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Pasha Zusmanovich

Publications and source records attributed to Pasha Zusmanovich.

At least 19 recordsLinked to original sources
arXiv2024

A $δ$-first Whitehead Lemma for Jordan algebras

We compute $δ$-derivations of simple Jordan algebras with values in irreducible bimodules. They turn out to be either ordinary derivations ($δ= 1$), or scalar multiples of the identity map ($δ= \frac 12$). This can be considered as a generalization of the "First Whitehead Lemma" for Jordan algebras which claims that all such ordinary derivations are inner. The proof amounts to simple calculations in matrix algebras, or, in the case of Jordan algebras of a symmetric bilinear form, to more elaborated calculations in Clifford algebras.

math.RA↗
arXiv2023

A variant of Baer's theorem

We provide a variant of Baer's theorem about isomorphism of endomorphism rings of vector spaces over division rings, where the full endomorphism rings are replaced by some subrings of finitary maps.

math.RA↗
arXiv2022

A $δ$-first Whitehead Lemma

We prove that $δ$-derivations of a simple finite-dimensional Lie algebra over a field of characteristic zero, with values in a finite-dimensional module, are either inner derivations, or, in the case of adjoint module, multiplications by a scalar, or some exceptional cases related to $\mathfrak{sl}(2)$. This can be viewed as an extension of the classical first Whitehead Lemma.

math.RA↗
arXiv2022

On regular Lie algebras

We study so called regular Lie algebras, i.e. Lie algebras in which each nonzero element is regular. We make a connection with an open problem whether any element of reduced trace zero in a simple associative algebra is a commutator.

math.RA↗
arXiv2022

On contact brackets on the tensor product

We study the behavior of contact brackets on the tensor product of two algebras, in particular, address the question of Martínez and Zelmanov about extension of a contact bracket on the tensor product from the brackets on the factors.

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arXiv2022

When Hom-Lie structures form a Jordan algebra

We are concerned with the question when Hom-Lie structures on a Lie algebra are closed with respect to the Jordan product. Somewhat unexpectedly, this leads us to certain questions connected with the Yang-Baxter equation, and with decomposition of a Lie algebra into the sum of subalgebras with given properties.

math.RA↗
arXiv2019

On $δ$-derivations of Lie algebras and superalgebras

We study $δ$-derivations -- a construction simultaneously generalizing derivations and centroid. First, we compute $δ$-derivations of current Lie algebras and of modular Zassenhaus algebra. This enables us to provide examples of Lie algebras having 1/2-derivations which are divisors of zero, thus answering negatively a question of Filippov. Second, we note that $δ$-derivations allow, in some circumstances, to construct examples of non-semigroup gradings of Lie algebras, in addition to the recent ones discovered by Elduque. Third, we note that utilizing the construction of the Grassmann envelope allows to obtain results about $δ$-(super)derivations of Lie superalgebras from the corresponding results about Lie algebras. In this way, we prove that prime Lie superalgebras do not possess nontrivial $δ$-(super)derivations, generalizing the recent result of Kaygorodov.

math.RA↗
arXiv2019

Commutative post-Lie algebra structures on Kac--Moody algebras

We determine commutative post-Lie algebra structures on some infinite-dimensional Lie algebras. We show that all commutative post-Lie algebra structures on loop algebras are trivial. This extends the results for finite-dimensional perfect Lie algebras. Furthermore we show that all commutative post-Lie algebra structures on affine Kac--Moody Lie algebras are "almost trivial".

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arXiv2019

Ado theorem for nilpotent Hom-Lie algebras

We prove an analog of the Ado theorem - the existence of a finite-dimensional faithful representation - for a certain kind of finite-dimensional nilpotent Hom-Lie algebras.

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arXiv2018

Yet another proof of the Ado theorem

We give a simple proof of the Birkhoff theorem about existence of a faithful representation for any finite-dimensional nilpotent Lie algebra of characteristic zero.

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