Minimal deformations of the commutative algebra and the linear group GL(n)

TL;DR

This paper introduces a new framework for minimally deforming the linear group and Lie algebra using generalized commutativity relations in quantum matrix algebras, connecting them with noncommutative matrix algebras.

Contribution

It proposes a novel construction of the symmetrized tensor product and defines the minimally deformed linear group QGL(n) and Lie algebra qgl(n), linking them to noncommutative matrix algebras.

Findings

Defined the deformed determinant in noncommutative matrix algebra.

Established the connection between QGL(n), qgl(n), and matrix algebra Mat(n,Q).

Presented exponential parametrization using Campbell-Hausdorff formula.

Abstract

We consider the relations of generalized commutativity in the algebra of formal series $ M_q (x^i ) $, which conserve a tensor $ I_q $-grading and depend on parameters $ q(i,k) $ . We choose the $ I_q $-preserving version of differential calculus on $ M_q$ . A new construction of the symmetrized tensor product for $ M_q $-type algebras and the corresponding definition of minimally deformed linear group $ QGL(n) $ and Lie algebra $ qgl(n) $ are proposed. We study the connection of $ QGL(n) $ and $ qgl(n) $ with the special matrix algebra $ \mbox{Mat} (n,Q) $ containing matrices with noncommutative elements. A definition of the deformed determinant in the algebra $ \mbox{Mat} (n,Q) $ is given. The exponential parametrization in the algebra $ \mbox{Mat} (n,Q) $ is considered on the basis of Campbell-Hausdorf formula.