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

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.

14 pages

Cited by in corpus (2)

• Quantum Symmetries and Marginal Deformations
• Differential Calculus on $q$-Deformed Light-Cone