = {x, y}. Denote by I the ideal of g generated by the elements E := (adẽ x ) j 0 +1ẽ y , F := (adf x ) j 0 +1f y . By Lemma 3.1.7 we have I ∩ h = 0. The homomorphism g → g = g/I is identity on h, so both claims of the proposition would follow from the similar claims for g. Since the first claim of the proposition tautologically holds for g, we have proven it also for g.
To prove the second claim for g, we will study the isotropic and the anisotropic cases separately.
The case a xx = 0. The rank one subalgebra defined by {x} ∈ X contains a copy of sl 2 . g is integrable as an sl 2 -module as it is generated by the elements on which e x ,f x act locally nilpotently, see [K2], Lemma 3.4. Therefore, the automorphism σ : g → g given by the formula is defined. Its restriction on h is given by the standard formula where the action of σ on h * is induced by its action on h. The latter implies the second claim of the proposition for the algebra g.
The case a xx = 0. If a xy = 0, the second claim is immediate. In the case a xy = 0 a direct calculation shows that [[ẽ x ,ẽ y ], [f x ,f y ]] = (−1) p(y) a xy (a(y) + a yx a xy a(x)).

3.2.
Coordinate systems and root algebras.
A v-coordinate system on a Lie superalgebra g is a surjective homomorphism g(v) → g whose kernel has zero intersection with h(v).
In other words, a v-coordinate system on g consists of an injective map of Lie superalgebras h → g (h is even commutative), and a collection of generators e x , f x such that the relations 1-4 of 3.1.1 hold.
Here is our main definition.
Let g be a root Lie superalgebra at R 0 . There is a weight space decomposition , and therefore induces an isomorphism We call the collection g KM = {g KM (v)} v∈R 0 the Kac-Moody Lie superalgebra supported at an admissible component R 0 . In other words, Kac-Moody Lie superalgebra g KM is the terminal object in the category of root Lie superalgebras supported at an admissible component R 0 .
The superinvolution θ of g defined in 3.1.4 induces an automorphism of g KM .
3.2.5. Rank one. The Lie algebra sl 2 plays a prominent role in Lie theory. A similar role in our setup will be played by root algebras of rank 1. Let us describe them all. Let X = {x}. In this case g(v) is described in 3.1.5. It is a root algebra. If a xx = 0 or p(x) = 0, then g KM = g. If a xx = 0 and p(x) = 1, the maximal ideal r of g having zero intersection with h is spanned by e 2 x , f 2 x and g KM = g/r ∼ = gl(1|1). The algebras g and g KM are exactly two root algebras in this case as only these two allow an automorphism lifting γ = t −1 • r x , see 2.6.3.
3.2.6. Decomposable root datum. Let X = X 1 ⊔ X 2 and let v i = (h i , a i : be two root data of ranks |X 1 | and |X 2 | respectively.
We define their sum v = v 1 + v 2 in an obvious way, as the root datum with h = h 1 ⊕ h 2 and a : X → h, b : X → h * and p : X → Z 2 defined by the conditions are the obvious embeddings. We will denote by R(X), R(X 1 ) and R(X 2 ) the groupoids of root data for the sets X, X 1 and of the corresponding components of R(X 1 ) and R(X 1 ). If g 1 and g 2 are root algebras supported on the components R ′ 0 and R ′′ 0 respectively, the product g = g 1 × g 2 is a root algebra of R 0 . In particular, It is not true in general that any root algebra supported on R 0 is a product.
Here is the best we can say.
Then any root algebra supported on R 0 uniquely decomposes as a product of a root algebra supported on R ′ 0 and a root algebra supported on R ′′ 0 . Proof. The algebra g = g(v) is generated by h, e x , f x , e y , f y where x ∈ X 1 and y ∈ X 2 . We have to verify that [e x , e y ] = 0 = [f x , f y ] for x ∈ X 1 and y ∈ X 2 . The reflexion r x : v → v ′ with respect to x ∈ X 1 carries, up to scalars, e x to f ′ x and f x to e ′ x , retaining e y and f y .
We can apply the sum of root data operation to an empty root datum ∅ V corresponding to X = ∅ and uniquely defined by a vector space V . For v = (h, a, b, p) the sum ∅ V + v has form (h ⊕ V, a, b, p) and any root algebra based on it is the direct product of a root algebra based on v with the commutative algebra V .
The following result is a corollary of 3.2.5.
3.2.8. Corollary. Let R 0 be an admissible component of R and g := g(v) be a root algebra. Fix x ∈ X and set α := b(x). We denote by g α the subalgebra of g generated by g α and g −α .
Proof. Clearly, g α is a quotient of the algebra [ g, g] where g is the corresponding algebra listed in 3.2.5; this gives (1), (2), (3) and shows that in (4) it is enough to verify g 2b(x) = 0. This follows from Lemma 3.1.7(b).

3.3.
Admissibility is just a weak symmetricity. In this subsection we prove the following result.
In order to construct ρ, we have to find ρ(ẽ ′ y ), ρ(f ′ y ), and verify the (very few) relations.
The latter is also possible by Proposition 3.1.8(2). It remains to verify that for some c ∈ Z. This is not a weight of g, so the bracket should vanish. ( where j 0 is defined as in 3.1.6. According to Lemma 3.1.7(c) the ideal generated by this weight space has no intersection with h, so this is not a weight of g and the bracket vanishes. Therefore, we have constructed a homomorphism ρ : g ′ → g for each reflexion r x : v ′ → v. It is identity on h, so it induces a homomorphism g ′ → g. Any reflexion has order two, so there is also a homomorphism g → g ′ in the opposite direction. Their composition preserves weight spaces, so it is invertible.
2. Assume now that R 0 is an admissible component. We will deduce that it is necessarily weakly symmetric. Assume that there exists v ∈ R 0 , a v-reflectable element x ∈ X and another y ∈ X such that a xy = 0. Let g be a root algebra.
Look at the x-reflexion r 3.4. Admissible components in rank two. In this subsection we show that any locally weakly symmetric root datum of rank two belongs to an admissible component (that is, a local weak symmetricity implies a weak symmetricity).
Classification of fully reflectable root data is available for all ranks. Fully reflectable admissible root data without isotropic real roots can be easily classified as all Cartan matrices in the component are D-equivalent. The classification of fully reflectable admissible root data with isotropic real roots was obtained in [H].
3.4.2. Symmetrizable. The cases a xy = a yx = 0 as well as a xy = 0 and a yx = 0 are symmetrizable, therefore, symmetrizable at all vertices by Lemma 2.3.1.

3.4.3.
Weakly symmetric but not symmetrizable. This is possible only if R 0 contains an object v having nonreflectable y ∈ X. Thus, the Cartan matrix should have form with a xy = 0. Since y is nonreflectable, a yy = 0 and p(y) = 0.
(a) Let a xx = 0 so p(x) = 1 since x is reflectable. Then A = 0 a xy 0 0 , that, after the reflexion, will become In this case the Cartan matrix is not changed and therefore the component is weakly symmetric.
Given a fibration f : G → H and h ∈ H, the fiber of f at h, G h , is defined as follows. • 3.5.2. Let R 0 be an admissible component of the root groupoid and let g be a root algebra on R 0 . Define the groupoid of symmetries of g, G 0 , together with a fibration π : G 0 → R 0 , as follows. The groupoids G 0 and R 0 have the same objects.
The fiber of π at v ∈ R 0 consists of automorphisms of g(v) that are identity on h(v). Any such automorphism a preserves the weight spaces, and so it is uniquely given by a collection of λ x . Therefore, the fiber of π at v identifies with the classifying groupoid 6 of the torus (C * ) X .
It is an equivalence as it induces an equivalence of fibers at any v ∈ R 0 .
3.6.1. In this subsection we will prove the existence of an initial object in the category of root algebras associated to an admissible component R 0 of R.
Let g be a root Lie superalgebra for the component R 0 . Fix v ∈ R 0 . The v-coordinate system for g is a Lie superalgebra epimorphism g(v) → g(v). Let k(v) be its kernel.
Choose an arrow α : v ′ → v in R presentable as a composition of reflexions. We denote g ′ = g(v ′ ) and g = g(v). The existence of isomorphism g ′ → g lifting α proves that , the sum being taken over all α : v ′ → v presentable as compositions of reflexions.
Let us verify that the collection We have to define, for each arrow α : This is enough to verify separately for reflexions, homotheties and tautological arrows. In the case when α is a tautological arrow or a homothety, it extends to an isomorphismα : g(v) → g(v ′ ). Since the homotheties and the tautological arrows commute with the reflexions,α carries s(v) to s(v ′ ), and this induces an isomorphism g U (v) → g U (v ′ ). It remains to define, for each reflexion r The algebra g U (v) is generated over h by the elements e y of weight b(y), f z of weight −b(z), subject to relations listed in 3.1.1 and factored out by s(v). Thus, in order to construct ρ, we have to choose ρ(e y ) ∈ g , ρ(f y )] = a(y). The latter is possible by Proposition 3.1.8(2). The rest of the relations say that, for any composition Any reflexion has order two, so there is also a homomorphism in the opposite direction. Their composition preserves weight spaces, so it is invertible.
This proves that the collection of algebras g U = { g(v)/s(v)} is the initial object in the category of root algebras based on R 0 . 3.6.2. Definition. The root algebra g U = { g(v)/s(v)} defined as above is called the universal root Lie superalgebra defined by the component R 0 7 .
The superinvolution θ of g defined in 3.1.4 induces an automorphism of the universal root algebra.
3.6.3. Serre relations. The classical Serre relations (ade x ) −axy+1 (e y ) = 0, (adf x ) −axy+1 (f y ) = 0, for x, y ∈ X such that a xx = 0 are among the most obvious relations defining the universal Lie superalgebra. They correspond to the summand , however, is usually not generated by the classical Serre relations.
The converse of this fact also holds; one has the following easy result.
Lemma. Let g U be the universal root algebra at a component 3.6.5. Remark. The lemma above implies that a root Lie superalgebra is canonically determined by any its component g 3.7. A side remark: groupoid extensions. The groupoid extension π : G 0 → R 0 has fibers isomorphic to classifying spaces of a torus. This very special type of extension admits a description in terms of gerbes. For This defines an automorphism of the fiber (G 0 ) v given by the formula α →γ • α •γ −1 . The result is independent of the choice ofγ as tori are abelian groups.
The above described action can be encoded into a groupoid extension p : T → R 0 that is a group over R 0 : one has a multiplication m : T × R 0 T → T corresponding to the fiberwise multiplication. Finally, π : G 0 → R 0 is a T-torsor: there is an action T × R 0 G 0 → G 0 . In more classical terms, we are talking about presenting an abelian group extension as a torsor over a split abelian group extension that is a semidirect product of the base and the fiber.
The group p : T → R 0 is easy to describe. The groupoid R 0 comes with the functor h : R 0 → Vect.
We define a functor T : R 0 → Gp into the category of groups assigning to v the factor group T The functor T gives rise to a groupoid extension p : is given by rescaling.
Note that the torsor G 0 is nontrivial as, for instance, for g = sl 2 the groupoid extension π : G 0 → R 0 is the projection N(T ) → W of the normalizer of the torus to the Weyl group that is not split.

Weyl group
Throughout this section we assume that R 0 is an admissible component of R.
The parity function p : X → Z 2 extends to a group homomorphism p : Q(v) → Z 2 that we denote by the same letter p.
Proof. The claim directly follows from the formulas for reflexions.
. By 3.2.8, for any root algebra g, ∆ re (v) ⊂ ∆(v) and all real root spaces of g are one-dimensional. Real roots coming as described above from γ : 4.2. Isotropic, anisotropic and nonreflectable real roots.
. We are going to extend these definitions to real roots. Since a real root at v is defined by a path γ : v → v ′ and a simple root at v ′ , the extension is possible if two simple roots at v ′ and v ′′ defining the same real root, are of the same type.
Then one of the following options holds.
Proof. We can assume, without loss of generality, that Let g be a root algebra and let y acts locally nilpotently on g(v ′ ), and this implies that y is reflectable at v ′ . Let now x be reflectable at v and y reflectable at v ′ . Then Corollary 3.2.8 describes possible options for g α . This implies the claim. 4.2.3. Proposition 4.2.2 allows one to extend the classification of simple roots to all real roots.
One has a decomposition is the set of isotropic real roots that are reflectable simple roots at some v ∈ R 0 . ∆ an (v) is the set of anisotropic real roots that are reflectable simple roots at some v ∈ R 0 . Any anisotropic real root α ∈ ∆ an (v) defines a coroot α ∨ ∈ h(v). ∆ nr (v) is the set of non-reflectable real roots, those that for any v ∈ R 0 and x ∈ X such that α = b(x), x is non-reflectable at v.
Remark. In our definition isotropic roots are necessarily real. In another tradition, a root of a symmetrizable Lie superalgebra is called isotropic if it has length zero. For the real roots both notions of isotropicity coincide.
For α ∈ ∆ an (v) the pair (α, α ∨ ) defines a reflection s α acting both on h(v) and on h * (v) by the usual formulas Proof. The first claim is a direct consequence of formula (5) and 4.2.2. The second claim directly follows from the formulas for s α . 4.2.5. Skeleton. We define Sk ⊂ R as the subgroupoid having the same objects as R; an arrow γ : v → v ′ is in Sk if it can be presented as a composition of reflexions. This is the skeleton groupoid.
We denote by Sk(v) the connected component of the skeleton containing v. Note that, by definition, any arrow in Sk(v) induces the identity map of h(v), so any two arrows with the same ends coincide. Therefore, Sk(v) is a contractible groupoid. Note that any arrow γ : v → v ′ in R can be decomposed γ = γ ′′ • γ ′ where γ ′ is in Sk and γ ′′ is a composition of a homothety and a tautological arrow. 4.2.6. Remark. As we prove later in 5.3.7, this decomposition is unique.
Thus, we have be an anisotropic root. Without loss of generality we can assume that there is an arrow γ : v → v ′ in Sk(v). Then the composition induces the reflection s α on h. This proves that generators of W (v) are in the image of the embedding Aut R (v) → GL(h(v)), so that the Weyl group identifies with a subgroup of Aut R (v). It is clear that any arrow γ : v → v ′ intertwines the canonoical embeddings Note that Aut R (v) acts on W (v) so that the embedding i commutes with this action. This means that W (v) is a normal subgroup of Aut R (v).
Proof. Immediate from the formulas 2.2.2 and (7).  (10) holds then x is reflectable at v if and only if it is reflectable at w. This is so as the Cartan matrices of v and of w coincide.
The proposition defines an action of the Weyl group W on Sk(v).
Proof. The uniqueness claim is obvious. For the existence, it is sufficient to verify the claim for w = s α . We can assume that α = b(x) is a simple root at v and let r x : v → u be the reflexion.
. Note that the composition ψ necessarily makes sense. Now a consecutive application of Lemma 4.3.4 yields the result. 4.3.7. Remark. The proof provides us with an explicit formula: can be easily expressed in terms of the action of W on Sk(v).

Corollary. For any
Proof. The composition t w • γ w is an endomorphism of v. The automorphism i(w) is uniquely defined by its action on h. The composition t w • γ w provides the same action.
We will show later (see 5.2.6) that the action of the Weyl group W (v) on Sk(v) is free. It is not transitive in general. Here is what we can say about the orbits of the action. 4.3.9. Lemma. For every v, v ′ ∈ Sk(v) there exists w ∈ W (v) and a sequence of isotropic reflexions If i is the first index for which φ i is an anisotropic reflexion, we can, as in the proof of Proposition 4.3.6, erase it, replacing reflexions φ j , j > i with their namesakes ψ j , so that the target of the composition satisfies the property s α (v ′′ ) = v ′ , for an anisotropic root α defined by φ i . Continuing parsing the decomposition of φ in this way, we end up with the required decomposition. 4.3.10. Principal reflections. In the case p(x) = 0 for all x and for all v ∈ R 0 , the Weyl group W is known to be generated by simple reflections s b(x) , x ∈ X for a fixed vertex v ∈ R 0 . This is not true in general, as, for instance, there may exist v ∈ R 0 for which all a xx = 0.
Here is what can be said in general.
and an element x ∈ X such that α = b ′ (x). A reflection s α with respect to a v-principal root is called a v-principal reflection.
One has is a composition of reflexions and α = b ′ (x). We will prove the claim by induction on length of the presentation of γ as a composition of reflexions.
If the sequence consists of isotropic reflexions only, α is principal and there is nothing to prove. Otherwise there is an anisotropic reflexion in the sequence. We denote below by φ ′ a composition of isotropic reflexions and by r y the first anisotropic reflexion.
Let v 1 = (h, a 1 , b 1 , p 1 ) and β = b 1 (y). By Proposition 4.3.6, s β carries v ′ to a vertex v ′′ obtained as the target of a composition of reflexions ψ : v 1 → v ′′ having the same indices as the components of φ : v 2 → v ′ . We denote v ′′ = (h, a ′′ , b ′′ , p ′′ ) and we get b ′ (x) = s β (b ′′ (x)). Therefore, s α = s b ′ (x) = s s β (b ′′ (x)) = s β s b ′′ (x) s β , the last equality by 4.2.4. Now s β is principal and v ′′ has a shorter sequence of reflexions connecting it to v. 4.3.13. Remark. The proof of 4.3.12 implies that any root α ∈ ∆ an (v) is Wconjugate to a principal root.

Modules over a root algebra.
4.4.1. Definition. Let g := g(v) be a root Lie superalgebra supported at R 0 . A weight g-module M is, by definition, an g(v)-module M whose restriction h is semisimple.
For a weight g-module M we denote by Ω(M) the set of weights of M. We will now define integrable g-modules.
4.4.2. Definition. Let g = g(v) be a root Lie superalgebra. We say that a weight g-module M is integrable if g α acts locally nilpotently on M for each anisotropic α ∈ ∆ re .
Note that the adjoint representation of any root Lie superalgebra is integrable.
Let g be a root Lie superalgebra and let M be an integrable g-module. Corollary 3.2.8 implies that Ω(M) is W -invariant. Moreover, the multiplicities of the weights µ and w(µ) coincide.
The adjoint representation of any root Lie superalgebra g is integrable. In particular, the set of roots ∆(g) of any root algebra is W -invariant.

Coxeter structures
5.1. Introduction. A Coxeter structure on a group G is a set of elements s i ∈ G such that (G, {s i }) is a Coxeter group. A Coxeter structure on a group provides its combinatorial description.
In this section we prove that the Weyl group of any admissible component R 0 has a Coxeter structure. A somewhat similar combinatorial description can be given to the components of the root groupoid.

Fix an indecomposable admissible component
In what follows we use the notation of 4.2.5, suppressing the parameter v from the notation. Thus, we will write h for h(v), Σ for Σ(v), and, for

5.2.
Coxeter structure of the Weyl group. Fix a vertex v ∈ R 0 . Let α 1 , . . . , α m be the set of v-principal roots and s i be the reflection s α i . The Weyl group W is generated by s i . We say that w = s i 1 . . . s i l is a reduced decomposition if it has a minimal length. In this case we say that ℓ(w) = l is the length of w.
5.2.1. Lemma. Let α be an anisotropic real root.
Proof. For (1) see 4.3.13. To prove (2) we notice that (Q + v ′ \ Q + v ) ∩ ∆ re consists of isotropic roots. Now let us show (3). By (1) and (2) it suffices to check that if α ∈ C and s i is a principal reflection then s i (α) ∈ C or s i (α) ∈ −C. Indeed, let v ′ be a vertex such that α i ∈ Σ v ′ . Then s i (α) ∈ Q + v ′ unless α = −α i . In the latter case α ∈ −C.

Lemma above claim (2) means that
This is the set of positive anisotropic roots (with respect to any v ′ ∈ Sp(v)).

5.2.2.
Lemma. Let w = s i 1 . . . s it and let α i be a principal root such that w(α i ) ∈ −C. Then there exists j such that ws i = s i 1 . . .ŝ i j . . . s it .

5.2.3.
Corollary. If w = s i 1 . . . s i l is a reduced decomposition and α i is a principal root then Proof. See [B], Théorème 6.1, §1, Ch. 4. If α and β are principal roots so that s 1 = s α and s 2 = s β , it is easy to see that the union W ′ (α) ∪ W ′ (β), where W ′ is the subgroup of W generated by s 1 and s 2 , is a classical root system of rank 2. This implies that m = 2, 3, 4, 6 or ∞.
Proof. Let w = s i 1 . . . s i l be a reduced decomposition. Set β j = s i 1 . . . s i j−1 (α i j ).

Skeleton as a graph.
A structure similar to the Coxeter structure on the Weyl group exists also on admissible components of the root groupoid. We fix v 0 ∈ R and study a combinatorial structure of Sk(v 0 ). Note that, from the algebraic point of view, Sk(v 0 ) is a contractible groupoid, so it may be seen as something lacking any interest. However, its arrows are compositions of reflexions, and remembering these reflexions makes a lot of sense. In this subsection we present a description of the shortest path length in this graph, similar to the one given in 5.2.8. It has a nice application to the description of the group Aut R (v) in 5.4. In Section 6 we study a Coxeter property of Sk(v).

5.3.1.
We look at the skeleton Sk(v 0 ) as the graph where the reflexions connect the vertices. Thus, the reflexions are the edges of our graph. We color the edges by elements of h * = h(v 0 ) * : a reflexion v rx − → v ′ is colored by the real root α = −b(x) = b ′ (x). Note that ∆ + re (v ′ ) is obtained from ∆ + re (v) by replacing the (existing) root −α with α.
For a path v 0 In particular, if a path is colored by the sequence (α 1 , . . . , α t ) with α t = α 1 , then there exists i such that α i = −α 1 .
We will start with an obvious remark.

Remark.
Let v rx − → v ′ be a reflection. If a xy = a yx = 0 and x = y, then a ′ (y) = a(y), b ′ (y) = b(y) and the yth rows (and the yth columns) of the Cartan matrices A v , A v ′ are equal. (1) α − β ∈ ∆ KM (the set of roots of g KM ).
(2) There exists a path v 0 Proof. If (2) is fulfilled, both α and β are simple roots at v 2 , so their difference is not a root. Let us prove that (1) implies (2). We have α, −β ∈ Σ v 1 , −α ∈ Σ v 0 and β ∈ Σ v 2 . We will denote by A i = (a i xy ) the Cartan matrix at v i and we will write a i (z) and b i (z) instead of a v i (z) and b v i (z).
We will show that x is reflectable at v 2 , y is reflectable at v 3 and r y • r x carries v 2 to v 0 . This will give the square in Sp(v) shown in the picture.
Reversing the lower reflexions, we get the required result. Note that reflectability of x ∈ X at v is determined by the x-th row of the Cartan matrix at v. By 5.3.2 the x-row of A 2 is equal to the x-row of A 1 , so x is reflectable at v 2 . Since b 1 (x) = b 2 (x), a 1 (x) = a 2 (x) and the xth row (resp., xth column) of A 2 is equal to the xth row (resp., xth column) of Once more, by 5.3.2 applied to r x : v 2 → v 3 , the y row of A 3 is equal to the y-row of A 2 , so y is reflectable at v 3 . It remains to show that r y carries v 3 to v 0 . Since b 2 (y) = b 3 (y), a 2 (y) = a 3 (y) and the yth row (resp., yth column) of A 3 is equal to the yth row (resp., yth column) of A 2 , we have Proof. Set β := α − α 1 . Let us show that β is not a root. Assume the contrary. Then β is even and β 2 is not a root. Since the set of even positive roots β such that β 2 is not a root is preserved by isotropic reflections, β = α − α 1 ∈ ∆ + vs . Therefore Now the assertion follows by induction in s.
The following result describes an exchange property for a sequence of isotropic reflections.

In 10.4 below we will see that
In particular, if a homothety can be presented as a composition of reflexions, it is the identity.
Proof. By Lemma 4.3.9 there exist v ′′ ∈ Sp(v) and w ∈ W such that v ′ = w(v ′′ ). The sets of positive anisotropic roots at v and at v ′′ coincide as none of them can become negative under an isotropic reflection. Therefore, w = 1 by 5.2.8. This be a sequence of isotropic reflections colored by a sequence (α 1 , . . . , α d ). Since ∆ + re (v ′ ) = ∆ + re (v), the formula (14) implies α i = −α j for some i, j. Then by 5.3.5 the sequence of isotropic reflections can be shortened.

Definition. For two vertices
is nonempty, it has an element that is a simple root α at v that can be replaced with −α by a reflection. Continuing this, we can get, after the required number of steps, a vertex v ′′ having the same ∆ + re (v ′′ ) as ∆ + re (v ′ ). Then by 5.3.7 v ′′ = v ′ . Note the following description of non-reflectable roots.

Corollary
Proof. Obviously, if α is isotropic or anisotropic, −α ∈ ∆ re . Let us assume that −α ∈ ∆ re , α ∈ Σ v and −α ∈ Σ v ′ . By formula (14) any path connecting v with v ′ contains an edge where α becomes negative. This proves reflectability of α. 5.3.11. Weyl vector. Choose ρ v ∈ h * such that for all x ∈ X. For each v ′ ∈ Sk(v) we define Note that the formula (15) holds for all v ′ ∈ Sk(v).

5.4.1.
Proposition. There is a unique action of Aut R (v) on Sk(v) such that for any u ∈ Sk(v) and γ ∈ Aut R (v), b γ(u) (x) = γ(b u (x)).
Proof. Uniqueness follows from 5.3.7. It is therefore sufficient to verify that for each u ∈ Sk(v) and γ ∈ Aut R (v) there is u ′ ∈ Sk(v) satisfying the property b u ′ (x) = γ(b u (x)). We proceed as follows. We present γ = γ ′′ •γ ′ where γ ′ : v → v ′ is a composition of reflexions and γ ′′ is a composition of a homothety with a tautological arrow. Choose a path ) which proves the claim.

5.4.2.
Corollary. The action of Aut R (v) on h * preserves ∆ re , as well as ∆ iso , ∆ an , ∆ nr .
Proof. The first claim follows from the formula b γ(u) (x) = γ(b u (x)). The rest follows from the fact that u and u ′ = γ(u) have D-equivalent Cartan data.
The group Aut R (v) has a trivial part which we now describe.

5.4.3.
Definition. An automorphism γ ∈ Aut R (v) is called irrelevant if it can be presented as a composition of a homothety and a tautological arrow.
The group of irrelevant automorphisms identifies with Proof. K(v) is the kernel of the action of Aut R (v) on ∆ re .
We can easily describe the image of Aut R (v) in the automorphisms of Sk(v). The description of the action presented above implies that the automorphism of Sk(v) defined by γ ∈ Aut R (v) is uniquely determined by the target v ′ of γ ′ : v → v ′ where γ ′ is the composition of reflexions appearing in the decomposition of γ. The vertex v ′ ∈ Sk(v) has the Cartan datum D-equivalent to that of v. This identifies the image of Aut R (v) with the set of the vertices on Sk(v) satisfying this property. 5.4.5. We denote by Sk D (v) the subset of (the vertices of) Sk(v) consisting of the vertices whose Cartan data are D-equivalent to that of v. The set Sk D (v) is endowed with the group structure induced from the group structure on Aut R (v). It is combinatorially described using "namesake path" construction described in the proof of Proposition 5.4.1. By construction we have an isomorphism is trivial by 4.2.6. 5.4.6. The group Sk D (v) has a subgroup Sp D (v) defined as the subset of Sk D (v) belonging to Sp(v). The following proposition summarizes what we know about the structure of the automorphism group.
where ψ is a composition of reflexions and η is a composition of a homothety with a tautological arrow. By 4.3.9 ψ decomposes as v and v ′′ are D-equivalent (actually, the same), so Sk D (v) is generated by W and Sp D .

and only if it is invariant with respect to
Sp D (v). In particular, if R 0 has no isotropic reflexions, any ideal of g U having zero intersection with h defines a root algebra.
Proof. By 3.6.4 one has to verify that J(v) is invariant with respect to any γ ∈ Aut R (v). We will verify that any ideal is invariant with respect to the action of W (v) and of K(v). The Weyl group is generated by reflections that are inner automorphisms by formula (4). So, the Weyl group preserves all ideals. Any γ ∈ K(v) preserves the weights, so it preserves the weight spaces. Thus, its multiplies by a constant each g U α where α is a simple root or its opposite. Since any root β of g U is either sum of simple roots or a sum of its opposites, γ acts on each g U β by multiplication by a constant. Since any ideal of g U (v) is a sum of its weight subspaces, any γ ∈ K(v) preserves it. Proposition 5.4.7 now implies the claim.
We will see in 10.1 that for all admissible fully reflectable indecomposable components R 0 , except for gl(1|1), any ideal J(v) of g U (v) having zero intersection with h is automatically Sp D (v)-invariant and therefore gives rise to a root algebra. 5.4.9. Corollary. Assume that no Cartan data at different vertices of Sp(v) are D-equivalent. Then Aut R (v) is the direct product of the Weyl group W and the subgroup K of irrelevant automorphisms. If, moreover, the Cartan matrix A(a, b) at v has no zero rows and dim h = 2|X| − rk A(a, b) is minimal possible, K(v) is a commutative unipotent group.
Proof. Under the assumption, Sp D (v) is trivial and so Sk

5.4.10.
Proposition. Let R 0 be an admissible component with finite dimensional Proof. By [Si] the conditions of Corollary 5.4.9 are fulfilled. The rest follows from triviality of the group K.
Note that for g KM = gl(n|n) one has Aut R (v) = W (v) ⋊ Z 2 , see 10.3.6. 5.4.11. Example. Look at the root datum containing the root algebra gl(1|2).

The Coxeter property of the skeleton
In this section we define Coxeter graphs and prove that the skeleton Sk(v) satisfies this property. The notion of Coxeter graph generalizes that of a Coxeter group. The Cayley graph of a group G with respect to a set of generators S = {s i } is Coxeter iff (G, S) is a Coxeter group. There are, however, Coxeter graphs that do not come from Coxeter groups. It is an interesting question to describe all finite Coxeter graphs. 6.1. Coxeter graphs. Let X be a finite set, G a graph with the set of vertices V and the set of edges E, endowed with a marking r : E → X. We assume that G is connected and that the edges having a common end, have different markings. We denote by r x : v → v ′ the edge connecting v and v ′ marked with x. By the assumption, for a chosen v such edge is unique, if exists. Note that r x comes with a choice of direction for the edge connecting v and v ′ .
The path φ −1 : v ′ → v is obtained from φ by changing the direction of all arrows.
6.1.1. Definition. A Coxeter loop φ : v → v is one of the following.
1. φ = r 2 x (These are called the trivial loops.) 2. φ = (r y • r x ) m . (These are called the loops of length 2m). 6.1.2. Definition. Let φ, ψ : v → v ′ be a pair of paths. If the concatenation ψ −1 • φ is a Coxeter loop, we will say that one has an elementary Coxeter modification φ ⇒ ψ.
and one has an elementary Coxeter modification φ 2 ⇒ ψ 2 . 6.1.4. Definition. A marked graph (X, G, r) is called Coxeter if any pair of paths from v to v ′ can be connected by a sequence of Coxeter modifications. 6.1.5. As an example, take a group Γ generated by a set S of elements with s 2 = 1. Let G be the corresponding Cayley graph, where the vertices are g ∈ Γ, X = S, and g and h are connected by the edge marked by s if g = hs. Then Γ is a Coxeter group iff G is a Coxeter graph.
Let v ∈ R. We look at the skeleton Sk(v) as marked graph, with the reflection r x marked with x ∈ X. Conversely, one has the following easy result. 6.1.6. Proposition. Let (X, G, r) be a Coxeter graph such that for any v ∈ V and x ∈ X there exists an edge r x : v → v ′ . Then (X, G, r) is the Cayley graph of a Coxeter group if and only if for any pair x, y ∈ X the length 2m xy of (x, y) Proof. The necessity of the condition is clear. Define Γ as the Coxeter group generated by s x , x ∈ X subject to the relations (s x s y ) mxy = 1. The isomorphism of (X, G, r) with the Cayley graph of Γ is defined by an arbitrary choice of a vertex v ∈ V and the assignment of s x to r x . Coxterity of the graph implies that any two paths v → v ′ in G define the same image in Γ.
Here is our main result. 6.1.7. Theorem.
The proof of the theorem is based on a presentation of the skeleton Sk(v) as the 1-skeleton of a convex polyhedron. In the following subsection we present basic facts about convex polyhedra. In 6.3 we construct a polyhedron having Sk(v) as its 1-skeleton. This easily implies Theorem 6.1.7. 6.1.8. Remark. Note that in the case when R 0 is fully reflectable and all reflexions are anisotropic the skeleton Sk(v) is isomorphic to the Cayley graph of the Weyl group. 6.2. Convex polyhedra: generalities.
6.2.1. Polytopes. Recall that a polytope P in a real finite dimensional affine space E is defined as the convex hull of a finite set of points. The dimension of P is, by definition, the dimension of the affine span of P .
A polytope P of dimension n has stratification P = P 0 ⊔ . . . ⊔ P n , where P n is the inerior of P in its affine span and P k for k < n consists of points v for which the intersection of all supporting hyperplanes at v has dimension k. Thus, P 0 is the set of vertices of P and P is the convex hull of P 0 . 6.2.2. Polyhedra. In this paper we use a slightly generalized notion of a convex polyhedron. We collect all necessary material here.
Definition. A polyhedron P in E is a closed convex set such that any v ∈ P has a neighborhood isomorphic to a neighborhood of a point of a polytope. 10 The dimension of a polyhedron is the dimension of its affine span. The stratification of points of a convex polytope extends to a stratification of a polyhedron: one has P = P 0 ⊔ · · · ⊔ P n where P n is the interior of P in its affine span and P k consists of the points for which the intersection of all supporting hyperplanes has dimension k. In particular, P 0 is the set of vertices of P. This is a discrete subset of E, not necessarily finite. Moreover, P is in general not a convex hull of P 0 .
For any v ∈ P n−1 there is a unique supporting hyperplane at v. Its intersection with P is a face of dimension n − 1. Each of them is a polyhedron of dimension n − 1 and their union is ∂P.
The following notation is used below. A linear hyperplane H ⊂ V and v ∈ E define an affine hyperplane v + H. The complement V \ H consists of two components; their closures are the halfspaces defined by H and denoted by H + and H − . In the same manner v + H + denotes the affine halfspace.
Note that P coincides with the intersection of the affine halfspaces v + H + defined by the faces of P of maximal dimension. 6.2.3. Definition. Let A be the set of supporting hyperplanes v α + H α of P and let v + H + α be the affine halfspaces containing P. The cone of P, C(P) is defined as the intersection In particular, C(P) is the intersection of the linear halfspaces H α defined by the (n − 1)-faces of P.
Note that by definition C(P) is a convex cone in V and P is invariant under the action of C(P): for ξ ∈ P and η ∈ C(P) one has ξ + η ∈ P.
Proof. Choose an interior point ζ ∈ P and define the projection from ∂P to the unit sphere S with the center at ζ by the formula Since P is convex, φ is injective. From the definition of P we see that ξ ∈ S is not in the image of φ iff ξ ∈ ζ − C(P). Set U = (ζ − C(P)) ∩ S. The restriction of φ to any (n − 1)-face is a stereographic projection. Since any point of P has a neighborhood isomorphic to a neighborhood of a point of a polytope, the map φ is an open embedding and so it defines a homeomorphism of ∂P with S \ U. If C(P) = {0}, U is a nonempty convex subset of S, so S \ U is contractible. This proves Claim 1.
To prove Claim 2, note that the C(P) acts of P: if c ∈ C(P) and p ∈ P then p − c ∈ P. Therefore, if C(P) = {0}, P cannot be compact. On the contrary, if C(P) = {0}, ∂P is homeomorphic to sphere, so it is compact. P is the convex hull of its boundary, so it is also compact.
Finally, if P is compact then it is a convex hull of its boundary that is a finite union of compact polyhedra of smaller dimension. This implies that P is the convex hull of the set of its vertices.
The only result we need in our study of Coxeter property of the skeleton is the following.
6.2.5. Corollary. For any polyhedron P of dimension > 2 one has H 1 (∂P) = 0. 6.3. A polyhedron defined by Sk(v). Let R 0 be an admissible component of a root groupoid, n = |X| and Sk(v) a skeleton. Let Q R := Q ⊗ Z R and for any vertex u of Sk(v) set Q + u,R := α∈Σu R + α. 6.3.1. Lemma. There exist an injective map λ : Proof. Choose λ v = 0, and set α.
Here we use Corollary 5.3.7 and Corollary 5.3.9 of the main text to check injectivity of λ.
We define The following claims are obvious: Q N is a convex polytope (compact, bounded by finitely many hyperplanes). We intend to show that P N = Q N and that the vertices of the polytope P N belonging to H + N \ H N are precisely {λ u |f (λ u ) > N}. This implies that P is a polyhedron. In fact, for µ ∈ P choose N so that f (µ) > N. Then µ ∈ P N = Q N , so µ has a neighborhood that is a neighborhood in a polytope.
Note that all λ u are vertices of P since there is a hyperplane in Q R intersecting P at one point λ u . For the same reason all λ u satisfying f (λ u ) > N are vertices of Q N . In order to show that Q N = P N , it is sufficient to verify that any vertex µ of Q N belongs to P. The 1-skeleton of Q N is connected, so it is enough to verify that any edge of Q N connecting λ u with another vertex µ, belongs to P. We know all edges of Q N in a neighborhood of λ u : they are just b u (x), x ∈ X. If x is reflectable at u, there is an arrow r x : u → u ′ , and µ lies on the segment The minus sign in the formula for C(P) is due to the minus sign in the formula (18). 6.3.3. Lemma. Let P be bounded. Then R 0 is fully reflectable, Sk(v) is finite.
Proof. Sk(v) embeds into the intersection of P with a lattice, therefore, it is finite. If x ∈ X is not reflectable at u ∈ Sk(v), the root b u (x) belongs to Q + u , and, therefore, to all Q + u ′ , u ′ ∈ Sk(v). This contradicts the condition Q ++ R = {0}.
We will now be able to describe the faces of P. Let Y ⊂ X, |Y | = k and u ∈ Sk(v). Let H Y (u) be the affine k-plane passing through λ u and spanned by where Sk Y (u) denotes the connected component of u ∈ Sk(v) in the subgraph spanned by the arrows r y for y ∈ Y .
Proof. The boundary ∂P of P by the proof of 6.3.2 lies in the union of hyperplanes That proves (2). Now for k = n−1 the statement (1) follows since (2) implies that F Y (u) has codimension 1. For general k it follows by induction in codimension. 6.3.5. Corollary. The map λ as in Lemma 6.3.1 establishes a one-to-one correspondence between Sk(v) and the set of vertices of P. Moreover, Sk(v) identifies with the 1-skeleton of P so that the reflexions r x : u → u ′ in Sk(v) identify with the edges connecting λ u with λ u ′ . 6.3.6. Corollary. The two-dimensional face F Y (u) of P defined by a two-element subset Y of X is bounded iff Sk Y (u) is the finite skeleton of a rank 2 fully reflectable component. In this case Sk Y (u) isomorphic to the Cayley graph of the dihedral group D m where m = 2, 3, 4 or 6 . The noncompact face F Y (u) has a non-compact contractible boundary. 11 Proof. The claim immediately follows from Lemma 6.3.4. The allowable values for m result from a well-known classification of rank 2 fully reflectable components with finite skeleton, see, for example, [S]. 6.3.7. Proof of Theorem 6.1.7. By 6.3.5 Sk(v) indentifies with the 1-skeleton of the polyhedron P. By 6.2.5 any pair of paths leading from u to u ′ in Sk(v) is connected by relations defned by compact 2-faces. Finally, by 6.3.6, compact 2-faces gives rise to Coxeter relations with m = 2, 3, 4, 6.
11 The set of vertices in it is linearly ordered. If there is a smallest (or greatest) vertex, it has a non-compact edge corresponding to a non-reflectable root.

7.
A trichotomy for admissible fully reflectable components 7.1. Overview. From now on we will consider only indecomposable admissible fully reflectable components.
In this section we define three types of such components: finite, affine and indefinite. We investigate the structure of the sets of roots of corresponding root algebras. Expectedly, the trichotomy for admissible components is closely connected to the trichotomy for the types of Cartan matrices defined by Kac in [K2], Theorem 4.3.
7.1.1. We keep the notation of 5.1.1. Fix an indecomposable admissible fully reflectable component R 0 and v ∈ R 0 . Let g be a root Lie superalgebra supported at R 0 . We denote by ∆ = ∆(g) the set of roots of g and by r the kernel of the canonical map g → g KM . Recall that r is the maximal ideal of g having zero intersection with h.
In this section we will deduce a certain information abot the ideal r for different types of components, see 7.4.1, 7.4.2. In particular, we will be able to deduce, for certain types of components, that they admit a unique root Lie superalgebra g KM .
Σ v ′ ⊂ ∆ and the root spaces g α , α ∈ ∆ re , are one-dimensional, in particular, are purely even or purely odd. This yields a decomposition of the family of real roots into even and odd part ∆ re = ∆ re,0 ⊔ ∆ re,1 . For anisotropic α ∈ ∆ re the elements α ∨ ∈ g α ∩h are defined so that α, α ∨ = 2.
We define For each v ′ ∈ Sk(v) we have the triangular decompositions (2) For any v ′ one has (3) Ω(r) ⊂ ∆ im , except for the rank one algebra g = g U with g KM = gl(1|1), see 3.2.5. (4) If R 0 has rank greather than one, then (1) is standard and (2) follows from (1). Claims (3) and (4) View an indecomposable Cartan matrix A as a linear operator on V . It is given by the formula By [K2], Thm.4.3, A satisfies exactly one of the following conditions • ∃v > 0 such that Av > 0 (type (FIN)).
• (IND) Au ≥ 0 with u ≥ 0 implies u = 0. It is proven there that the Kac-Moody Lie algebras of type (FIN) are all simple finite-dimensional Lie algebras, the Kac-Moody Lie algebras of type (AFF) have finite growth: they are always symmetrizable and can be obtained as (twisted) affinizations of simple finite-dimensional Lie algebras. The Kac-Moody algebras of indefinite type have infinite growth.
We present below a version of this trichotomy in terms of connected components of root groupoids. The component is required to be indecomposable and fully reflectable. Note that both conditions hold in the context of [K2], Thm. 4.3. 7.3.2. Let R 0 be a component of the root groupoid with a fixed vertex v and indecomposable A(v). Set Obviously, Q ++ = Q ++ R ∩ Q. Note that the sets ∆ re and Q ++ depend on the component Sk(v) only. One has Q ++ ∩ Qα = 0 for each α ∈ ∆ re .
In the remaining type (IND), assume W ν ⊂ Q + is finite and ht ν is maximal. Then ν(α ∨ ) ≥ 0 for all α and, therefore, Aν ≥ 0. Hence ν = 0. By the assumption there exists v > 0 such that Av < 0. Then W v ⊂ Q + by [K2], Lemma 5.3 and, by above, this is an infinite orbit. 7.3.6. Purely anisotropic components of finite and affine types. If p(x) = 0 for each x, then g KM is a Kac-Moody Lie algebra. In this case g KM is finite-dimensional if and only if the Cartan matrix A if of type (FIN) and a (twisted) affine Lie algebra if A is of type (AFF).
If we do not require all generators to be even, we have an extra requirement saying that the x-row of A consists of even entries if p(x) = 1. Therefore, to every anisotropic component one can associate a Kac-Moody Lie algebra by changing the parity of all generators to 0. As we showed in the previous subsection, this operation does not change the type of the corresponding components. We call all Kac-Moody Lie superalgebras obtained in this way from a Kac-Moody Lie algebra g the cousins of g.
The Cartan matrices of types (FIN) and (AFF) are well-known. Let us describe the cases when such a matrix has a row with even entries.
In the type (FIN) the only such case is the type B n and it has exactly one row with even entries. The Kac-Moody Lie algebra with Cartan matrix B n is so(2n+1) and its cousin is a finite-dimensional simple Lie superalgebra osp(1|2n).
7.5.2. Hoyt's classification. Indecomposable Kac-Moody superalgebras with at least one simple isotropic root were classified in [H]. In this subsection we review the results of C. Hoyt classification that will be used in the following sections. Exactly one of the following options holds in this case: (1) dim g KM < ∞.
(2) dim g KM = ∞ and ∆ im = Zδ, ∆ ⊂ Zδ + ∆ ′ for some finite set ∆ ′ ⊂ h * and some δ ∈ ∆ + 12 . In this case all symmetrizable Kac-Moody algebras are twisted affinizations of simple finite-dimensional Lie superalgebras. They also appear in Van de Leur classification of symmetrizable Kac-Moody superalgebras of finite growth. In addition, there is one-parameter Kac-Moody superalgebra S(2, 1; a) and the twisted affinization q(n) (2) of the strange superalgebra psq(n) for n ≥ 3. By direct inspection one can check that there exists m ∈ Z such that if α ∈ ∆ then α ± mδ ∈ ∆. (3) The algebra g KM = Q ± (m, n, t) with dim(h) = 3 where m, n, t are negative integers, not all equal to −1, with non-symmetrizable and nondegenerate Cartan matrices. There are three linearly independent principal roots, therefore the Weyl group has no non-zero fixed vectors in h * . Hence Q ± (m, n, t) are of type (Ind). Little is known about Lie superalgebras of this type. 7.5.3. Let R 0 be a component of R of type (2) in Hoyt's classification 7.5.2. We will prove that it is of type (Aff).
Proof. Choose a vertex u ∈ R 0 . Let then γ, α + sδ > 0 for all s > M. Thus, and hence T u (γ) is finite. Suppose that u does not satisfy the conditions of the lemma. Then there is Repeating the argument several times, we end up with a vertex v such that T v (γ) = ∅. 7.5.5. Corollary. If R 0 is of type (2), then Q ++ = Z ≥0 δ and hence R 0 is of type (Aff).

Proof. Let
7.6. Combining the results of [H] with 7.3.4 we obtain the following result.
(1) The following conditions are equivalent: (2) The following comditions are equivalent: • R 0 of type (Aff); • W is infinite and h * contains a non-zero trivial W -orbit. (3) The following conditions are equivalent: • R 0 of type (Ind); • g has an infinite Gelfand-Kirillov dimension.
7.6.2. Remark. Cartan matrices of components of type (Fin) are usually nondegenerate. The only exception is gl(n|n). Cartan matrices of type (Aff) are always degenerate, usually of corank one. The only exception is sl(n|n) (1) where corank is two.

Symmetrizable root data
We retain the notation of Section 7. We continue to assume that all x ∈ X are reflectable at all v ∈ R 0 . In this section we prove, following a method of Gabber-Kac [GK], that if R 0 has a symmetric Cartan matrix (and, therefore, all Cartan matrices associated to R 0 are symmetrizable) then g KM is the only root algebra, except for the cases g KM = gl(1|1) and (ρ|δ) = 0 where (−|−) is the nondegenerate symmetric bilinear form on h * introduced in 3.1.3 and ρ is as in 5. 3.11. Fix v ∈ R 0 , an admissible component of R. We keep the notation of Section 3 for the half-baked algebra g = n − ⊕ h ⊕ n + , a root algebra g and the Kac-Moody algebra g KM = g/r. We set b := n + + h, its image b in g and r ± := r ∩ñ ± . Note that r ± are ideals of g.
The Proof. This result is the main part of the proof of Proposition 9.11 of [K2].
Recall that g U denotes the universal root algebra.
Assume that r − /s − = 0. This Lie superalgebra is a semisimple h-module with the weights belonging to −Q + \{0}. This implies that it does not coincide with its commutator, that is, that r − /(s − + [r − , r − ]) = 0. Since the adjoint representation of g is integrable, r − /(s − + [r − , r − ]) is a nonzero integrable g-module. Using 8.2.1 we get a nonzero integrable subquotient in α∈Σ M KM (−α) which contradicts the conditions. Thus, s − = r − , so automatically s + = r + as the automorphisms θ, see 3.1.4, defined on g, g U and g KM , identifies s + with s − and r + with r − . 8.3. Main result. In this subsection we assume that the Cartan matrix for r is symmetric, i.e. ∀x, y ∈ X b(x), a(y) = b(y), a(x) . Note that by 2.3.1 all Cartan matrices at r ′ ∈ R 0 are symmetrizable.
By Proposition 3.1.3 g admits an invariant bilinear form such that the restriction of this form on h is non-degenerate and and (a(x)|h) = b(x), h for each h ∈ h.
8.3.1. Let us show that r coincides with the kernel of this form. Indeed, since the kernel is an ideal and the restriction of (−|−) on h is non-degenerate, the kernel lies in r. Since ( g α | g β ) = 0 for α + β = 0, one has (h|r) = 0. Thus is an ideal containing h, so r ⊥ = g, that is r lies in the kernel of (−|−). Thus, the algebra g KM inherits a non-degenerate invariant bilinear form having the properties listed in 3.1.3.
Let us first notice that for S(2, 1, b) Lemma 7.5.1 (3) and 10.4.5 imply g U = g KM . In all other cases we define for any root algebra g its subfactorḡ := [g, g]/Z(g).
Thenḡ KM = [g KM , g KM ]/Z(g KM ) is isomorphic to the twisted loop algebra L(s) σ for some simple superalgebra s and an automorphism σ of finite order m. In particular,ḡ KM is perfect. The superalgebra s is basic classical, exceptional or psq n . Its even part s0, therefore, is a reductive Lie algebra.
Let h ′ be the even part of the Cartan subalgebra of s. One can choose σ so that σ(h ′ ) = h ′ . Furthermore, if kδ is an even root and ε = e 2πi m then g KM kδ = {h ⊗ t k | h ∈ h ′ , σ(h) = ε k h}. The cohomology group H i (ḡ KM , C) has a natural structure of h-module. We write H i (ḡ KM , C) µ for the cohomology group of weight µ with respect to h-action.
Letĝ be the graded central extension ofḡ KM given by the exact sequence and then extend to the exact sequence using the semidirect product decomposition g KM = t ⋉ [g KM , g KM ] where t ⊂ h is a suitable abelian subalgebra. We claim that g is a root algebra. Indeed, we just have to check the relations 3.1.1 at every vertex v ∈ R 0 . The only non-trivial relation is [ẽ x ,f y ] = 0 for x = y. This is equivalent to b(x) − b(y) = kδ and the latter follows from kδ ∈ Q + (v) for positive k and kδ ∈ −Q + (v) for negative k.
Finally, let us prove that g = g U . Indeed, by 7.5.1 the kernel k of the map g U → g lies in the center of [g U , g U ] and is a direct sum k =0 k kδ . Therefore g U = g. 9.1.3. Let δ have degree d in the standard grading of L(s) σ . The base change It is given on 2-cocycles by the formula (24)c(x ⊗ t a , y ⊗ t b ) = δ kd,a+b c(x, y).
Letḡ KM = L(s) σ and h be a Cartan subalgebras of g KM . Set h • := ker δ. Then h • acts on s and therefore on H 2 (s, C). We denote by H 2 (s, C) • the h • -invariant subspace. 13 The automorphism σ acts on H 2 (s, C) • and induces a Z/mZ-grading.
9.1.4. Lemma. If kδ is an even root and kd ≡ p mod m then the homomorphism (23) induces an isomorphism H 2 (s, C) • p ≃ H 2 (L(s) σ , C) kδ . Proof. The correspondence between the weight spaces follows from formula (24). Injectivity of the map is straightforward. To prove surjectivity it suffices to show that every class in H 2 (L(s) σ , C) kδ is represented by a cocycle ϕ such that for all a, b ∈ Z and x, y ∈ s. The Lie algebra s ′ = [s0, s0] is semisimple. The corresponding twised affine Lie algebraŝ ′ is symmetrizable and, therefore, (ŝ ′ ) U = (ŝ ′ ) KM . By Lemma 9.1.2 H 2 (L(s ′ ) σ , C) kδ = 0. On the other hand L(s) σ 0 = L(s ′ ) σ ⊕ a for some abelian Lie algebra a. Thus, we can choose ϕ so that ϕ(L(s ′ ) σ , L(s) σ 0 ) = 0. Since kδ is an even root, ϕ is an even cocycle, so ϕ(L(s ′ ) σ , L(s) σ ) = 0. In particular, for every h ∈ (h ′ ∩ s ′ ) σ we have ϕ(h ⊗ t m , L(s) σ ) = 0. Let α be a non-zero weight of s with respect to (h ′ ∩ s ′ ) σ and x ∈ s α , y ∈ s −α , we can choose h so that α(h) = 0. Then the cocycle condition dϕ(x ⊗ t a−m , y ⊗ t b , h ⊗ t m ) = 0 implies (25) for x ∈ s α , y ∈ s −α . Since the s α for all nonzero weights α generate s and ϕ(x ⊗ t a , y ⊗ t b ) = 0 for x ∈ s α and y ∈ s β with α + β = 0, one proves the desired identity for all x, y using linearity and the cocycle condition.

Description of root algebras. Examples
In Subsection 10.1 we describe root algebras in the indecomposable fully reflectable case. In the rest of this section we compute some of the groups Aut R (v).
In this section we identify admissibile components of R by root Lie superalgebras supported on them. Then the graph Sp(v) is a star with v at the center and three other vertices v i with r x i : v → v i and the Cartan matrices v 1 : We have three principal reflections s α k , where for {i, j, k} = {1, 2, 3}. The Weyl group is generated by these reflections (this group is isomorphic to the affine Weyl group A 2 ). The group K(v) is the additive group C. If we choose h of dimension greater than 4, the Weyl group will remain the same, but K(v) will be different. Regardless of h, Aut R (v) = W (v) × K(v) by 5.4.9.
10.2.2. Example: B(1|1) (1) , D(2|1, a), D(2|1, a) (1) , Q ± (m, n, t). All these cases are similar to 10.2.1. We can (and will) choose a vertex v such that p(x) = 1 for all x ∈ X. We always have a xy = 0 if x = y. The graph Sp(v) is a star with the center at v. The other vertices are v x with the edges r x : v → v x . If a xx = 0 then p ′ (y) = 0 for each y = x. Hence Sp(v) consists of v and all v x such that a xx = 0. Cartan data at all vertices of Sp(v) are not D-equivalent, so 5.4.9 is applicable. This gives Aut(v) = W × K. (1) n , its relatives and friends. There is a number of components of the root groupoid whose Cartan matrices satisfy common properties listed below in (26) and whose automorphism groups allow a more or less uniform description. We call them "relatives and friends of sl (1) n " and they consist of the types sl(k|ℓ) (1) for k, ℓ such that k + ℓ = n and q

sl
(2) n . We take X = {x i } i∈Zn . Let v ∈ R 0 be a vertex with the Cartan matrix of the following form: (26) a ij = 0 for j = i, i ± 1; a i,i±1 ∈ {±1}, a i,i−1 + a ii + a i,i+1 = 0 p(x i ) = 1 ⇐⇒ a ii = 0. 10.3.2. Let R 0 be the component of R corresponding to sl (1) n ; we will use bar notationv etc. for the objects connected to R 0 . Fix a linear isomorphism ι : Qv ∼ → Q v given by ι(bv(x i )) := b v (x i ).
Let v → v ′ be a path in R 0 andv →v ′ be its namesake in R 0 . It is easy to see that b This provides a bijection between the sets of real roots ∆ re = ∆ re . Note that all roots of ∆ re are anisotropic. Since the set {b v (x i )} i∈Zn determines a vertex in Sk(v) by 5.3.7, this gives a bijection between Sk(v) and Sk(v).

10.3.3.
We identify Q v and Qv via ι. By 5.2.6 the Weyl group W (sl (1) n ) acts freely on Sk(v). By 4.3.9 this action is transitive. This gives a simply transitive action of W (sl (1) n ) on Sk(v). Note that the Weyl group W can be identified with a subgroup of W (sl (1) n ) as it is generated by a part of the reflections belonging to W (sl (1) n ). Let us compute 10.3.4. Action of W (sl (1) n ). By 10.3.1, the vector does not depend on the choice of v ′ ∈ Sk(v). View Q v as a subset of V = Span Z (ε 1 , . . . , ε n , δ) by setting b(x i ) = ε i − ε i+1 for i = 1, . . . , n − 1; b(x n ) = δ + ε n − ε 1 .
Note that W (sl (1) n ) stabilizes δ. By 10.3.1, A w(v) is D-equivalent to A v if and only if p v (x i ) = p w(v) (x i ) for all i. Therefore, (27) w ∈ Sk D (v) ⇐⇒ p(wε i ) − p(ε i ) is independent of i.
We will now compute the groups Sk D (v) using the formula (27). 10.3.5. Case sl(k|ℓ) (1) , k, ℓ = 0. We can choose v in such a way that p(x i ) = 0 for i = k, n and p(x n ) = p(x k ) = 1. Note that p(δ) = 0. Denote by S k ⊂ S n (resp., S ℓ ⊂ S n ) the group of permutations of {ε i } k i=1 (resp., of {ε i } n i=k+1 ). In this case p(wε i ) = p(ε i ) for w ∈Q, so Sk D (v) ⊃Q.
If k = ℓ this gives S n ∩ Sk D (v) = S k × S ℓ . In the case k = ℓ we have S n ∩ Sk D (v) = (S k × S k ) ⋊ Z 2 , where Z 2 interchanges the two copies of S k . Hence Note that the Weyl group has the form W = (S k Observe that W has an infinite index in Sk D (v).
10.3.6. Remark. For R 0 of type A(k − 1|ℓ − 1) a similar reasoning (replacing the index set X = {x i } i∈Zn with the set X = {x 1 , . . . , x n }) shows that S k+ℓ acts transitively on Sk(v) and that Note that the Weyl group is in both cases S k × S ℓ .
If k = l then K(v) = C and Aut(v) is a nontrivial semidirect product of C and Sk D (v).
In this case p(wε i ) = p(ε i ) for w ∈ S n , so S n ⊂ Sk D (v). Hence . Take ν ∈Q. One has If n is odd this gives Q ′ = 2Q, so Sk D (v) = W and Aut R (v) = W × K. If n is even, 2Q has index 2 in Q ′ . Thus W has index two in Sk D (v), so that W × K is an index 2 subgroup of Aut R (v). 10.4. A deformation of sl(2|1) (1) . A very interesting relative of sl(2|1) (1) is the root Lie superalgebra S(2|1, b) defined in [S]. We will recall some of the results of [S] below. Set X := {x 1 , x 2 , x 0 } and fix h with dim h = 4.
Let R(b), b = 0, be the component of R containing a vertex v such that p v (x 1 ) = p v (x 2 ) = 1, p v (x 0 ) = 0 and the Cartan matrix A v is equal to In studying skeleta of R(b) it is convenient to allow permutations of the elements of X. This leads to the action of S 3 on the components of R with the index set X and, as we will see soon, carries components R(b) to components of the same type.
Permuting x 1 and x 2 in A(b) we obtain A(−b), so R(b) is mapping to R(−b). In particular, each root algebra for S(2|1; b) is isomorphic to a root algebra for S(2|1; −b). Proof. It is enough to verify what happens to the Cartan datum under an isotropic reflexion r x : v → v ′ . Since permuting x 1 and x 2 in A(b) yields A(−b), it is enough to verify the assertion for x = x 1 . In this case we have Taking the homothety h λ : v ′ → v ′′ with λ = (−1, −b −1 , b − 1) we get Applying now the cyclic permutation carrying x 2 to x 1 , we get the Cartan matrix A(b − 1). It is easy to see that going along the other isotropic reflection would produce in the same way the matrix A(b + 1).
Proof. It is enough to check that all Cartan matrices in Sp(v) are not D-equivalent. Note that Sp(v) can be seen as the infinite graph Consider the equivalence relation on the set of 3 × 3 matrices generated by the action of A 3 (the group of even permutations in S 3 ) and B ∼ DB for a diagonal invertible matrix D. Observe that In the proof of Lemma 10.4.1 we showed that if A v ∼ A(b), then for an isotropic reflexion v rx → v ′ we have A v ′ ∼ A(b ± 1). This implies that A v k ∼ A(b − k), so A v k ∼ A v 0 for any k = 0. Hence the group Sp D (v 0 ) is trivial, so Aut(v 0 ) = W × K.