СЕМИНАР „АЛГЕБРА И ЛОГИКА”

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Доклад на тема

LOCALLY NILPOTENT LINEAR DERIVATIONS OF FREE METABELIAN ASSOCIATIVE ALGEBRAS

ще изнесе Dr SEHMUS FINDIK (Cukurova University, Adana, Turkey).
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Abstract:
This is a joint work with Rumen Dangovski and Vesselin Drensky.

A nonzero locally nilpotent linear derivation of the polynomial algebra in d variables over a field K of characteristic 0 is called a Weitzenboeck derivation. The classical theorem of Weitzenboeck states that the algebra of constants of the derivation is finitely generated. Similarly one may consider the algebra of constants of a locally nilpotent linear derivation acting on a d-generated algebra F which is relatively free in a variety of algebras over K. Now the algebra of constants in F is usually not finitely generated.

In the case of associative algebras there is a dichotomy. If the variety of algebras satisfies a polynomial identity which does not hold for the algebra of 2×2 upper triangular matrices, then the algebra of constants in F is finitely generated (Drensky, 2004). Otherwise, if the derivation is not zero, then the algebra of constants is not finitely generated (Drensky and Gupta, 2005). From this point of view the free associative metabelian algebra F(M) is crucial for the investigation.

We show that the vector space of the constants in the commutator ideal F(M)′ is a finitely generated module of the constants in the polynomial algebra in 2d variables. For small d, we calculate the Hilbert series of the constants in F(M)′ and find the generators of the module of the constants in the polynomial algebra in 2d variables.