Irreducible representations of quantum linear groups of type [formula omitted

Abstract

A Hecke symmetry is an invertible operator defined on the tensor square of a finite dimensional vector space, which satisfied the (quantized) Yang-Baxter equation and the Hecke equation (x + 1)(x − q) = 0, see [4, 6] for more details. Given such a matrix one can construct a Hopf algebra which is “the function algebra” over an (algebraic) matrix quantum group. One is interested in the representations of this quantum group, that is the comodules over the Hopf algebra. It turns out that the dimension of the vector space on which the Hecke symmetry is defined does not play a significant role but rather a certain “inner” characteristics of the Hecke symmetry, the birank – a pair of non-negative integers. A Hecke symmetry is called even if its birank is of the form (r, 0) and odd if its birank is of the form (0, r). In these cases, the representation category of the corresponding quantum group is like the representation category of the matrix group GL(r), it is semisimple and generally does not depend on the Hecke symmetry but only on the birank. If the Hecke symmetry is neither even nor odd the representation category of the corresponding quantum group seems to be similar to the representation category of the super group GL(r|s). This problem has not been settled yet. The aim of this paper is to classify irreducible representation of the matrix quantum group corresponding to a Hecke symmetry of birank (2, 1). Such a quantum group is usually called matrix quantum group of type (1, 0). In this work we show that irreducible representations of a quantum group of type (1, 0) can be indexed by tuples (m,n, p) of integers with m ≥ n. We exhibit basic decomposition rules of the tensor product of these representations and compute their dimensions. Recall that representations of a quantum group are by definition comodules over the corresponding Hopf algebra (“of functions”). Using the Koszul komplex we construct for each tuple (m,n, p),m ≥ n a comodule and prove that they are simple. The difficulty here is that the comodule category is not semisimple. Our main technique is based on the theory of Hopf algebras with integral (co-Frobenius Hopf algebras). For such Hopf algebras there is a special class of simple comodules that split in any comodule. We show that our Hopf algebra is co-Frobenius and use a criterion for a comodule to be splitting in any comodule (which is equivalent to being injective and projective). (In Kac’s theory of representation of Lie superalgebras irreducible representations that split in any representations are called typical [9].)