engleski

The kernel of multidegree operator on generic subspaces of algebra B

In this presentation we consider a free unital associative complex algebra B equipped with a multiparametric q-differential structure given by linear operators that act as twisted derivations on B. The algebra B is naturally graded by total degree and more generally it has a finer decomposition into multigraded components BQ called weight subspaces. Of particular interest are generic weight subspaces corresponding to a set Q of the cardinality n. We define a multidegree operator @ on B and special we consider its restriction @Q to BQ. Our motivation is to determine the kernel of the operator @Q, where we will show that the operator @Q can be factorized in terms of simpler operators. In order to simplify this computation, we have used a twisted group algebra approach. Therefore, first we have introduced a twisted group algebra A(Sn) of the symmetric group Sn with coefficients in the polynomial ring Rn in n2 commuting variables Xab, and then we have used a natural representation of A(Sn) on the generic weight subspaces BQ of the algebra B. Then we have studied factorization of certain canonical defined elements in the algebra A(Sn). In this way, we have obtained the corresponding factorizations that we have used in the factorization of the operator @Q. Here we will also show that the elements in the kernel of the operator @Q can be expressed in terms of certain iterated q-commutators.

q-algebras; twisted derivations; twisted group algebra; representation; iterated \textbf{;;; \em q};;; -commutators

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