Quantum Determinants and Quasideterminants
نویسنده
چکیده
Introduction The notion of a quasideterminant and a quasiminor of a matrix A = (a ij) with not necessarily commuting entries was introduced in GR1-3]. The ordinary determinant of a matrix with commuting entries can be written (in many ways) as a product of quasiminors. Furthermore, it was noticed in GR1-3, KL, GKLLRT, Mo] 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 deened in Section 1.4. Such quantum groups arise when Belavin-Drinfeld classical r-matrices BD] 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. Acknowledgments We are indebted to Israel Gelfand for inspiring us to do
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