QUANTUM SUPERGROUPS OF GL(n|m) TYPE: DIFFERENTIAL FORMS, KOSZUL COMPLEXES AND BEREZINIANS

We introduce and study the Koszul complex for a Hecke R-matrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke R-matrix. Their behaviour with respect to Hecke sum of R-matrices is studied. Given a Hecke R-matrix in n-dimensional vector space, we construct a Hecke R-matrix in 2n-dimensional vector space commuting with a differential. The notion of a quantum differential supergroup is derived. Its algebra of functions is a differential coquasitriangular Hopf algebra, having the usual algebra of differential forms as a quotient. Examples of superdeterminants related to these algebras are calculated. Several remarks about Woronowicz’s theory are made. 0.1. Short description of the paper. 0.1.1. In this paper we will be concerned with differential Hopf algebras generated by sets of matrix elements. We start (Section 1) by giving a construction of such algebras, generalising the construction [31, 27] of a bialgebra generated by a single set of matrix elements (without differential) as a universally coacting bialgebra preserving several algebras generated by a set of coordinates. In our generalisation the data are morphisms in the category of graded differential complexes. 0.1.2. Given a Hecke R-matrix for a vector space V , we construct in this paper another Hecke R-matrix R for the space W = V ⊕ V equipped with the differential d = ( 0 1 0 0 ) and the grading σ : W → W , σ = ( 1 0 0 −1 ) . The matrix R is distinguished by the property R(d⊗ 1 + σ ⊗ d) = (d⊗ 1 + σ ⊗ d)R. 0.1.3. The algebra H of functions on the quantum supergroup constructed from R is a Z-graded differential coquasitriangular Hopf algebra (Section 2). In brief, it defines a differential quantum supergroup. A quotient of H is the Z>0-graded differential Hopf algebra Ω of differential forms defined via R in [25, 26, 30, 35]. The classical version (q = 1) of this construction is: take a vector space V , add to it another copy of it with the opposite parity and consider the general linear supergroup of the Z/2-graded space so obtained. Date: 26 May 1995 : February 21, 2008. 1991 Mathematics Subject Classification. Primary 16W30, 17B37, Secondary 17A70. Research was supported by EPSRC research grant GR/G 42976. 1 2 V. LYUBASHENKO AND A. SUDBERY 0.1.4. We introduce Koszul complexes for Hecke R-matrices in Section 3. They are Z>0 × Z>0-graded algebras with two differentials, D of degree (1, 1) and D ′ of degree (−1,−1). We calculate their anticommutator, called the Laplacian. The cohomology space of D is called the Berezinian. It generalizes the determinant, coinciding in the even case with the highest exterior power of V . The behaviour of Koszul complexes and Berezinians with respect to the Hecke sum [16] is described: we prove that the Berezinian of a Hecke sum is the tensor product of Berezinians. 0.1.5. The Berezinian is used to define the quantum superdeterminant in Section 4. We calculate the superdeterminant in several examples. In particular, in the algebra Ω of differential forms on the standard quantum GL(n|m) the superdeterminant equals 1. This confirms the idea that there are no central group-like elements in Ω, explaining why it has not been possible to construct a differential calculus on special linear groups with the same dimension as the classical case. 0.1.6. We make several remarks on Woronowicz’s theory [32] in Section 5. In particular, each first order differential calculus is extended to a differential Hopf algebra. 0.2. Notations and conventions. k denotes a field of characteristic 0. In this paper a Hopf algebra means a k-bialgebra with an invertible antipode. Associative comultiplication is denoted by ∆x = x(1) ⊗ x(2), the counit by ε, the antipode in a Hopf algebra by γ. If H is a Hopf algebra, H denotes the same coalgebra H with opposite multiplication, Hop denotes the same algebra H with the opposite comultiplication. When X is a graded vector space, â denotes the degree of a homogeneous element xa ∈ X. The braiding in a braided tensor category C [7] (e.g. in the category of representations of a quasitriangular Hopf algebra) is denoted by cX,Y : X ⊗ Y → Y ⊗ X, where X, Y ∈ Ob C. By definition, the maps 1V ⊗k−1 ⊗ cV,V ⊗ 1V ⊗n−k−1 : V ⊗n → V ⊗n obey the braid group relations. Given σ ∈ Sn, we denote by (c)σ and (c )σ the maps V ⊗n → V ⊗n coming from the liftings of σ to the elements of the braid group representing reduced expressions of σ. The first case is a word in the generators 1⊗ c⊗ 1, the second is a word in 1⊗ c ⊗ 1. We often use tangles to describe maps constructed from an R-matrix and pairings. In the conventions of [13] we denote c : X ⊗ Y → Y ⊗X by J