The next meeting of the Algebra and Logic Seminar will be held on March 128, 2025 (Friday) at 1:00 pm (UTC+2) in Room 578 of IMI-BAS and online via Zoom.
A talk on:
Graded algebras that are the sum of two homogeneous subalgebras
will be delivered by
Plamen Koshlukov (State University of Campinas, Brazil).
Abstract. Let A be an algebra over a field F, graded by a group G, and let B and C be two homogeneous subalgebras of A such that A=B+C. We study the following problem: If B and C satisfy graded identities, does the same also hold for A?
The analogous problem for algebras without any grading was proposed in 1994 by Beidar and Mikhalev; in implicit form it appeared in a paper by O. Kegel, in 1963. Several particular cases were considered in a series of papers by various authors. In 2016, Kępczyk gave an affirmative answer to this problem (without grading).
We show that if B and C satisfy graded identities, and also B is a (one-sided) ideal of A then A=B+C also satisfies graded identities. We also study the situation where A satisfies specific graded semi-identities. In this case, if C satisfies some graded identity in neutral variables, we show that A satisfies graded identities. We also find upper bounds for the degrees of such identities. Here we use methods that go back to the classical Regev theorem on the growth of the codimensions of an associative algebra.
Finally we exhibit an example that shows that the graded version of the Kępczyk theorem is no longer valid.
This is a joint work with P. S. Fagundes.
Zoom link for the seminar:
Everybody is invited.
Algebra and Logic Department, IMI – BAS
http://www.math.bas.bg/algebra/seminarAiL/
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