NEW ZEALAND JOURNAL OF MATHEMATICS Volume 27 (1998), 277-291 SYMMETRIC AND ANTISYMMETRIC TENSORS OVER FREE SUPERMODULES C. V i c t o r i a (Received May 1996) Abstract. A ^-graded algebra of ^-symmetric Z2-tensors associated with a free supermodule of finite rank is introduced as a suitable generalization of the usual symmetric and antisymmetric tensors. 1. Introduction The concepts of multilinearity, symmetric and antisymmetric tensor products in superalgebra are treated in [2], [3], [4], [5] and [6]. We introduce here the Z 2-graded algebra of Z 2 -symmetric Z2-tensors associated with a free supermodule of finite rank (Definition 3.1). For any (po,pi) 6 Z2, the supermodule of Z 2 -symmetric (Po,Pi)-tensors is proved to be free of finite rank and we give a basis for it. This basis has two descriptions, one in terms of tensor products (Theorem 3.6) and the other in terms of differential operators (Theorem 3.10). This first description needs ‘sign functions’ (Definition 3.4) which although in some cases coincide with the usual symmetric or trivial and antisymmetric representations of the group of permuta­ tions of po +p\ objects (Proposition 3.13), they are not, in general, representations of it (Proposition 3.12). As a consequence, the tensor product does not induce any product of Z2 -symmetric Z 2-tensors, as it does in the case of modules over commutative algebras. Thanks to the description of Z 2 -symmetric Z2-tensors in terms of certain differential operators (Theorem 3.10), however, their product may be defined as the composition of their corresponding differential operators (Defini­ tion 3.17). The usual Z-graded algebras of symmetric and antisymmetric tensors of free modules over commutative algebras are proved to be natural subalgebras of the Z2-graded algebra of Z 2 -symmetric Z2-tensors (Propositions 3.19 and 3.20). Then, although the usual symmetric and antisymmetric products have independent definitions, our constructions make them merge both as particular cases of the same composition law (i.e. composition of differential operators). The Z-graded algebra of symmetric tensors of free supermodules introduced in [2], [3] and [4], is in a natural way, a subalgebra of the Z 2-graded algebra of Z 2 -symmetric Z2-tensors (Propositions 3.21). Since the functoriality of all of our constructions with respect to the general linear supergroup is proved in Proposition 3.22, there is no loss of generality by stating the 1991 AMS Mathematics Subject Classification: 15A69, 15A75, 16S32, 16W50, 16W55, 17A70, 17C70. Key words and phrases: Multilinear algebras, tensor products, exterior algebra, grassmann al­ gebras, rings of differential operators, graded rings and modules, “super” (or “skew” ) structure, super algebras, super structures . 278 C. VICTORIA definitions and results over which is the model of free A-supermodules of rank (mo, m i) (see [4]). In relation with forthcoming articles (see [8]), let us say that the functoriality of our constructions allows us to consider the Z2-symmetric Z 2-tensors on super­ manifolds, as the analogues of differential forms (that is, antisymmetric Z-graded tensors) in manifolds. Once the exterior differential is suitably generalized to these Z2-forms, a Z 2-graded algebra cohomology is obtained. This cohomology ring the­ ory is proved to be functorial with respect to all morphisms of supermanifolds (in contrast with the cohomology group of [1] and [9]) and is different from the coho­ mology ring of the underlying manifold (in contrast with the cohomology defined in [2], [3] and [4]). The contents of this paper are part of the author’s Ph.D. dissertation [7]. 2. Previous Results 2.1. Generalities and Notation. Let S = S[o] + S[i] be a Z2-graded algebra or a Z2-graded module. A homo­ geneous element s G S is called even if s G S[o] and odd if s G S[ij. The value of the parity function Xs on a homogeneous element s G S, Xs(s), is defined to be zero if s is even and 1 if s is odd. As it is sometimes used in the literature, s denotes the parity, A^(s), of s G S. All superalgebras (that is, all Z 2-graded associative algebras with unit 1 G A) A are assumed to be supercommutative; that is x y — (—1 )xVyx, \/x,y homogeneous elements in A (see [4]). A Z2-graded module M on a superalgebra A is a supermodule if A^jMfj] C M[i+J-], i, j G Z2 = {0,1}. Supermodule isomorphisms are always assumed to be homogeneous of degree zero. Besides the parity function As associated with a superalgebra or with a super­ module, we will also need to consider the parity functions defined on Z*, resp. on N, as Jo if A > 0 fo i f F < m 0 X*'iA) = \lif A < 0 reSP' = \ l ifF>m„ where (mo, m i) G N2 has been fixed. In expressions labeled by superindices and subindices, like X £ , the first group of capital letters are used to denote generic superindices in Z* and the second one F,G , . to denote generic subindices in {1 ,... , mo + m i}. So, A,B,... denote A z* (A), X (B ) ,... and F,G,... denote V o,n »i)(^ )i ^(m0,mi)(Gr), .. , and so on. Moreover, the parity j of small latin letters j G N means \ Po,Pl){j) for the pair (po,pi) fixed in the text. We warn the reader about the fact that the Einstein convention about summation over repeated indices will be assumed throughout this work, unless stated otherwise. The parity change functor assigns to a supermodule M = M[0] + M[i] the super­ module IIM defined as follows: (1) (IIM)^] = M [i+i], Vi G {0,1} = Z2. (2) If x ,y are homogeneous elements in M and lire, IIy are the same elements but with reversed parity in IIM, then by definition, IIx + II y = II(x 4- y). (3) If a G A[i], IIx G IIM, then by definition, allx = (—l) aIIax. SYMMETRIC AND ANTISYMMETRIC TENSORS OVER FREE SUPERMODULES 279 For m = (m o,m i) G N2 and M an A-supermodule, M m is defined as M®m° © (IIM)®mi. An A-supermodule is free of rank m = (mo,mi) G N2 if it is isomorphic to Am. We refer to [4] for more details. 2.2. Symmetric and Antisymmetric Z-Graded Tensor Algebras. In this section we recall the definition of the Z-graded algebras of symmetric and antisymmetric tensors of a supermodule introduced in [2], [3] and [4] as a general­ ization to supermodules of the Z-graded algebra of symmetric and antisymmetric tensors of a module. The tensor product of two A-supermodules M and N is the A-supermodule defined as follows: as A-module, the tensor product of M and N is M <S>a N ; the Z 2 ~graduation is obtained by defining as homogeneous elements the product of homogeneous elements of M and N, with parity m <8> n — m + n; the product is defined as a(m ® n ) = (am) ® n = (—l)amm 0 (an). Similarly the tensor product of a number p G N of A-supermodules can be defined and one can check that the A-supermodules constitute a tensor category (see [4]). Let M be an A-supermodule. The A-supermodule ® n^0 M ® n, where M®° = A, has a natural structure of Z-graded algebra with the tensor product ® M ® b —> M ® (a+b\ This is the Z-graded tensor algebra of M. In this alge­ bra the ideal Is, resp. I a, is the ideal generated by mi ® m2 + (—l)^*^m2 <8> mi, Vm i,m 2 homogeneous elements in M, where (*) = 1 + m im 2, resp. (*) = m im 2. The Z-graded algebra of symmetric, resp. antisymmetric, tensors is defined as the quotient ( 0 n>o M ® n) /I , where I = Is, resp. I = 7a- They are denoted by SymM and A m respectively, and as usually Sym?,, resp. A?,, denotes the A-supermodule M ® P /Is n M®P, resp. M ® p /Ia n M®*>. When a transposition acts on a symmetric, resp. antisymmetric, Z-graded tensor one has mi <g>... (8) rrii <g>... (g) rrij <g>... <g> mp = (—l)^**^mi ® ® m^ ® ... <g> m* <g>... ® mp where (**) - rhifhj + (m* + resp. (**) = 1 + fhirhj + (fhi + ™j ) Y /i<k< j™ k- 2.3. The General Linear Supergroup. For a superalgebra A and m = (mo, m i) G N2, the general linear supergroup of rank m with coefficients in A is defined to be the group of all even automorphisms of A m [4]. An exhaustive description of the general linear supergroup in terms of matrices and coordenates can be found in [5] and [6]. It is easy to check the functoriality of the given definitions with respect to the general linear supergroup. 3. Z 2 —Symmetric Z2—Tensors 3.1. Definitions. Fix p — (po,Pi) G N2 and let © = { A i,... ,A Po+Pl} C Z* be a subset with A i , ... , Ap0 even superindices and with A Po+1,... , ^4Po+Pl odd ones. Recall from Section 2.1 that the parity A = Xz*{A) of A G Z* is to be 0, resp. 1, if A > 0, resp. A < 0. When the parity of A is 0, resp. 1, A is called an even, resp. odd, superindex. The elements of the canonical basis X\,... , X mo+mi of Am, m — (m o,m i) G N2, have parities X p — Am(F) = F. S(Am) is by definition the 280 C.