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A class of quadratic deformations of Lie superalgebras

We study certain Z_2-graded, finite-dimensional polynomial algebras of degree 2 which are a special class of deformations of Lie superalgebras, which we call quadratic Lie superalgebras. Starting from the formal definition, we discuss the generalised Jacobi relations in the context of the Koszul property, and give a proof of the PBW basis theorem. We give several concrete examples of quadratic Lie superalgebras for low dimensional cases, and discuss aspects of their structure constants for the `type I' class. We derive the equivalent of the Kac module construction for typical and atypical modules, and a related direct construction of irreducible modules due to Gould. We investigate in detail one specific case, the quadratic generalisation gl_2(n/1) of the Lie superalgebra sl(n/1). We formulate the general atypicality conditions at level 1, and present an analysis of zero-and one-step atypical modules for a certain family of Kac modules.


Introduction
The classical problem of investigation of the correct mathematical structures within which to frame physical models is, in the case of quantum mechanical systems, intimately related to underlying symmetry principles. Until recently, the classical mathematics of discrete groups and Lie groups and their associated algebras and representations were the framework within which regularities of energy levels and matrix elements were organised, even in the absence of microscopic theories. The role of such underlying symmetry principles has not been supplanted even with the advent of supersymmetry and Lie superalgebras to handle fermionic states. The revolutions of quantum groups and integrable models in one dimensional quantum field theory or two dimensional lattice models have shown in recent decades that more subtle mathematical structures, such as braiding for tensor products of composite state vector spaces, can and should have a natural place in the physicist's tool-kit. These objects, centring around the famous Yang-Baxter equation, are definitively 'nonlinear' in their structure, and often entail one or more additional 'deformation' parameters which can be new physical constants, presaging the breakdown of standard symmetry relations in the appropriate asymptotic regime.
The present work deals with some classes of polynomial algebras. These are mild generalisations of Lie algebras, but are surprisingly universal, in that they find applications in many different fields, and also contain special cases of some of the above-mentioned quantum groups. Our study is restricted to certain polynomial, Z 2 -graded algebras of degree 2, which are a special class of deformations of Lie superalgebas, which we term 'quadratic Lie superalgebras'. In our previous work, we studied these objects in the context of the problem of enumerating gauge invariant fields (observables) in a Hamiltonian lattice formulation of QCD. In [14] we introduced and studied 'polynomial super gl(n) algebras' heuristically via free-fermion and -boson constructions. The methodology resembled the intention of very early attempts to define 'baryonic currents' for the purpose of generalising the applicability of scalar and vector meson current algebras in particle physics [5]. In [13] our constructions were used to study the structure of the observable algebra of lattice QCD and to prove a classification theorem for its irreducible representations (superselection structure).
In the present work we are at pains to establish a systematic approach to quadratic Lie superalgebras in their own right. They are introduced in §2 by way of formal definitions via appropriate tensor algebras and defining relations. For completeness we refer to the abstract context of Koszul complexes and Koszulness. Although the latter attribute is needed, it is not further explored in this paper as our notation is anchored in concretely enumerated structure relations. We cite the important structure theorem (Theorem 2.7) establishing the existence and significance of generalised Jacobi identities, which if true together with Koszulness, establishes the existence of the PBW basis theorem. In Lemma 2.9 we give a direct proof of the PBW property (detailed in the appendix, §A.1) which guarantees Koszulness. In §3, we introduce a natural specialisation to a sub-class of quadratic Lie superalgebras, the so-called type I class, which admits a refined grading with a specific structure. Although it is beyond the scope of our present techniques to formulate a classification of quadratic Lie superalgebras, nonetheless some features of the structure constants are generic, with an intimate relation to invariants of the even subalgebra, as shown by Lemma 3.12 and its proof. The results of §2 are used together with the grading to establish useful alternative decompositions of the universal enveloping algebra. In turn, this sets the scene in §3 for the definition of induced representations following the Kac module [15,16] construction in the Lie superalgebra case, and as modified by Gould [8] to give a presentation of all finite dimensional irreducible Kac modules, including those of atypical type. In §4 we turn to a specific family of quadratic Lie superalgebras which we identified previously, the quadratic generalisations gl 2 (n/1) of the classical Lie superalgebra sl(n/1). Using the methods of characteristic identities for generators of classical groups [10,1] as applied to the Lie superalgebra case [8,9] we provide an explicit characterisation of atypicality conditions for irreducible modules at level one. For a specific class of highest weight modules we also examine zero-and one-step atypicals, and give some explicit cases. In the concluding §5 we recall some of the previously-identified quadratic Lie superalgebras motivated from lattice QCD, and note how the phenomenon of atypicality arises there. One of the features of our analysis, both here and in §4, is the remarkable fact that type I quadratic Lie superalgebras generically admit atypical modules of depth zero -that is, they are representable on a single, nontrivial irreducible module of the even subalgebra.

Formal constructions and the PBW basis theorem
Let L = L0 + L1 be a finite-dimensional Z 2 -graded complex vector space. Take the even and odd subspaces L0 and L1 to be spanned by basis elements x i , i = 1, 2, · · · , n, and y r , r = 1, 2, · · · , m, respectively. The tensor algebra T (L) = ∞ n=0 ⊗ n (L) ∼ = C + L + L ⊗ L + · · · inherits the Z 2 -grading in the natural way, in that for T n := ⊗ n T (L) we have with eachᾱ i =0 or1, and T n ∼ = T n |0 + T n |1.
We now introduce the concrete defining relations of the graded algebras of interest. The explicit Definition 2.1 below will be justified by a discussion of the structure of the enveloping algebra, to be given presently in a slightly more general context, after which the definition and the motivation for taking up this class as a natural object of investigation will become clearer (see Remarks 2.3 below).
Consider the following arrays of complex numbers: d p,q k,ℓ = d q,p k,ℓ = d p,q ℓ,k ; b p,q k = b q,p k ; a p,q = a q,p , as well as c i,j k = −c j,i k , andc k,p q , for indices in the specified ranges above, namely i, j, k, ℓ = 1, 2, · · · , n, and p, q, r = 1, 2, · · · , m. They are required to satisfy the following bilinear relations (summation on repeated indices, and for ease of writing, the comma −,− separating suffixes is omitted where there is no ambiguity): Let I be the linear subspace of L ⊗ L + L + C spanned by the set and let J(I) be the corresponding two-sided ideal in T (L) generated by I.
Definition 2.1: Quadratic Lie superalgebra: Given quantities d p,q k,ℓ , b p,q k , a p,q , c i,j k andc k,p q with properties (1) above, and the corresponding subspace I as in (2) and ideal J(I), the subalgebra of the tensor algebra defined by U (I) = T (L)/J(I) is called the quadratic Lie superalgebra associated with I.
Associated with the tensor algebra is the filtration defined by T n = n k=0 T k , in such a way that C ∼ = T 0 ⊂ T 1 ⊂ T 2 ⊂ · · · , that is, T n ⊂ T n+1 . U ≡ U (I) inherits this filtration in the natural way, so that we have C ∼ = U (I) 0 ⊂ U (I) 1 ⊂ U (I) 2 ⊂ · · · and we define the associated graded algebra as the direct sum: Definition 2.2: Graded algebra associated to quadratic Lie superalgebra: The graded algebra gr(U (I)) = C + ∞ n=1 U (I) n /U (I) n−1 is called the graded superalgebra associated with the quadratic Lie superalgebra U (I).
Where the context is clear, we omit the bracket (I) and simply denote the quadratic Lie superalgebra by U , and the associated graded algebra by gr(U ). I is Z 2 -graded, and hence so also is the ideal J(I), spanned by elements of T (L) of the form (at degree p + q + 2, for example) with p prefactors of L and q postfactors. Thus U inherits the natural Z 2 grading from T (L). Note that U is strictly speaking a (Z 2 -graded) quadratic-linear algebra in the notation of [18], because I L ⊗ L but contains elements of degree 1. However, just as in the case of Lie superalgebras, the appellation quadratic Lie superalgebra which we adopt here suggests an intimate relation with an underlying Lie algebra, which is easily seen from the above definitions as follows.
Consider the subspace I 0 ⊂ L0 ⊗ L0 + L0 + C spanned by the set given just by the first line of (2) and the corresponding ideal J(I 0 ) in T (L0). Clearly U 0 := T (L0)/J(I 0 ) ⊂ U0 simply by inclusion.
On the other hand, the linear space L0 spanned by x i , subject to the constraint given by the first line of (1), is indeed a Lie algebra, with the bracket [ the corresponding universal enveloping algebra. By abuse of notation we refer to L0 simply as L 0 , the underlying Lie algebra of the quadratic Lie superalgebra U .

Remarks 2.3:
As in the Lie superalgebra case, the basis elements x i are called the even generators, the y p are the odd generators, and the arrays d pq kℓ , b pq k , a pq , c ij k , andc kp q are called the structure constants of U (indeed, if d pq kℓ = 0, then L is itself a Lie superalgebra). Note that we do not include additional quantities, say e ij pq , which might provide quadratic modifications to the first line of (2); nor do we admit additional scalars c ij corresponding to central extensions of L 0 . The structure constants can be regarded as fixed, numerical components of some complex tensors of the appropriate contravariant and covariant ranks. If needed, generic homogeneous basis elements are denoted w a , w b , a, b = 1, 2, · · · , m + n; alternatively w i ≡ x i and w n+r ≡ y r , for indices in the specified ranges. Algebraic relations can then be written with the help of the usual notation for the grading, namely |a| =0 for a = 1, 2, · · · , n, and |a| =1 for a = n + 1, n + 2, · · · , n + m (see for example § §A.1, A.2).
Further important objects associated with L and U (I) are as follows. Let S be the ideal generated by the symmetric and antisymmetric tensor products L0∧L0 and L1∨L1 in L⊗L (that is, the spaces spanned by the elements x i ⊗ x j + x j ⊗ x i , i ≤ j, and y p ⊗ y q − y q ⊗ y p , p < q, for i, j = 1, 2, · · · , n and p, q = 1, 2, · · · , m, respectively), together with the subspace of L0 ⊗ L1 + L1 ⊗ L0 spanned by x i ⊗ y p + y p ⊗ x i , for i = 1, 2, · · · , n and p = 1, 2, · · · , m ).

Definition 2.4: Super symmetric tensor algebra:
The super symmetric tensor algebra S(L) = T (L)/S is the tensor algebra modulo these symmetric and antisymmetric relations.
More generally, let I 2 = I ∩ L ⊗ L be the projection of I onto L ⊗ L, and J(I 2 ) be the corresponding two-sided ideal of T (L) generated by I 2 .
Definition 2.5: Homogeneous quadratic Lie superalgebra: The homogeneous quadratic Lie superalgebra U (I 2 ) = T (L)/J(I 2 ) is the tensor algebra modulo these homogeneous quadratic relations.
To complete these introductory remarks, we refer to the notion of the Koszul complex associated with U , although we do not require this for the present work. Let I ⊥ ⊂ L * ⊗ L * ∼ = (L ⊗ L) * be the quadratic annihilator of I, and J(I ⊥ ) the corresponding two-sided ideal. Define U ! := T (L * )/J(I ⊥ ). Note that (U ! ) * is a right U * -module by the action x · η(ζ) := x(ηζ). Taking a basis w a for L, and the corresponding dual basis w a for L * , it can be shown that the mapping d ∈ End(U ⊗ (U ! ) * ) defined by d · u ⊗ v := a w a u ⊗ vw a is a differential, that is, d • d = 0 [18].
Definition 2.6: Koszul quadratic Lie superalgebra: The quadratic Lie superalgebra U is Koszul iff the cohomology associated with the complex defined by the operator d is trivial.
The role of the notion of Koszulness is shown in the following key theorem. Firstly note that the direct sum decomposition I ⊂ L ⊗ L + L + C enables maps α : I 2 → L and β : I 2 → C to be defined such that Theorem 2.7: Poincaré Birkhoff Witt basis theorem ( [18], Ch 5, Theorem 2.1): is Koszul, and the following conditions are satisfied: we have the isomorphism gr(U (I)) ∼ = U (I 2 ) The relations (J1-J3) are called the generalised Jacobi identities. As mentioned above, proving Koszulness requires an analysis of the structure of the tensor ideal J(I), and as shown in [18] requires certain distributivity conditions to hold on the subspaces (3). Here instead we firstly (Lemma 2.8 below) establish (J1), (J2) and (J3), and secondly (Lemma 2.9) prove that the homogeneous quadratic algebra U (I 2 ) is a PBW algebra, or of PBW type, that is, it admits an ordered basis of monomials (a PBW basis) in the generators w a , under mild conditions. Now a result of Priddy ([19], Theorem 5.3; see also [20]) states that a homogeneous quadratic algebra of PBW type is Koszul, and so the conditions for Theorem 2.7 are established, and the isomorphism gr(U (I)) ∼ = U (I 2 ) is guaranteed. Therefore, the PBW basis also provides a basis for gr(U (I)), and hence, via the isomorphism of spaces, a basis also for U itself -the normal statement of the PBW basis theorem. Lemma 2.8: Generalised Jacobi Identities for quadratic Lie superalgebras: The bilinear relations (1) of a quadratic Lie superalgebra are equivalent to the generalised Jacobi identities (4). Proof: The first task is to identify generic elements of I 2 ⊗L∩L⊗I 2 , by finding linear combinations of tensor products of basis elements from each space which match. In the standard Lie algebra case, the required combinations are easily found. Given basis elements x i and x j x k − x k x j (dropping the tensor product symbol ⊗ for ease of writing), the cyclic sum can be written in two ways: from which it is evident that the space spanned by such elements is simply the third rank exterior product. For Lie superalgebras, suitably graded cyclic sums with y p y q + y q y p are also included. For quadratic Lie superalgebras however, such elements of I 2 include additional terms (for example with y p y q + y q y p becoming y p y q + y q y p − d pq kℓ x k x ℓ as above). Cyclic combinations of an arbitrary basis element tensored with such terms no longer produce an equality. However, in the quadratic Lie superalgebra case, it turns out that one can explicitly add and subtract terms on each side to compensate for these pairs.
To go further we investigate (J1-J3) for a particular grading of the triple of basis elements from111,110,100 and000. Given (2) the maps α and β are (repeated indices summed) where the symbols ad i x j and b pq are introduced to save space. Let us consider for example110; other cases can be handled similarly. The following identity shows that the z pqi span this grading of the intersection and rewrite this so as to expose the intersection with I 2 : where new terms d ip,q kℓ :=c ip r d rp kℓ , d p,iq kℓ :=c iq r d pr kℓ have been added and subtracted. Obviously the last term belongs to the symmetric product L 0 ∨ L 0 and so must vanish if (J1) is to be fulfilled. It is readily checked that the component form of is nothing but the second equation of (1) above. Proceeding with (J2), Vanishing of this combination produces the component form corresponding to the third of equations (1).
The first and second equations of (1) are derived by considering (J1-J3) for the000 and001 sectors respectively, that is, rewriting combinations of x i yx j x k and x i x j y p ; the fourth and fifth entries are derived similarly by examining the111 sector, that is, rewriting combinations of y p y q y r .
The remaining requirement for the PBW basis theorem, the existence of a PBW basis, can be established under fairly general conditions on the partial ordering of ordered pairs of indices. The definition of the ordering, together with the proof of the following property in a slightly more general context is given in the Appendix, §A.1 as Theorem A.21.
Lemma 2.9: PBW basis for quadratic Lie superalgebras: A homogeneous quadratic Lie superalgebra is of PBW type under any index ordering such that only those d pq kℓ are nonvanishing for which k, ℓ precedes p, q.

Type I quadratic Lie superalgebras and Kac modules
The class of quadratic algebras is much richer than that of Lie algebras. The same is true for the restricted class of quadratic Lie superalgebras that we study, and it is beyond the scope of this work to investigate their enumeration and classification. However, some remarks bearing on the structure can easily be derived in the notation that we have given, and this leads to a binary typology of quadratic Lie superalgebras analogous to that well known for Lie superalgebras [15], as well as explicit methods for building the d pq kℓ . This is discussed in §3.1 below, and in §3.2 we take up the study of 'type I' quadratic Lie superalgebras. This results in the analogue of the Kac module construction for superalgebras [16], followed by a variant of this due to Gould [8], which gives an explicit construction of finite dimensional irreducible modules, both typical and atypical.

Remarks on the structure of quadratic Lie superalgebras
We assume from now on that the underlying Lie algebra L 0 is reductive. For brevity we drop explicit reference to the tensor product ⊗ when working with U (I) or its subalgebras. Given the significance of the generalised Jacobi identities established above, we also adopt the familiar Lie algebra and Lie superalgebra notation for the structure relations in the form of brackets which have 'nonlinear' parts on the right-hand side. Thus the second line of (2) now reads there is no natural adjoint action ad w : L → L. However, (2) makes it clear that for x ∈ L 0 we can still define ad x : L1 → L1 by A further simplification is to assume this is a representation equivalent to its contragradient. If the matrix representation of the even generators in the odd submodule is π(x i ) p q = −c ip q , by assumption there is a nondegenerate bilinear form Ω rs with inverse Ω rs such that which implements the equivalence between π and its contragredient −π ⊤ . We adopt the following nomenclature: Definition 3.10: Balanced quadratic Lie superalgebras: A quadratic Lie superalgebra is called balanced if the odd submodule L1 is a real representation of the even Lie subalgebra L0.
Definition 3.11: Typology of balanced quadratic Lie superalgebras: Let λ be the highest weight of an irreducible representation of L 0 with contragredient λ * , and V 0 (λ), V 0 (λ * ) the corresponding L 0 -modules. Balanced quadratic Lie superalgebras are categorised into the following types according to the odd submodule L1: for λ a complex representation, with {L ± , L ± } = 0; or Type II: L1 ∼ = V 0 (λ), for λ a real representation.
Note that the extra assumption {L ± , L ± } = 0 for Type I ′ is stronger than for Type I Lie superalgebras, where the classification theorems are enough to guarantee it. Here it is introduced in the absence of classification results to avoid pathologies. As a final comment on the general nature of quadratic Lie superalgebras, it is clear that the structure constants must be natural tensors of L 0 which bear a close relation to known invariants [3]. Consider a Lie superalgebra or quadratic Lie superalgebra of balanced type with even and odd generators x i , y p and structure constants d pq kℓ , b pq k in the notation adopted above (d ≡ 0 in the Lie superalgebra case). Take quadratic and cubic invariants of L 0 to be of the form The following result is easily proven and provides a guaranteed explicit construction of d pq kℓ , as an extension of the (known) result for Lie superalgebras, which is here recalled: Lemma 3.12: Structure constants for Lie superalgebras and quadratic Lie superalgebras: The following forms for the structure constants fulfil the Jacobi and generalised Jacobi identities for Lie superalgebras and quadratic Lie superalgebras: Proof : The required Jacobi identities are tantamount to proving the invariance of the given Casimir operators. For the superalgebra case we use the Casimir invariant C 2 and the following universal object [2] in U 0 ⊗ U 0 , the cut coproduct where ∆x = x⊗id+id⊗x is the usual coproduct Lie algebra homomorphism. The object π⊗id•∆C ′ 2 commutes with π ⊗ id • ∆x = π(x) ⊗ id + id ⊗ x in End(L1) ⊗ U 0 ; note that the form b pq i is the matrix element of the bilinear form Ω composed with this map, namely C ij π(x j ) p r Ω rq : We identify b pq i as above, and also π(x k ) p t = −c kp t . For the second term we need the intertwining property of Ω pq , namely The invariance condition thus becomes which is just the second line of (1) above.
In the quadratic Lie superalgebra case, the same procedure works starting with the invariance of the cut coproduct of a cubic Casimir C 3 = C kℓm x k x ℓ x m , namely (taking account of the symmetry of C ijk and the mixed nature of the terms arising from the coproduct after removing C 3 ⊗ id and id ⊗ C 3 ), The new feature arising is that this expression in End(L1) ⊗ U 0 is graded by degree in U 0 , and invariance requires both linear and quadratic parts to vanish separately. Taking matrix elements in the projection onto the quadratic part, and using the intertwining property of the map Ω as above, yields the first line of (1).
For a more complete analysis along these lines, further considerations on a case-by-case basis, for example whether C 3 is given in irreducible form, would be needed. The object (11) will play a crucial role in the projection operator techniques to be used in §4.2.

Enveloping algebra for type I quadratic Lie superalgebras and Kac modules
The definition of type I ′ quadratic Lie superalgebras means that with each of dimension 2 d if dim(V 0 (λ)) = d. We note also the canonical elements which are, in particular, one dimensional L 0 -modules under the natural extension of the adjoint action (compare (9)). Let the respective (highest) weights be ∓2ρ 1 , that is, we take [h, S] = 2ρ 1 (h) = −[h, S] for h belonging to a Cartan subalgebra of L 0 . From the PBW basis theorem it follows that This is easily checked by considering basis elements of U 0 , U ± and re-ordering the appropriate products into the standard ordered monomial basis of U , or vice versa by showing that the latter monomials can be expressed as combinations of the re-ordered product forms. We introduce finite-dimensional induced representations as follows [15,16]. Let V 0 (Λ) be a finite-dimensional L 0 -module with (dominant integral) highest weight Λ. Make V 0 (Λ) into a U + -module by declaring yv = 0 for y ∈ L + , for all states v ∈ V 0 (Λ). Define A crucial point in relation to this construction is due to Gould [8]: namely, that the (necessarily irreducible) module cyclically generated by the states S ⊗ v, v ∈ V 0 (Λ), is a factor module of every invariant submodule of V (Λ) . By appropriate reorganisation of weight labels, this observation leads to the following direct construction of V (Λ) itself: Theorem 3.14: Modified Kac module construction by levels: Proof: See Gould [8].
Each of the above subspaces of V (Λ) for k fixed is referred to as the subspace at level k, 0 ≤ k ≤ n. By analogy with the Lie superalgebra case, the module V (Λ) is said to be typical if the level k = n subspace is nontrivial, The quadratic Lie superalgebra gl 2 (n/1) and atypicality conditions In this section we wish to apply the results of § §2,3, and especially §3.2, Theorem 3.14, to a specific quadratic Lie superalgebra, as a concrete case study. Several instances of (quadratic versions of) 'polynomial super gl(n)' algebras were given in [14] using the notation gl 2 (n/λ + λ * ), to indicate that the even subalgebra is gl(n), and the odd submodule L 1 ≃ V 0 (λ) + V 0 (λ * ) has the type I structure as in definition 3.11 above. Specific examples of λ given in [14] are Λ 1 , Λ 3 , or 3Λ 1 , where the odd generators are either in the fundamental defining representation, or antisymmetric or symmetric rank 3 tensor representations of gl(n), respectively (denoted {1}, {1,1,1} and {3}, in partition form), together with their contragredients. The last two cases will be discussed briefly in the concluding remarks, §5. Here we take up the case of odd generators in the fundamental representation, which we denote here simply gl 2 (n/1), by analogy with the Lie superalgebra case.
4.1 General results and structure for gl 2 (n/1) The even generators are the generators of gl(n) in Gel'fand form, E i j , i, j = 1, 2, · · · , n, and the odd generators are denoted Q i , Q j , i, j = 1, 2, · · · , n. In the notation of §3.2, we have gl 2 (n/1) ∼ = L ∼ = L − +L 0 +L + with L 0 ∼ = L0 ∼ = gl(n), L1 ∼ = L + +L − and L − , L + spanned by Q i , Q j respectively. The E i j have the standard commutation gl(n) relations while the components of Q form a vector, and the Q a contragredient vector, under the adjoint action of gl(n), Finally the quadratic bracket relations between Q and Q read where In [14] it has been established that the structure constants implicit in these relations (17) obey (1). As far as the factorisation of the enveloping algebra is concerned (see §3.2) (12) applies directly so that each of U ± is finite-dimensional with 2 n components.
We now proceed with the concrete steps elaborated in §3.2 for the construction of irreducible modules of gl 2 (n/1). Let Λ be a dominant integral gl(n)-weight and consider the corresponding weight Λ ′ := Λ − 2ρ 1 (which is also dominant integral). Take a lowest-weight gl(n) module V 0 (Λ ′ ). Introduce as before S in the form with the help of the Levi-Civita tensor ε i 1 i 2 ···in . Then from §3.2, Theorem 3.14 and following [8,9], we have for the irreducible gl 2 (n/1) Kac module associated with Λ, with the tensor product of type ⊗ U − . This equation (18) is the key construct from which the structure of the irreducible module can be derived. As we shall see, tensor projection methods can be applied, which explicitly give rise to patterns of vanishing of certain combinations of operators Q i 1 Q i 2 · · · Q i k applied to S ⊗ v, depending on the components of the highest weight vector Λ = (Λ 1 , Λ 2 , · · · , Λ n ), and central charge c. Firstly we record important basic relations between the generating multinomials of (12) above. In addition to S above, introduce the following auxiliary quantities:

Lemma 4.16: Calculus of multinomial odd generators:
The quantities (19) satisfy the following relations: whence [ E , S i 1 i 2 ···i k ] = (n−k)S i 1 i 2 ···i k . Moreover, we have where the quantities A, B are Proof : (20) is proven by explicit calculation using the anticommutation and commutation relations. For the first step of (21), (22) we have from Q m S = 0, the bracket from which the form of A m i , and subsequently A i m , can be read off by appropriate shifting, for example E k ℓ S = S E k ℓ + nδ k ℓ , and similarly S k E k i = (n − 1)δ i k − E i k S k . The second step proceeds analogously, starting firstly with [Q i , S j ] = S kℓ B kℓ ij and hence Q m [Q i , S j ] = 2S ℓ B mℓ ij ; on the other hand noting we can rearrange using also Q m S j = δ m j S to get which allows B mℓ ij to be inferred by shifting the terms in even generators to the right hand side as before.

Review of polynomial characteristic identities and shift operator analysis for gl(n)
In §3.1, Lemma 3.12, (11), we gave following [2] a very general way of identifying objects in the enveloping algebra U (gl(n)) when acting on an arbitrary gl(n)-module V 0 (λ). In the case of the standard gl(n) Gel'fand generators, the −E i j are the matrix elements of the object (11) in End(V 0 (Λ * 1 ) ⊗ End(V 0 (λ))). As has been shown in [10,2] (see also [7] and [8,9] for applications to Lie superalgebras), these objects satisfy a certain polynomial characteristic identity, of the form s (E − α s ) = 0, where general matrix powers of the array E are defined (extending the notation used already) by The roots are dependent on the components of the highest weight λ via α s = λ s + n − s, s = 1, 2, · · · , n. Dually there is an analogue identity s (E − α s ) = 0 for the array E i j = −E j i with its matrix powers defined analogously, and α s = n − 1 − λ s .
Let V , W be gl(n)-modules, and consider a tensor operator T -that is, a map T : V → End(W ) which intertwines the action of the Lie algebra, in the sense that [x, x · w, the space T (V )W is isomorphic to the tensor product V * ⊗W . If V = C n , identified as the fundamental representation Λ 1 , the object T is termed a vector operator, and the T i := T (e i ) with respect to the standard basis form the components of T .
Correspondingly if V = C n is identified with the dual Λ * 1 rather than the defining representation Λ 1 of gl(n), the corresponding object, say T , is called a contravariant vector operator, and its components are T i := T (f i ) with respect to the standard dual basis.
For (contravariant) vector operators, the polynomial characteristic identity can be used to provide a resolution into a sum of shift operators, themselves vector operators, using projection operators, with similar expressions for T i , where each projection operator is built from the appropriate monomial factors of the polynomial identity, namely .
only has components in the gl(n) module with highest weight λ+ δ s , where δ s is a weight of Λ * 1 (or Λ 1 , respectively). An important aspect of the polynomial identities is the following [10]. Although generically the identity is of degree n, in practice an identity of reduced degree may hold. If δ s is a weight of V 0 (Λ * 1 ), then the factor with root α s is retained only if λ + δ s is a dominant integral weight. Correspondingly, the projectors P [s], P [s] for non-dominant weights λ + δ s vanish identically, and the decomposition of T i is only over the remaining nonzero shifts. With these enveloping algebra methods in hand we turn to the analysis of the structure of the Kac modeules for gl 2 (n/1).

Shift operator analysis and atypicality conditions for gl 2 (n/1)
We return to (18) following (14), Theorem 3.14. Consider the first level, the states Q i S ⊗ v for v ∈ V 0 (Λ ′ ) and i = 1, 2, · · · , n. It is evident from (16) above that their span is isomorphic to the gl(n)-module V 0 (Λ * 1 ) ⊗ V 0 (Λ ′ ), the tensor product of the starting module V 0 (Λ ′ ) with the contragredient vector representation, with highest weight Λ * 1 = (0, 0, · · · , 0, −1). The following analysis relies on the resolution of the structure of this tensor product as a direct sum of irreducibles. Abstractly, this is of course done by inserting the corresponding projection operators; however, it is clear from §4.2 above that we can identify the Q i formally as the components of a vector operator Q which admits a covariant decomposition as a sum of shift components Q = r Q[r], each of which is also a vector operator, but which maps to states of only one specific irreducible summand. Let δ s be a weight of V 0 (Λ * 1 ). The irreducible Kac module V (Λ) contains the gl(n) submodule V 0 (Λ + δ s ) iff Λ + δ s is a dominant integral weight, and a s (Λ 1 , Λ 2 , · · · , Λ n ) = 0 for a certain polynomial function in the highest weight labels.
Proof : We follow the method used in Gould [8] for the Lie superalgebra gl(n/1). Apply the tensor projection method outlined above to establish that the level one subspace Q i S ⊗ v is the direct sum of projected parts Q[s] i S ⊗ v . Using the definitions and rearrangement identities, and the tensor product ⊗ U − , we manipulate the spanning states as follows: where in the last step, thanks to the characteristic identity, and the properties of the projectors noted above, ) the corresponding polynomial in α ′ s = Λ s − 1 + n − s, and the Casimir eigenvalues C ′ 1 , C ′ 2 in the representation V 0 (Λ ′ ) carried by the states v. Denote the corresponding polynomial of the components of Λ = Λ ′ + 2ρ 1 by a s (Λ) ≡ a s (Λ 1 , Λ 2 , · · · , Λ n ).
For the second level states, the projected parts of the antisymmetric tensor Q i Q j S acting on V 0 (Λ ′ ) are needed. The decomposition is known to be [8,9] Rearrangement of the tensor projections acting on B kl ij leads to a formulation equivalent to (24), giving criteria for the presence of the contributing even modules. For the purposes of this paper we give in the appendix, §A.3, a summary of the combinatorial manipulations which would be needed explicitly for this (and higher) levels, and provide a conjecture (Theorem A.23 ) on the nature of polynomials a rs (Λ 1 , Λ 2 , · · · , Λ n ), (a rst (Λ 1 , Λ 2 , · · · , Λ n ), · · · ) governing the structure of atypicality conditions in such cases.
Although we have not given the complete structure of atypical modules for gl 2 (n/1) in closed form, the general approach is clear. The general level one result, Theorem 4.17, demonstrates the combinatorial complication of the quadratic case. A unique feature which does not exist in the same way for Lie superalgebras is the emergence of classes of truncated atypical irreps wherein all potential even irreducible modules at a given level are annihilated as a result of polynomial identities satisfied by the even generators. For gl 2 (n/1), nontrivial cases associated with quadratic identities can be sought at level one and two, denoted zero-and one-step atypicals respectively (Theorem 4.20). To finish our examination of gl 2 (n/1) we turn to a more systematic examination of these cases. In the same vein we give a complete formulation of atypicality criteria at level 1 (Corollary 4.19) for a class of Kac modules V (Λ) for which the even generators satisfy a quadratic characteristic identity on the top even module V 0 (Λ).

Classes of atypical modules for gl 2 (n/1)
The above results are complete but rather implicit. We have not given a s (or equivalents such as a rs at higher levels, §A.3) explicitly as polynomials in the components of the highest weight vector Λ, as the expressions are tedious and non-transparent in this form. We finish this section with some examples of atypical modules, for the family of Kac modules with gl(n) highest weight Λ = (µ, µ, · · · , µ; ν, ν, · · · , ν) ≡ (µ r , ν n−r ) in Cartesian components, where µ − ν ∈ Z + and r = 1, 2, · · · , n − 1. An alternative parametrisation is µ = w + k, ν = w where k ∈ Z + , which makes it clear that the gl(n) modules V 0 (Λ) that we consider are rational (tensor density) representations, equivalent to the direct product of a finite-dimensional tensor representation with highest weight (k, k, · · · k, 0, · · · , 0) ≡ (k r , 0 n−r ) (represented by a rectangular r×k Young diagram), with the one dimensional character det w . Define µ := µ + n − r, ν := ν, for r = 1, 2, · · · , n − 1.
The characteristic identity in this case is quadratic, so that there are generically only two projections at each level, only two roots to deal with, and simple expressions can be given for the a s , and related polynomials. The case r = n (or equivalently r = 0) requires a separate discussion as the characteristic identity becomes linear (with only one root). Firstly we collect some relevant data about the characteristic identity and invariants. Lemma 4.18: Characteristic identity and Casimir eigenvalues for Λ = (µ r , ν n−r ): Consider the class of Kac modules of gl 2 (n/1) with Λ as above. In terms of the quantities µ, ν defined above, we have the following data for operators acting on states v ∈ V 0 (Λ ′ ): (i) The Gel'fand array satisfies the quadratic matrix identity (ii) The linear and quadratic Casimir invariants are (iii) The adjoint operator A takes the form (iv) The operator B takes the form B kℓ ij = (r−1)µ + (n−r−1)ν −r(n−r)−1 (Eδ) kℓ ij − rµ+(n−r)ν −r(n−r)−3 (δE) kℓ ij + + c−n+1+ 1 2 (n−1)(µ−1)(ν −1)+ + 1 2 rµ+(n−r)ν −r(n−r)−n (r−1)µ+(n−r−1)ν −r(n−r)−1 (δδ) kℓ ij , where we define the antisymmetric tensor combination (XY ) kℓ Proof : (i),(ii) : µ, ν are by definition the roots of the quadratic identity satisfied by E. C ′ 1 is simply the sum n r=1 Λ ′ r , while C ′ 2 follows by taking the matrix trace of the characteristic identity (or directly from the standard expression for the Casimir invariant in terms of the highest weight labels).
Proof : (i) This case cannot be obtained from case (ii) as the characteristic identity is linear -that is, Making these substitutions directly into A i j given in (27) leads to (33).
(ii) From Theorem 4.17 and (24), V (µ r , ν n−r ) ∼ = V 0 (µ r , ν n−r ) entails the simultaneous satisfaction of both level one atypicality conditions, namely a r (µ, ν) = a s (µ, ν) = 0. If µ = ν the situation is equivalent to the vanishing of the adjoint operator A i j on V 0 (Λ ′ ) in the induced module construction. From (27), the numerical coefficients of both E i j and δ i j must vanish. The second of (34) arises from the vanishing of the δ i j coefficient by use of the first condition, and rearranging (assuming n ≥ 3).
We give an alternative demonstration of the existence of zero-step modules in a slightly more general context. Although derived from the vanishing of Q i S ⊗ v ≡ S j ⊗ A j i v, that is, the vanishing of the operator A j i acting on the states v belonging to V 0 (Λ ′ ), the outcome is of course that the atypical Kac module V (Λ) is in fact isomorphic to V 0 (Λ) itself, with the odd generators represented as identically zero. This can only be the case if the anticommutator . We can confirm this in a general way as follows. Let the quadratic characteristic identity be Here s ′ , p ′ are the sum and product of the roots, respectively. Identifying Q ′i j and A i j we have . On the other hand, on V 0 (Λ) itself we have C 1 = C ′ 1 + n, C 2 = C ′ 2 + 2C ′ 1 + n, the characteristic identity entails the sum and product s ′ + 2, p ′ + s ′ + 1, respectively. With these interrelations it is easily shown that, if the identification between Q ′ and A (acting on V 0 (Λ ′ )) is made as above, then the appropriately adjusted quadratic identity Q, and the anticommutator bracket C (as given in (17), are indeed also identical on V 0 (Λ). Thus at least for the zero-step atypical case, the constraints are available directly from the structure relations without the need for the induced module construction.
(ii) From the discussion following Theorem 4.17, one-step atypical representations require the vanishing of all tensor projections of the states Q i Q m S ⊗ v, that is, we require where in the first line we have used SQ m ⊗ v = S ⊗ Q m v ≡ 0, and in the second line for some even elements C im jk . Thus we require simply B kℓ ij A j m v = 0. Using the explicit expressions for the forms of the tensors B and A given earlier, and exploiting the quadratic characteristic identity for E, this condition can be reduced to a set of numerical constraints on certain operator coefficients as follows. Recall the schematic notation introduced previously, n 3 4 5 6 7 7 7 8 9 9 10 10 r 2 2 2 2 2 3 4 2 2 5 2 4 k 1 2 3 4 5 2 1 6 7 1 8 2 Table 1: Zero-step atypicals: gl 2 (n/1) Kac modules V (k r , 0 n−r ) for n = 3, 4, · · · , 10 Expanding and using the quadratic identity leads to Taking the vanishing of the numerical coefficients in turn, for the (EE) term we have either s ′ = a 1 or s ′ = b 1 . Setting s ′ = a 1 means b 1 = −1 and so the second term requires a 0 −p ′ = 0, and we trivially recover the zero-step case. If s ′ = b 1 , the (δE) coefficient becomes a 0 −b 0 = s ′ or, C ′ 1 − 1 = C ′ 1 + n − 3 which leads to the degenerate Lie superalgebra case n = 2.
As a final illustration we give in Table 1 concrete instances of zero-step atypical modules for various low-dimensional cases. For these purposes assume w = 0, that is, we take µ = k and ν = 0 for true tensor representations. Further, we can regard the second of (34) as the defining condition for the central charge c given the choice of other labels, and so we present solutions to (34a), namely (r − 1)(k + n − r) = r(n − r), r = 2, · · · , n − 1 (see Table 1). More generally it can be seen from both parts of (34) that simultaneous solutions may exist for µ and ν for arbitrary fixed parameters n, r, c, but that the requirement µ − ν ∈ Z + is a severe restriction.

Discussion and Conclusions
In this paper we have developed the structure and representation theory of a distinguished class of Z 2 -graded polynomial algebras of degree 2, which we have termed quadratic Lie superalgebras. Their defining relations and PBW property have been formulated and proven along the same lines as for Lie superalgebras (but using the property of Koszulness in the quadratic case). The 'type I' class includes several examples studied earlier inspired by various physical contexts. Some general observations on the structure of the structure constants and the generalized Jacobi identities have been made, although a classification is not feasible with present methods. Given the PBW Theorem for the enveloping algebra, the balanced type I case admits a formulation of induced modules following Kac and especially the modified construction of Gould. A specific case, the quadratic modification of the simple Lie superalgebra sl(n/1), denoted here gl 2 (n/1), has been considered in detail, and tensor projection techniques used to expose the structure of level 1 atypicality conditions. An outline of the application of these methods has been provided for the analysis of induced modules at level ≥ 2, and a conjecture given for atypicality conditions at level 2.
A new feature not present for the Lie superalgebra case but which distinguishes the quadratic (and presumably all polynomial cases) is the existence of truncated atypicals, where all states of the Kac module at a certain level can vanish due to polynomial identities satisfied by the even generators at a given level. These have been investigated at level one and two for a class of Kac modules of gl 2 (n/1) where the even Gel'fand generators satisfy a quadratic polynomial identity; the corresponding truncated modules are denoted as zero-step and one-step modules, respectively. It has been found that zero-step modules are a generic feature of this class, whereas there are in fact no one-step modules. This is in contrast to gl(n/1). In that case, appropriate graded tensor products of the defining representation lead to atypical modules with a simple combinatorial structure, for example with decomposition into gl(n) of the form kΛ 1 + (k − 1)Λ 1 in highest weight notation (or {k} + {k − 1} in partition notation).
Let us recall the context of constructions of quadratic superalgebras arising from the study of gauge invariant fields in Hamiltonian lattice QCD [14,13] and the relevance of atypicality conditions therein. Examples of composite operators arising in those studies were the baryonic type fields ψ ijk = ε abc c ai c bj c ck and ψ ijk = ε abc c † ai c † bj c † ck , where i, j, k = 1, 2, · · · , n are spatial lattice, spin and flavour labels, and a, b, c = 1, 2, 3 are colour labels, for fermionic creation and annihilation operators c, c † . On the other hand, a related construction of gauge invariant fermions a, a † led to the colourless composites Q ijk = a i a j a k , Q ijk = a † i a † j a † k . Clearly, the objects ψ ijk , ψ ijk and Q ijk , Q ijk have anticommutator brackets quadratic in the corresponding even gl(n) generators E i j = c † ai c aj and a † i a j , respectively, yielding instances of the class I type quadratic superalgebras with odd generators in the non-fundamental representations with highest weights 3Λ 1 and Λ 3 , respectively -that is, the quadratic Lie superagebras gl 2 (n/3Λ 1 +3Λ * 1 ) and gl 2 (n/Λ 3 +Λ * 3 ) (see also §4, introduction).
In conclusion we note that a considerable literature exists on the subject of (homogeneous) algebras and superalgebras which could be drawn upon to elucidate some of the representationtheoretical questions encountered in this study. We cite in particular [11,12] for homological aspects, and [6] for constructions related to free differential algebras (see also [19]).
If either |a| = 0 or |b| = 0 then it is clear from the inductive assumptions that the above are well defined since d ab cd = 0. For the case |a| = |b| = 1 then (c, d) < (a, b) implies that the quadratic term is well defined by inductive assumption also. We need only show that for all a, b and N . Due to the (graded) antisymmetry of (A-2) we may assume a > b and a ≥ b for the case |a| = |b| = 1. By induction on ℓ(N ) we may assume that holds for all w a , w b for ℓ(L) < ℓ(N ) (since (A-2) is true for N = ∅). If |a| = 0 or |b| = 0 then d cd ab w b w a = 0 and (A-2) is true. If |a| = |b| = 1 then (A-2) is true for b < N , this follows from the second case of the inductive definition (A-1), and we need only consider the case b ≥ N . Let N = cL, thus we have a ≥ b ≥ c and (A-2) becomes (a, b, c) y a y b y c z L + y b y a y c z L = d ef ab x e x f y c z L . We permute the a, b, c cyclically to obtain the equations (b, c, a) y b y c y a z L + y c y b y a z L = d ef bc x e x f y a z L , (c, a, b) y c y a y b z L + y a y c y b z L = d ef ca x e x f y b z L .
By inspection of the ordering of cases in (A-1) we see that equations (b, c, a) and (c, a, b) are true under the inductive assumption; and similarly, by (A-3), the right hand side of (a, b, c) becomes d ef ab x e x f y c z L = y c d ef ab x e x f z L = y c (y a y b + y b y a )z L = (y c y a y b + y c y b y a )z L .
We apply this to the right hand sides of equations (b, c, a) and (c, a, b), and it is easily checked that (a, b, c) + (b, c, a) + (c, a, b) becomes a trivial identity; hence (a, b, c) is true.
only if : Let us assume that dĉd ab is non zero for some |a| = |b| = 1, |ĉ| = |d| = 0 where at least one ofĉ ord is greater than a or b. It follows from y a y b = −y b y a + d cd ab x c x d that both (a, b) and (b, a) belong to S, and we shall show that this leads to a linear dependence between the PBW monomials generated by S. We have 0 = (y a y a )y b − y a (y a y b ) and it remains to prove that right hand side is a non trivial sum of PBW monomials. It is clear that (a, b, a) ∈ S 3 . The terms of the form y a x c x d and x c x d y b may be reordered in any way due to the commutativity of any even-even or odd-even pair, so one needs only to show that some permutation of y a x c x d is a PBW monomial for all |a| = 1, |c| = |d| = 0. Since x c x d = x d x c and d cd ab is c-d-symmetric for all a, b it follows that (i, j) ∈ S whenever i ≤ j and |i| = |j| = 0. Similarly, the only relations involving pairs of odd and even indices are y a x c = x c y a , ∀a, c, hence a < c ⇒ (a, c) ∈ S and c < a ⇒ (c, a) ∈ S. Therefore at least one permutation of (a, c, d) and (c, d, b) will always belong to S 3 . Since none of these are equal to (a, b, a) the right hand side of (A-4) is a non trivial sum of PBW monomials.
Lemma A.22: : Let A be a PBW-algebra. The ordered monomials generated by any total ordering of the basis elements of X are a basis for A.