**The International Center for Mathematical Sciences – Sofia (ICMS-Sofia)**

**presents**

## ICMS Seminar

9.04.2024, 14:00 Sofia time,

ICMS-Sofia, Room 403

#
**The Specht property for varieties of $Z_n$-graded Lie algebras**

**The Specht property for varieties of $Z_n$-graded Lie algebras**

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Daniela Martinez Correa

Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”

Daniela Martinez Correa

Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”

**Abstract:**

Let $UT_n(K)$ be the algebra of the $n\times n$ upper triangular matrices and denote $UT_n(K)^{(-)}$ the Lie algebra on the vector space of $UT_n(K)$ with respect to the usual bracket (commutator), over an infinite field $K$. In this talk, we give a positive answer to the Specht property for the ideal of the $\mathbb{Z}_n$-graded identities of $UT_n(K)^{(-)}$ with the canonical grading when the characteristic $p$ of $K$ is 0 or is larger than $n-1$.

Moreover, if $K$ is a field of positive characteristic $p$, we construct three varieties of $Z_{p+1}$-graded Lie algebras which do not have a finite basis of their graded identities and satisfy the graded identities which in the case of infinite field define the variety generated by $UT_{p+1}^{(-)}(K)$. The first variety contains the other two. The second one is locally finite. The third variety is generated by a finite dimensional algebra over an infinite field.

This is a joint work with Vesselin Drensky (IMI-BAS) and Plamen Koshlukov (Unicamp, Brazil).