Quantum determinants and quasideterminants

Abstract

The notion of a quasideterminant and a quasiminor of a matrix A=(a_{ij}) with not necessarily commuting entries was introduced recently by I.Gelfand and the second author. The ordinary determinant of a matrix with commuting entries can be written (in many ways) as a product of quasiminors. Furthermore, it was noticed by a number of authors that such well-known noncommutative determinants as the Berezinian, the Capelli determinant, the quantum determinant of the generating matrix of the quantum group U_h(gl_n) and the Yangian Y(gl_n) can be expressed as products of commuting quasiminors. The aim of this paper is to extend these results to a rather general class of Hopf algebras given by the Faddeev-Reshetikhin-Takhtajan type relations -- the twisted quantum groups. Such quantum groups arise when Belavin-Drinfeld classical r-matrices are quantized. Our main result is that the quantum determinant of the generating matrix of a twisted quantum group equals the product of commuting quasiminors of this matrix.