arXiv subjects

Matej Brešar

Publications and source records attributed to Matej Brešar.

26 records · Page 2Linked to original sources

arXiv2019

Centrally Stable Algebras

We define an algebra $A$ to be centrally stable if, for every epimorhism $φ$ from $A$ to another algebra $B$, the center $Z(B)$ of $B$ is equal to $φ(Z(A))$, the image of the center of $A$. After providing some examples and basic observations, we consider in somewhat greater detail central stability in tensor products of algebras, and finally establish our main result which states that a finite-dimensional unital algebra $A$ over a perfect field $F$ is centrally stable if and only if $A$ is isomorphic to a direct product of algebras of the form $C_i\otimes_{F_i}A_i$, where $F_i$ is a field extension of $F$, $C_i$ is a commutative $F_i$-algebra, and $A_i$ is a central simple $F_i$-algebra.

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arXiv2018

Biderivations and commuting linear maps on Lie algebras

Let $L$ be a Lie algebra over a field of characteristic different from $2$. If $L$ is perfect and centerless, then every skew-symmetric biderivation $δ:L\times L\to L$ is of the form $δ(x,y)=γ([x,y])$ for all $x,y\in L$, where $γ\in{\rm Cent}(L)$, the centroid of $L$. Under a milder assumption that $[c,[L,L]]=\{0\}$ implies $c=0$, every commuting linear map from $L$ to $L$ lies in ${\rm Cent}(L)$. These two results are special cases of our main theorems which concern biderivations and commuting linear maps having their ranges in an $L$-module. We provide a variety of examples, some of them showing the necessity of our assumptions and some of them showing that our results cover several results from the literature.

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arXiv2017

Skolem-Noether algebras

An algebra $S$ is called a Skolem-Noether algebra (SN algebra for short) if for every central simple algebra $R$, every homomorphism $R\to R\otimes S$ extends to an inner automorphism of $R\otimes S$. One of the important properties of such an algebra is that each automorphism of a matrix algebra over $S$ is the composition of an inner automorphism with an automorphism of $S$. The bulk of the paper is devoted to finding properties and examples of SN algebras. The classical Skolem-Noether theorem implies that every central simple algebra is SN. In this article it is shown that actually so is every semilocal, and hence every finite-dimensional algebra. Not every domain is SN, but, for instance, unique factorization domains, polynomial algebras and free algebras are. Further, an algebra $S$ is SN if and only if the power series algebra $S[[ξ]]$ is SN.

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arXiv2014

Functional identities on matrix algebras

Complete solutions of functional identities $\sum_{k\in K}F_k(\bar{x}_m^k)x_k = \sum_{l\in L}x_lG_l(\bar{x}_m^l)$ on the matrix algebra $M_n(\mathbb{F})$ are given. The nonstandard parts of these solutions turn out to follow from the Cayley-Hamilton identity.

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arXiv2014

Quasi-identities on matrices and the Cayley-Hamilton polynomial

We consider certain functional identities on the matrix algebra $M_n$ that are defined similarly as the trace identities, except that the "coefficients" are arbitrary polynomials, not necessarily those expressible by the traces. The main issue is the question of whether such an identity is a consequence of the Cayley-Hamilton identity. We show that the answer is affirmative in several special cases, and, moreover, for every such an identity $P$ and every central polynomial $c$ with zero constant term there exists $m\in\mathbb{N}$ such that the affirmative answer holds for $c^mP$. In general, however, the answer is negative. We prove that there exist antisymmetric identities that do not follow from the Cayley-Hamilton identity, and give a complete description of a certain family of such identities.

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arXiv2013

Functional identities of one variable

Let $A$ be a centrally closed prime algebra over a characteristic 0 field $k$, and let $q:A\to A$ be the trace of a $d$-linear map (i.e., $q(x)=M(x,...,x)$ where $M:A^d\to A$ is a $d$-linear map). If $[q(x),x]=0$ for every $x\in A$, then $q$ is of the form $q(x) =\sum_{i=0}^{d} μ_i(x)x^i$ where each $μ_i$ is the trace of a $(d-i)$-linear map from $A$ into $k$. For infinite dimensional algebras and algebras of dimension $>d^2$ this was proved by Lee, Lin, Wang, and Wong in 1997. In this paper we cover the remaining case where the dimension is $ \le d^2$. Using this result we are able to handle general functional identities of one variable on $A$; more specifically, we describe the traces of $d$-linear maps $q_i:A\to A$ that satisfy $\sum_{i=0}^m x^i q_i(x)x^{m-i}\in k$ for every $x\in A$.

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arXiv2012

Group gradings on finitary simple Lie algebras

We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.

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arXiv2012

Central polynomials for matrices over finite fields

Let $c(x_1,...,x_d)$ be a multihomogeneous central polynomial for the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of positive characteristic $p$. We show that there exists a multihomogeneous polynomial $c_0(x_1,...,x_d)$ of the same degree and with coefficients in the prime field $F_p$ which is central for the algebra $M_n(F)$ for any (possibly finite) field $F$ of characteristic $p$. The proof is elementary and uses standard combinatorial techniques only.

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