Characteristic Relations for Quantum Matrices

Abstract

General algebraic properties of the algebras of vector fields over quantum linear groups $G{L}_q(N)$ and $S{L}_q(N)$ are studied. These quantum algebras appears to be quite similar to the classical matrix algebra. In particular, quantum analogues of the characteristic polynomial and characteristic identity are obtained for them. The $q$-analogues of the Newton relations connecting two different generating sets of central elements of these algebras (the determinant-like and the trace-like ones) are derived. This allows one to express the $q$-determinant of quantized vector fields in terms of their $q$-traces.

11 pages, latex, an important reference [16] added.

References in corpus (4)

• Algebraic Structures Related to Reflection Equations
• Differential Geometry on Linear Quantum Groups
• Bicovariant Quantum Algebras and Quantum Lie Algebras
• Deformed Traces and Covariant Quantum Algebras for Quantum Groups $G{L}_{qp}(2)$ and $G{L}_{qp}(1|1)$

Cited by in corpus (9)

• On quantum matrix algebras satisfying the Cayley-Hamilton-Newton identities
• Baxterized Solutions of Reflection Equation and Integrable Chain Models
• Hecke algebraic properties of dynamical R-matrices. Application to related quantum matrix algebras
• Generalized Cayley-Hamilton-Newton identities
• Baxterization of solutions to reflection equation with Hecke R-matrix
• Quantum matrix algebra for the SU(n) WZNW model
• Manin Matrices for Quadratic Algebras
• The field of quantum $GL(N,\mathbb{C})$ in the C${}^{\ast }$-algebraic setting
• Multiparameter quantum Cauchy-Binet formulas