arXiv subjects

Plamen Koshlukov

Publications and source records attributed to Plamen Koshlukov.

23 records · Page 2Linked to original sources

arXiv2020

Automorphisms and superalgebra structures on the Grassmann algebra

Let $F$ be a field of characteristic zero and let $E$ be the Grassmann algebra of an infinite dimensional $F$-vector space $L$. In this paper we study the superalgebra structures (that is the $\mathbb{Z}_{2}$-gradings) that the algebra $E$ admits. By using the duality between superalgebras and automorphisms of order $2$ we prove that in many cases the $\mathbb{Z}_{2}$-graded polynomial identities for such structures coincide with the $\mathbb{Z}_{2}$-graded polynomial identities of the "typical" cases $E_{\infty}$, $E_{k^\ast}$ and $E_{k}$ where the vector space $L$ is homogeneous. Recall that these cases were completely described by Di Vincenzo and Da Silva in \cite{disil}. Moreover we exhibit a wide range of non-homogeneous $\mathbb{Z}_{2}$-gradings on $E$ that are $\mathbb{Z}_{2}$-isomorphic to $E_{\infty}$, $E_{k^\ast}$ and $E_{k}$. In particular we construct a $\mathbb{Z}_{2}$-grading on $E$ with only one homogeneous generator in $L$ which is $\mathbb{Z}_{2}$-isomorphic to the natural $\mathbb{Z}_{2}$-grading on $E$, here denoted by $E_{can}$.

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arXiv2020

$\mathbb{Z}$-graded polynomial identities of the Grassmann algebra

Let $F$ be an infinite field of characteristic different from 2, and let $E$ be the Grassmann algebra of an infinite dimensional $F$-vector space $L$. In this paper we study the $\mathbb{Z}$-graded polynomial identities of $E$ with respect to certain $\mathbb{Z}$-grading such that the vector space $L$ is homogeneous in the grading. More precisely, we construct three types of $\mathbb{Z}$-gradings on $E$, denoted by $E^{\infty}$, $E^{k^\ast}$ and $E^{k}$, and we give the explicit form of the corresponding $\mathbb{Z}$-graded polynomial identities. We show that the homogeneous superalgebras $E_{\infty}$, $E_{k^\ast}$ and $E_{k}$ studied in \cite{disil} can be obtained from $E^{\infty}$, $E^{k^\ast}$ and $E^{k}$ as quotient gradings. Moreover we exhibit several other types of homogeneous $\mathbb{Z}$-gradings on $E$, and describe their graded identities.

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arXiv2016

GK-dimension of the Lie algebra of generic $2\times 2$ matrices

Recently Machado and Koshlukov have computed the Gelfand-Kirillov dimension of the relatively free algebra $F_m=F_m(\text{var}(sl_2(K)))$ of rank $m$ in the variety of algebras generated by the three-dimensional simple Lie algebra $sl_2(K)$ over an infinite field $K$ of characteristic different from 2. They have shown that $\text{GKdim}(F_m)=3(m-1)$. The algebra $F_m$ is isomorphic to the Lie algebra generated by $m$ generic $2\times 2$ matrices. Now we give a new proof for $\text{GKdim}(F_m)$ using classical results of Procesi and Razmyslov combined with the observation that the commutator ideal of $F_m$ is a module of the center of the associative algebra generated by $m$ generic traceless $2\times 2$ matrices.

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arXiv2013

On the polynomial identities of the algebra $M_{11}(E)$

Verbally prime algebras are important in PI theory. They were described by Kemer over a field $K$ of characteristic zero: 0 and $K $ (the trivial ones), $M_n(K)$, $M_n(E)$, $M_{ab}(E)$. Here $K $ is the free associative algebra of infinite rank, with free generators $T$, $E$ denotes the infinite dimensional Grassmann algebra over $K$, $M_n(K)$ and $M_n(E)$ are the $n\times n$ matrices over $K$ and over $E$, respectively. The algebras $M_{ab}(E)$ are subalgebras of $M_{a+b}(E)$, see their definition below. The generic (also called relatively free) algebras of these algebras have been studied extensively. Procesi described the generic algebra of $M_n(K)$ and lots of its properties. Models for the generic algebras of $M_n(E)$ and $M_{ab}(E)$ are also known but their structure remains quite unclear. In this paper we study the generic algebra of $M_{11}(E)$ in two generators, over a field of characteristic 0. In an earlier paper we proved that its centre is a direct sum of the field and a nilpotent ideal (of the generic algebra), and we gave a detailed description of this centre. Those results were obtained assuming the base field infinite and of characteristic different from 2. In this paper we study the polynomial identities satisfied by this generic algebra. We exhibit a basis of its polynomial identities. It turns out that this algebra is PI equivalent to a 5-dimensional algebra of certain upper triangular matrices. The identities of the latter algebra have been studied; these were described by Gordienko. As an application of our results we describe the subvarieties of the variety of unitary algebras generated by the generic algebra in two generators of $M_{11}(E)$. Also we describe the polynomial identities in two variables of the algebra $M_{11}(E)$.

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arXiv2012

The centre of generic algebras of small PI algebras

Verbally prime algebras are important in PI theory. They are well known over a field $K$ of characteristic zero: 0 and $K $ (the trivial ones), $M_n(K)$, $M_n(E)$, $M_{ab}(E)$. Here $K $ is the free associative algebra with free generators $T$, $E$ is the infinite dimensional Grassmann algebra over $K$, $M_n(K)$ and $M_n(E)$ are the $n\times n$ matrices over $K$ and over $E$, respectively. Moreover $M_{ab}(E)$ are certain subalgebras of $M_{a+b}(E)$, defined below. The generic algebras of these algebras have been studied extensively. Procesi gave a very tight description of the generic algebra of $M_n(K)$. The situation is rather unclear for the remaining nontrivial verbally prime algebras. In this paper we study the centre of the generic algebra of $M_{11}(E)$ in two generators. We prove that this centre is a direct sum of the field and a nilpotent ideal (of the generic algebra). We describe the centre of this algebra. As a corollary we obtain that this centre contains nonscalar elements thus we answer a question posed by Berele.

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