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Inverses of Cartan matrices of Lie algebras and Lie superalgebras

The inverses of indecomposable Cartan matrices are computed for finite-dimensional Lie algebras and Lie superalgebras over fields of any characteristic, and for hyperbolic (almost affine) complex Lie (super)algebras. This enables one to express the fundamental weights in terms of simple roots corresponding to the Cartan matrix. We discovered three yet inexplicable new phenomena, of which (a) and (b) concern almost affine complex Lie (super)algebras: (a) several of the inverses of Cartan matrices have all their elements negative (not just non-positive, as they should be according to an a priori characterization due to Zhang Hechun); (b) the 0s only occur on the main diagonals of the inverses; (c) the determinants of inequivalent Cartan matrices of the simple Lie (super)algebra may differ (in any characteristic). We interpret most of the results of Wei Yangjiang and Zou Yi Ming, Inverses of Cartan matrices of Lie algebras and Lie superalgebras, Linear Alg. Appl., 521 (2017) 283--298 as inverses of the Gram matrices of non-degenerate invariant symmetric bilinear forms on the (super)algebras considered, not of Cartan matrices, and give more adequate references. In particular, the inverses of Cartan matrices of simple Lie algebras were already published, starting with Dynkin's paper in 1952, see also Table 2 in Springer's book by Onishchik and Vinberg (1990).

1. Introduction 1.1. General remarks. The problem "find the inverses of Cartan matrices" might look as a topic of a somewhat boring course work for a first year student taking linear algebra.
However, the explicit answer is needed in a number of situations, so the problem formulated above was solved -for finite-dimensional simple Lie algebras over C -long ago, see Dynkin's paper [D1,Ch.1].
Here are examples where these inverses are used.
• The inverse of the Cartan matrix A (or rather of its transposed, A T ) of a given simple finite-dimensional Lie algebra over C is a tool to obtain the set of fundamental weights from the set of simple roots, see, e.g., [Bbk,Ch.6,§1,Subsect. 10,eq. (14)], where the result of using the inverse of A T is given, but not the inverse of A T itself (being considered, perhaps, well-known by that time not only to experts for more than a decade). The inverses of matrices A T are contained, among other very useful tables, in the book by Dynkin's former Ph.D. students, see [OV, Table 2]. Any sufficiently comprehensive textbook on representations of Lie algebras reproduces the statement of Bourbaki because the fundamental weights are important: (1) they form a basis in the weight lattice, (2) any finite-dimensional irreducible representation of a simple finite-dimensional Lie algebra over C is a direct summand in the tensor product of tensor powers of fundamental representations.
• We found out that we need the results of this paper over fields of positive characteristic in our study of the notions related with the Duflo-Serganova functor, a newly discovered powerful tool in representation theory of Lie superalgebras, see [KLLS, BGKL].
• Another natural motivation to invert the Cartan matrices that springs to mind is the fact that the Cartan matrix of sl(n + 1) is the matrix of the difference operator corresponding to −( d dx ) 2 . Under the name Toeplitz matrix K n it is known to experts in the method of finite differences. Together with the Cartan matrix T n of the Lie superalgebra osp(1|2n), which is the Cartan matrix of sp(2n) with the last row divided by 2, these are the invertible two of the "four special matrices" (actually, series of matrices) spoken about in [Str,Section 1.1]. (The two non-invertible of the "four special matrices" correspond to an affine Lie algebra and an affine Lie superalgebra, respectively; for the classification of affine Lie (super)algebras, see [CCLL].) • For inverses of partial Cartan matrices, and some examples of their usage, see [Stk,]; the inverses of Cartan matrices are also recalled there (Tables A.14-A.15).
• For several more instances where the inverses of Cartan matrices of Lie algebras are considered, see the bibliography in [WZ]. In particular, there is a reference to a paper by Lusztig and Tits which, though interesting, is rather unclear at places, so we give an elucidation. Lusztig and Tits. The title of the paper [LT] is misleading since the paper contains a much stronger result than an already known one. Lusztig and Tits established certain common properties of the inverses of matrices much more general than the Cartan matrices of simple finite-dimensional Lie algebras over C, see Proposition 1.2.1 (not explicitly formulated in [LT], but follows from a more general statement formulated in [LT]).

A result due to
Recall that a forest is a disjoint union of trees, the latter being undirected graphs in which any two vertices are connected by exactly one path. A graph loop is an edge going in and out of the same vertex. A simple graph is an unweighted, undirected graph containing no graph loops or multiple edges. For a finite simple graph, the adjacency matrix A is an n × n matrix, where n is the quantity of vertices, with 0s on its diagonal and A ij = A ji = 1 if the ith and jth vertices are connected. Recall, see [V], that B is a main submatrix of a given matrix C if B is obtained by deleting from C any number of pairs (a row, a column) each pair intersecting on the main diagonal.
Given the adjacency (not incidence, it is a misprint in [LT]) matrix A of a forest, the matrices C of the same size as A are constructed in [LT] by replacing any of the diagonal elements and non-zero off-diagonal elements of A with elements in R -conditions a) and b) in (1) -provided two more conditions (c) and d) are met: (1) a): C ij = 0 ⇐⇒ C ji = 0; b): if i = j, then A ij = 1 ⇐⇒ C ij = 0; c): C ij ≤ 0 if i = j; d): det C > 0 and det B > 0 for all main submatrices B of C; in particular, this means that C ii > 0 for all i.
More exactly, conditions a) and b) are stronger than the conditions imposed on C in [LT]. Conditions c) and d) (or something equivalent to them) are implicit in [LT], but used in the proof of the following statement (in the form formulated by A. Lebedev).
1.2.1. Proposition (A Corollary of [LT]). For any tree, let A be the adjacency matrix; let C be a matrix satisfying conditions (1). Then all elements of C −1 are positive.
1.3. Two interpretations of det C. For the Cartan matrix C of the simple finite-dimensional Lie algebra over C, we have, see [OV]: (2) a): det C is equal to the order of the center of the simply connected compact group corresponding to C. b): det C = |P/Q|, where P and Q denote the weight lattice and root lattice, respectively.
1.4. Cartan matrices of hyperbolic Lie algebras and almost affine Lie superalgebras. The Cartan matrix of any hyperbolic Lie algebra is the one that, its ith row and its ith column being deleted for any i, becomes the Cartan matrix of the direct sum of finite-dimensional or affine Kac-Moody algebras. For the definition of Lie (super)algebras g(A) with Cartan matrix A, see §2; for superization of hyperbolic Lie algebras, called almost affine Lie superalgebras, and classification of both types, see [CCLL].
For Cartan matrices C of hyperbolic Lie algebras Zhang Hechun [ZH] established that (3) "the entries of C −1 are non-positive rational numbers".
Recall that a given Lie superalgebra g(A) with Cartan matrix A is almost affine if it is not finite-dimensional or affine Kac-Moody, but any its main submatrix corresponds to a direct sum of finite-dimensional or affine Kac-Moody superalgebras, and all Cartan matrices of g(A) are almost affine. The almost affine Lie superalgebras also have the property (3); this follows from their classification and the explicit form of their Cartan matrices when invertible.
Explicit results given in § 8, allow us to sharpen the description (3). We discovered two new phenomena: all elements of the matrices C −1 marked !!! in § 8 are negative; (4) in each of the remaining matrices C −1 , the 0s occur only on the main diagonal.
1.5. Natural generalizations and sharpenings of the problems posed in [WZ].
A) For any isotropic reflection r acting on Cartan matrix A −→ r(A), see Subsect. 2.6, describe an algorithm for the passage A −1 −→ (r(A)) −1 . In other words, find inverses of not one -selected by a random criteria (for a description of numerous possibilities in the infinite-dimensional case, see [Eg]) -Cartan matrix of a given Lie superalgebra g(A), but of all its Cartan matrices (for dim g(A) < ∞).
B) For the exceptional simple Lie (super)algebras, give the complete explicit answer in the following two cases: Ba) for finite-dimensional ones (in any characteristic); Bb) for hyperbolic Lie algebras and almost affine Lie superalgebras. C) Investigate what can be said about inverses of Cartan matrices, if exist, for various other types of Lie (super)algebras corresponding to the cases of Vinberg's theorem, see [V, ZH].
1.5.1. Disclamer. The "fundamental weights" are not as important in the representation theories of modular Lie algebras and of Lie superalgebras (in any characteristic) due to the lack of complete reducibility of their representations, and due to existence of deforms of vacuum (highest or lowest) weight modules, and of modules without vacuum weight vector.
However, even for p > 0, these fundamental weights are as good as their namesakes over C if we confine ourselves to the restricted representations, the very first (if not the only) ones to be studied from a point of view of the geometer (P. Deligne, see his Appendix to [LL]). 1.6. Our results. In § 4 and formula (23) we solved Problem A, see Subsection 1.5; in § § 6-8 we solved Problem B. More explicitly, we computed the inverses of indecomposable Cartan matrices for finite-dimensional Lie algebras and Lie superalgebras over fields of positive characteristic and for almost affine (hyperbolic) infinite-dimensional complex Lie (super)algebras. (For the classifications of Lie (super)algebras with indecomposable Cartan matrices over C in the almost affine (hyperbolic) case, and finite-dimensional over algebraically closed fields of characteristic p > 0, see [CCLL] (with prerequisites in [Se, HS]) and [BGL], respectively.) We have discovered phenomena (4) and (5). We have discovered that the determinants of inequivalent Cartan matrices of the simple Lie (super)algebra may differ (in any characteristic).
For the serial Lie superalgebras, the inverses of Cartan matrices are explicitly given for certain "basic" Cartan matrices from which all the other Cartan matrices of the given Lie superalgebra are obtained by means of isotropic (odd) reflections as explained in [CCLL].
We also corrected and widened the list of references (and the range of applications) of inverses of Cartan matrices as compared with those given in [WZ], e.g., the explicit form of inverse Cartan matrices of finite-dimensional simple Lie algebras is reproduced in [WZ] as if new, though well-known, see [D1,OV,Stk]. These omissions and the desire to solve the problem considered, but not solved, in [WZ], except for occasional coincidences of Cartan matrices with Gram matrices of non-degenerate invariant symmetric bilinear forms (NISes) on the (super)algebras considered, is what prompted our work.
2) In characteristic p > 0, the determinants of Cartan matrices of exceptional simple Lie superalgebras are mostly equal to 1, but not always; over C, we see that these determinants are different for inequivalent Cartan matrices of the same Lie superalgebra. What is the meaning of these determinants?
3) How to interpret the determinants of the matrices described in Proposition 1.2.1; the determinants of the Cartan matrices of hyperbolic Lie algebras, and almost affine Lie superalgebras, cf. with properties (2)? 1.6.2. Remark. V. Kac was the first to realize that certain finite-dimensional Lie superalgebras have analogs of Cartan matrices and defined them imitating the definition for Lie algebras, compare [K] with [Kapp], where the exceptional simple Lie superalgebras first appeared. In these cases, V. Kac listed the indecomposable Cartan matrices (with a gap, corrected in [Se, vdL], where infinite-dimensional generalizations with symmetrizable Cartan matrices were also considered). For further improvements of the definitions, see [HS, CCLL, BGL].
In [WZ], the Gram matrices of the non-degenerate symmetric bilinear form on the space of roots, that never explicitly appeared before, but can be recovered from the data in [Se1], or by symmetrizing Cartan matrices, were misattributed to works of V. Kac and called Cartan matrices. Kac never used such matrices (and would hardly apply the term Cartan matrix to a matrix with both 2 and −2 appearing simultaneously on the main diagonal). Analogs of Cartan matrices A with A ii ≤ 0 were introduced by Borcherds, see [B, R] and references in [CCLL].
2. Chevalley generators, Cartan matrices, reflections (from [CCLL, BGL]) 2.1. Chevalley generators and Cartan matrices. Let us start with the construction of Lie (super)algebras with Cartan matrix. Let A = (A ij ) be an n × n-matrix whose entries lie in the ground field K. Let rk A = n − l. It means that there exists an l × n-matrix T = (T ij ) such that (6) a) the rows of T are linearly independent; b) T A = 0 (or, more precisely, "zero l × n-matrix").
Indeed, if rk A T = rk A = n − l, then there exist l linearly independent vectors v i such that Let the elements e ± i and h i , where i = 1, . . . , n, generate a Lie superalgebra denotedg(A, I), where I = (p 1 , . . . p n ) ∈ (Z/2) n is a collection of parities (p(e ± i ) = p i ), free except for the relations ±A ij e ± j and [h i , h j ] = 0 for any i, j. Let Lie (super)algebras with Cartan matrix g(A, I) be the quotient ofg(A, I) modulo the ideal explicitly described in [GL,BGL3,BGLL].
By abuse of notation we denote by e ± j and h j -the elements ofg(A, I) -also their images in g(A, I) and g (i) (A, I) and call these images, and their pre-images, the Chevalley generators ofg(A, I), g(A, I), and g (i) (A, I), cf. Subsection 2.5.1.
2.1.1. In small font. The additional to (7) relations that turng(A, I) into g(A, I) are of the form R i = 0 whose left sides are implicitly described, for the general Cartan matrix with entries in K, as (8) the R i that generate the ideal r maximal among the ideals ofg(A, I) whose intersection with the span of the above h i and the d j described in eq. (12) is zero. Set Then, from the properties of the matrix T , we deduce that (10) a) the elements c i are linearly independent; b) the elements c i are central, because The existence of central elements means that the linear span of all the roots is of dimension n−l only. (This can be explained even without central elements: The weights can be considered as column-vectors whose i-th coordinates are the corresponding eigenvalues of ad h i . The weight of e i is, therefore, the i-th column of A. Since rk A = n − l, the linear span of all columns of A is (n − l)-dimensional just by definition of the rank. Since any root is an (integer) linear combination of the weights of the e i , the linear span of all roots is (n − l)-dimensional.) This means that some elements which we would like to see having different (even opposite if p = 2) weights, actually, have identical weights. To remedy this, we do the following: let B be an arbitrary l × n-matrix such that (11) the (n + l) × n-matrix A B has rank n.
Let us add to the algebra g =g(A, I) (and hence g(A, I)) the grading elements d i , where i = 1, . . . , l, subject to the following relations: = 0 (the last two relations mean that the d i lie in the Cartan subalgebra, and even in the maximal torus which will be denoted by h).
Note that these d i are outer derivations of g(A, I) (1) , i.e., they can not be obtained as linear combinations of brackets of the elements of g(A, I) (i.e., the d i do not lie in g(A, I) (1) ).
2.2. Roots and weights. In this subsection, g denotes one of the algebras g(A, I) org(A, I).
Let h be the span of the h i and the d j . The elements of h * are called weights. For a given weight α, the weight subspace of a given g-module V is defined as Any non-zero element x ∈ V is said to be of weight α. For the roots, which are particular cases of weights if p = 0, the above definition is inconvenient because it does not lead to the modular analog of the following useful statement.
2.2.1. Statement ( [K]). Over C, the space of any Lie algebra g can be represented as a direct sum of subspaces Note that if p = 2, it might happen that h g 0 . (For example, all weights of the form 2α over C become 0 over K.) To salvage the formulation of Statement in the modular case with minimal changes, at least for the Lie (super)algebras g with Cartan matrix -and only this case we will have in mind speaking of roots, we decree that the elements e ± i with the same superscript (either + or −) correspond to linearly independent roots α i , and any root α such that g α = 0 lies in the Z-span of {α 1 , . . . , α n }, i.e., (14) g = α∈Z{α 1 ,...,αn} g α .
Thus, g has a R n -grading such that e ± i has grade (0, . . . , 0, ±1, 0, . . . , 0), where ±1 stands in the i-th slot (this can also be considered as Z n -grading, but we use R n for simplicity of formulations). If p = 0, this grading is equivalent to the weight grading of g. If p > 0, these gradings may be inequivalent; in particular, if p = 2, then the elements e + i and e − i have the same weight. (That is why in what follows we consider roots as elements of R n , not as weights.) Any non-zero element α ∈ R n is called a root if the corresponding eigenspace of grade α (which we denote g α by abuse of notation) is non-zero. The set R of all roots is called the root system of g.
Clearly, the subspaces g α are purely even or purely odd, and the corresponding roots are said to be even or odd.
2.3. Systems of simple and positive roots. In this subsection, g = g(A, I), and R is the root system of g.
For any subset B = {σ 1 , . . . , σ m } ⊂ R, we set (we denote by Z + the set of non-negative integers): The set B is called a system of simple roots of R (or g) if σ 1 , . . . , σ m are linearly independent and R = R + B ∪ R − B . Note that R contains basis coordinate vectors, and therefore spans R n ; thus, any system of simple roots contains exactly n elements.
A subset R + ⊂ R is called a system of positive roots of R (or g) if there exists x ∈ R n such that (α, x) ∈ R\{0} for all α ∈ R, (Here (·, ·) is the standard Euclidean inner product in R n .) Since R is a finite (or, at least, countable if dim g(A) = ∞) set, so the set is a finite/countable union of (n − 1)-dimensional subspaces in R n , so it has zero measure. So for almost every x, condition (15) holds.
By construction, any system B of simple roots is contained in exactly one system of positive roots, which is precisely R + B . 2.3.1. Statement. Any finite system R + of positive roots of g contains exactly one system of simple roots. This system consists of all the positive roots (i.e., elements of R + ) that can not be represented as a sum of two positive roots.
We can not give an a priori proof of the fact that each set of all positive roots each of which is not a sum of two other positive roots consists of linearly independent elements. This is, however, true for finite dimensional Lie algebras and Lie superalgebras of the form g(A) if p = 2.
The rescaling affects only the matrix A B , not the set of parities I B . The Cartan matrix A is said to be normalized if (17) A jj = 0 or 1, or 2.
We let A jj = 2 only if i j =0; in order to eliminate possible confusion, we write A jj =0 or1 if i j =0, whereas if i j =1, we write A jj = 0 or 1. Normalization conditions correspond to the "natural" Chevalley generators of the most usual "building blocks" of finite-dimensional Lie (super)algebras with Cartan matrix: sl(2) if A jj = 2, sl(1|1) if A jj = 0, and osp(1|2) if A jj = 1, respectively. (In this paper we do not need A jj =0 or1, see [BGL, CCLL].) We will only consider normalized Cartan matrices; for them, we do not have to indicate the set of parities I.

2.4.1.
Warning. Unlike the case of simple finite-dimensional Lie algebras over C, where the normalized Cartan matrix A is uniquely defined, generally this is not so: each row with a 0 or0 on the main diagonal can be multiplied by any nonzero factor; usually (not only in this paper) we multiply the rows so as to make A B symmetric, if possible. Which version of the Cartan matrix should be considered as its "normal form"? The defining relations give the answer: The normalized Cartan matrix is used, for example, to describe presentation of the given Lie superalgebra (relations between, or rather among 1 the Chevalley generators).
2.5. Equivalent systems of simple roots. Let B = {α 1 , . . . , α n } be a system of simple roots. Choose non-zero elements e ± i in the 1-dimensional (by definition) superspaces g ±α i ; set where the entries A ij are recovered from relations (7), and let I B = {p(e 1 ), · · · , p(e n )}. Lemma 2.7.1 claims that all the pairs (A B , I B ) are equivalent to each other.
Two systems of simple roots B 1 and B 2 are said to be equivalent if the pairs (A B 1 , I B 1 ) and (A B 2 , I B 2 ) are equivalent.
It would be nice to find a convenient way to fix some distinguished pair (A B , I B ) in the equivalence class. For the role of the "best" (first among equals) order of indices we propose the one that minimizes the value |i − j| (i.e., gather the non-zero entries of A as close to the main diagonal as possible). Observe that this numbering differs from the one that Bourbaki use for the e type Lie algebras.
2.5.1. Chevalley generators and Chevalley bases. We often denote the set of generators of g(A, I) and g (i) (A, I) corresponding to a normalized Cartan matrix by X ± 1 , . . . , X ± n instead of e ± 1 , . . . , e ± n ; and call these generators, together with the elements H i := [X + i , X − i ], and the derivations d j , see (12), the Chevalley generators.
For p = 0 and normalized Cartan matrices of simple finite dimensional Lie algebras, there exists only one (up to signs) basis containing X ± i and H i in which A ii = 2 for all i and all structure constants are integer, cf. [St]. Such a basis is called the Chevalley basis.
Observe that, having normalized the Cartan matrix of sp(2n) so that A ii = 2 for all i = n, but A nn = 1, we get another basis with integer structure constants. We think that this basis also qualifies to be called Chevalley basis; for Lie superalgebras, and if p = 2, such normalization is a must.
Conjecture. If p > 2, then for finite dimensional Lie (super)algebras with indecomposable Cartan matrices normalized as in (17), there also exists only one (up to signs) analog of the Chevalley basis.
From [BGL]: "We had no idea how to describe analogs of Chevalley bases for p = 2 until recently; it seems, the methods of the recent paper [CR] should solve the problem." (Now, more than a decade ago, this problem is still open.) 2.6. Reflections. Let R + be a system of positive roots of Lie superalgebra g over a field K of characteristic p > 0, and let B = {σ 1 , . . . , σ n } be the corresponding system of simple roots with some corresponding pair (A = A B , I = I B ). Then for any k ∈ {1, . . . , n}, the set (R + \{σ k }) {−σ k } is a system of positive roots. This operation is called the reflection in σ k ; it changes the system of simple roots by the formulas where we consider Z/pZ as a subfield of K.
The name "reflection" is used because in the case of simple finite-dimensional complex Lie algebras this action, extended on the whole R by linearity, is a map from R to R, and it does not depend on R + , only on σ k . This map is usually denoted by r σ k or just r k . The map r σ k extended to the R-span of R is reflection in the hyperplane orthogonal to σ k relative the bilinear form dual to the Killing form.
The reflections in the even (odd) roots are referred to as even (odd ) reflections. A simple root is called isotropic, if the corresponding row of the Cartan matrix has zero on the diagonal, and non-isotropic otherwise. The reflections that correspond to isotropic or non-isotropic roots will be referred to accordingly.
If there are isotropic simple roots, the reflections r α do not, as a rule, generate a version of the Weyl group because the product of two reflections in nodes not connected by one (perhaps, multiple) edge is not defined. These reflections just connect a pair of "neighboring" systems of simple roots and there is no reason to expect that we can multiply such two distinct reflections. In the case of modular Lie algebras or of Lie superalgebras for any p, the action of a given isotropic reflection (19) can not, generally, be extended to a linear map R −→ R. For Lie superalgebras over C, one can extend the action of reflections by linearity to the root lattice, but this extension preserves the root system only for sl(m|n) and osp(2m + 1|2n), cf. [Se1].
We would like to draw attention of the reader to an under-appreciated paper [SkB], where the analog of Weyl group was considered in relation with br(3).

2.7.
How reflections act on Chevalley generators. If σ i is an isotropic root, then the corresponding reflection r i sends one set of Chevalley generators into a new one: The Cartan matrix r i (A) corresponding to the Chevalley generators (21) should be obtained as described above: setH 2.7.1. Lebedev's lemma. Serganova [Se] proved (for p = 0) that there is always a chain of reflections connecting B 1 with some system of simple roots B ′ 2 equivalent to B 2 in the sense of definition in Subsection 2.6. Here is the modular version of this statement due to Serganova.
Lemma (Lebedev, unpublished). For any two systems of simple roots B 1 and B 2 of any finite dimensional Lie superalgebra with indecomposable Cartan matrix, there is always a chain of reflections connecting B 1 with B 2 .

2A kj
A kk in (20) are elements of K, while the roots are elements of a vector space over R. Therefore these expressions in the first case in (20) should be understood as "the smallest non-negative integer congruent to − This convention is important in describing Serre relations and their analogs for p > 0, see [BGL3,BGLL]. We do not know yet what is an equally "natural" or correct way of presenting the elements of the inverse Cartan matrices.
There is known just one exception: if p = 2 and A kk = A jk , then the expression −

2A jk
A kk should be understood as 2, not 0. (If dim g < ∞, the expressions −

2A jk
A kk are always congruent to integers.) 2.7.3. How reflections act on Cartan matrices ( [CCLL]). Let A be a Cartan matrix of size n and I = (p 1 , . . . , p n ) the vector of parities. If p k =1 and A kk = 0, then the reflection in the kth simple odd root sends A to r k (A), where This can be expressed in terms of matrices as where all columns of the matrix B, except the kth one, are zero, whereas the ith coordinate of the kth column-vector is b i , the ith coordinate of the kth row-vector of C is c i , the other rows of C being zero; E is the unit matrix. Therefore, B 2 = C 2 = 0, and The reflected matrix r k (A) might have to be normalized; the new parities are p j ≡ p j + c j (mod 2).
3. The matrices considered in [WZ] are not Cartan matrices For any simple finite-dimensional Lie algebra over C, we know two ways to introduce its Cartan matrix A = (A ij ).
First approach: take 3n Chevalley generators and compute [h i , e ± j ] = ±A ij e ± j , see (7). Second approach: take an auxiliary space -the space spanned by the roots or, sometimes, a bit larger space, see tables in [Bbk, OV]; for sl (V ), this auxiliary space is the space V or its dual. Let it be spanned by column-vectors ε i := (0, . . . , 0, 1, 0 . . . , 0) T with a 1 on the ith place.
Assume that there is a Euclidean inner product on V given by The Gram matrix of this inner product in the basis of simple roots is precisely the Cartan matrix of sl (V ). This inner product in the space spanned by roots is induced by the restriction of the Killing form from sl(V ) on the maximal torus, i.e., the space of coroots. For the Lie algebras of series sl, o and sp, this inner product can be also defined by means of the form proportional to the Killing form, but much easier to compute: (X, Y ) = tr(XY ) for any X, Y ∈ sl(V ).
The above description yields the Gram matrix of a nondegenerate invariant symmetric bilinear form on the Lie algebras whose roots are of equal length. For the simple Lie algebras whose roots are of different lengths, the matrix thus obtained is a symmetrization of the Gram matrix, see [BKLS].
Passing to the Lie superalgebras, we can also follow either of these two procedures. The first approach does indeed lead to Cartan matrices, as described in the beginning of this section.
The second approach uses a basis of the superspace V , such that the vectors ε i span V0, while the vectors δ j span V1, with the pseudo-Eucledian inner product For the Lie superalgebras gl(V ) and psl (V ), this inner product is induced by the supertrace Serganova showed, see [Se1], that a natural generalization of the axioms of root systems to the case of pseudo-Euclidean inner product leads precisely to the root systems of Lie superalgebras with indecomposable Cartan matrices and their simple relatives over C, see [CCLL].
It seems that the Gram matrix of this inner product in the basis of simple roots first appeared in [WZ]: Serganova never wrote it explicitly for any Lie superalgebra. For Lie superalgebras, no Gram matrix of the above inner product in the root space of g is equivalent (in any conventional way) to any Cartan matrix of g, except for osp(1|2n). However, For the root space of g = sl(m + 1|n + 1), the Gram matrix of the above inner product calculated in the standard supermatrix format of g turns into the Cartan matrix (28) if its bottom n + 1 rows are multiplied by −1.
3.1. Remark. To describe defining relations of g(A) in terms of Chevalley generators, we need the Cartan matrix A corresponding to the selected supermatrix format of elements of g(A), see [GL], not the Gram matrix of the inner product in the space spanned by roots.
When we learn what the inverse of any of these Gram matrices is needed for, we will probably have to compute it in just one basis: the NIS on a simple finite-dimensional Lie superalgebra of characteristic = 2 is unique, up to a proportionality; for a recipe for constructing NIS from the Cartan matrix, see [BKLS]; the Gram matrix of its restriction onto the space spanned by the h i is the dual of the Gram matrix of the inner product on the space spanned by roots.
4. Finite-dimensional serial Lie (super)algebras over C 4.1. Remark. Remember that the Cartan matrix with a 0 on the main diagonal is not uniquely defined: the line with this 0 can be multiplied by any non-zero number, e.g., by a −1, see Subsection 2.4.1. For any invertible matrix, its inverse is uniquely defined, of course.
For the series sl and osp, there are 9 types of pairs of "basic" Cartan matrices connected by odd reflections, see [CCLL,Subsection 4.1, Table 1]. From one such Cartan matrix all the other Cartan matrices of the given Lie superalgebra of the given type are obtained by means of odd reflections, see (22). It is a matter of taste which one in the pair of "basic" matrices is most simple; we selected (any) one with the smallest number of 0s on the main diagonal.
In [WZ], Gram matrices are inverted; this sometimes gives the answer for the Cartan matrices as well, thanks to the fact (26). However, inverting Cartan matrices of the sl cases, we have to consider 2 more types of cases as compared with [WZ]; for the osp series, we have to consider 7 types of cases, not 3 as in [WZ].
Let A m denote the normalized Cartan matrix of sl(m + 1). Below, in formulas (27)- (40), instead of A m there can be any Cartan matrix of sl(a|b) with a + b = m + 1. To derive the explicit expression of the inverse matrix C −1 in this general case, apply an odd reflection (23).

4.2.
The case of sl. For sl(m + 1|1), the "basic" Cartan matrix and its inverse (27) is a particular cases of the expression (28), up to an occasional minus sign, see Remark 4.1.
For sl(m + 1|n + 1), where mn = 0, there are 2 "basic" types of Cartan matrices, see (28) and (31): where, as shown in [WZ], L 1 and L 4 are (m + 1) × (m + 1) and n × n matrices, respectively, defined, together with L 2 , by the following formulas To invert the Cartan matrix of the other type, namely (31), we set We are seeking the matrix N −1 in the form Denote the i th row (resp. column) by r(i) (resp. c(i)). Consider the following transformations: Observe that τ n is obtained from T n , see (34), by transposing with respect to the side diagonal. Applying transformations a)-d) in lexicographic order we obtain N from M.

From the equation
The case of osp. For osp(a|2b), there are 6 pairs of series and one single series of "basic" types of Cartan matrices which, together with their inverses, are (34) -(40): (34) the matrix T n , see [Str,Section 1.1], is inverted, e.g., in [WZ], where it is denoted S n ;

Proof of formulas (35)-(40). Answers: eqs. (41), (42)
There are several formulas for the inverse of various (invertible) block 2 × 2 matrices. In our particular cases, we can use the following ad hoc transformations and the known expressions of A −1 n and W n = τ −1 n , see (33). 1) Let m = 2, or 3, or 4, and U is the m × n-matrix with only non-zero element, 1, in the bottom left corner. Invert the matrix of the form Note that V U = α, see (30). It is easy to verify that 2) In the above notation, consider the matrix of the form Set P := Q −1 . In what follows, for the cases (35)-(40), we prove the existence of matrices L and R and give the matrices P .
Since rl = Q −1 = P , we have the following uniform answer for eqs. (35)-(40): where (recall that m is the size of the upper left block of N) Eq. (35). We perform the following transformations (recall eq. (32)): Eq. (36). We perform the following transformations: Eq. (37). We perform the following transformations: Eq. (38). We perform the following transformations: Eq. (39) We perform the following transformations: Eq. (40) We perform the following transformations: 6. Finite-dimensional exceptional Lie (super)algebras over C In this section, the determinant of the Cartan matrix is equal to the minus denominator of the fraction serving as a factor of the "matrix part" of the inverse matrix; if the factor is equal to 1, then the determinant is equal to 1.
7.1. p = 2; Lie (super)algebras. In this Subsection, * on the main diagonal stands for either0 or 0; both can be viewed as 0 for our purposes.
Below, the symbol !!! marks the cases where the inverse of the Cartan matrix has no zero elements, cf. [ZH].
6.1. Cartan matrices A, their inverses, and det A for almost affine Lie superalgebras