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Minimum deformations of commutative algebra and linear group $GL(n)$
B. M. Zupnik
Abstract:
In the algebra of formal series
$M_q(x^i)$, the relations of generalized commutativity that preserve the tensor
$I_q$ grading and depend on parameters
$q(i,k)$ are considered. A norm of the differential calculus on
$M_q$ consistent with the
$I_q$ grading is chosen. A new construction of a symmetrized tensor product of algebras of the type
$M_q(x^i)$ and a corresponding definition of the minimally deformed linear group
$QGL(n)$ and Lie algebra
$qgl(n)$ are proposed. A study is made of the connection of
$QGL(n)$ and
$qgl(n)$ with the special matrix algebra
$\operatorname {Mat}(n,Q)$, which consists of matrices with noncommuting elements. The deformed determinant in the algebra
$\operatorname {Mat}(n,Q)$ is defined. The exponential mapping in the algebra
$\operatorname {Mat}(n,Q)$ is considered on the basis of the Campbell–Hausdorff formula.
Received: 07.04.1992
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