Searcharxiv⌕ Search

arXiv subjects

Alexey Petukhov

Publications and source records attributed to Alexey Petukhov.

16 recordsLinked to original sources
arXiv2025

Countable separation property for associative algebras

For an associative algebra $A$ with a simple module $M$ with trivial endomorphisms and trivial annihilator we verify the countable separation property (CSP), i.e. we prove that there exists a list of nonzero elements $a_1, a_2,\ldots$ of $A$ such that every two-sided ideal of $A$ contains at least one such $a_i$. Based on this result we verify the countable separation property for a free associative algebra with finite or countable set of generators over any field. The countable separation property was studied before in the works of Dixmier and others but only in the context of Noetherian algebras (and a free associative algebra is very far from being Noetherian).

math.RA↗
arXiv2025

Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types

Let $\mathfrak g$ be a classical simple Lie algebra over an algebraically closed field $\mathbb F$ of characteristic zero or large enough, and let $\mathfrak n$ be a maximal nilpotent subalgebra of $\mathfrak g$. The main tool in representation theory of $\mathfrak n$ is the orbit method, which classifies primitive ideals in the universal enveloping algebra ${\rm U}(\mathfrak n)$ and unitary representations of the unipotent group $N=\exp(\mathfrak n)$ in terms of coadjoint orbits on the dual space $\mathfrak n^*$. In the paper, we describe explicitly coadjoint orbits of low dimension for $\mathfrak n$ as above. The answer is given in terms of subsets of positive roots. As a corollary, we provide a way to calculate the number of irreducible complex representations of dimensions $q$, $q^2$ and $q^3$ for a maximal unipotent subgroup $N(q)$ in a classical Chevalley group $G(q)$ over a finite field $\mathbb F_q$ with $q$ elements. It turned out that this number is a polynomial in $q-1$ with nonnegative integer coefficients, which agrees with Isaac's conjecture.

math.RT↗
arXiv2020

Spherical actions on isotropic flag varieties and related branching rules

Let $G$ be a symplectic or special orthogonal group, let $H$ be a connected reductive subgroup of $G$, and let $X$ be a flag variety of $G$. We classify all triples $(G,H,X)$ such that the natural action of $H$ on $X$ is spherical. For each of these triples, we determine the restrictions to $H$ of all irreducible representations of $G$ realized in spaces of sections of homogeneous line bundles on $X$.

math.AG↗
arXiv2020

The orbit method for locally nilpotent infinite-dimensional Lie algebras

Let $\mathfrak{n}$ be a locally nilpotent infinite-dimensional Lie algebra over $\mathbb{C}$. Let $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$ be its universal enveloping algebra and its symmetric algebra respectively. Consider the Jacobson topology on the primitive spectrum of $\mathrm{U}(\mathfrak{n})$ and the Poisson topology on the primitive Poisson spectrum of $\mathrm{S}(\mathfrak{n})$. We provide a homeomorphism between the corresponding topological spaces (on the level of points, it gives a bijection between the primitive ideals of $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$). We also show that all primitive ideals of $\mathrm{S}(\mathfrak{n})$ from an open set in a properly chosen topology are generated by their intersections with the Poisson center. Under the assumption that $\mathfrak{n}$ is a nil-Dynkin Lie algebra, we give two criteria for primitive ideals $I(λ)\subset\mathrm{S}(\mathfrak{n})$ and $J(λ)\subset\mathrm{U}(\mathfrak{n})$, $λ\in\mathfrak{n}^*$, to be nonzero. Most of these results generalize the known facts about primitive and Poisson spectrum for finite-dimensional nilpotent Lie algebras (but note that for a finite-dimensional nilpotent Lie algebra all primitive ideals $I(λ)$, $J(λ)$ are nonzero).

math.RT↗
arXiv2019

Branching rules related to spherical actions on flag varieties

Let $G$ be a connected semisimple algebraic group and let $H \subset G$ be a connected reductive subgroup. Given a flag variety $X$ of $G$, a result of Vinberg and Kimelfeld asserts that $H$ acts spherically on $X$ if and only if for every irreducible representation $R$ of $G$ realized in the space of sections of a homogeneous line bundle on $X$ the restriction of $R$ to $H$ is multiplicity free. In this case, the information on restrictions to $H$ of all such irreducible representations of $G$ is encoded in a monoid, which we call the restricted branching monoid. In this paper, we review the cases of spherical actions on flag varieties of simple groups for which the restricted branching monoids are known (this includes the case where $H$ is a Levi subgroup of $G$) and compute the restricted branching monoids for all spherical actions on flag varieties that correspond to triples $(G,H,X)$ satisfying one of the following two conditions: (1) $G$ is simple and $H$ is a symmetric subgroup of $G$; (2) $G = \mathrm{SL}_n$.

math.RT↗
arXiv2017

On annihilators of bounded $(\frak g, \frak k)$-modules

Let $\frak g$ be a semisimple Lie algebra and $\frak k\subset\frak g$ be a reductive subalgebra. We say that a $\frak g$-module $M$ is a bounded $(\frak g, \frak k)$-module if $M$ is a direct sum of simple finite-dimensional $\frak k$-modules and the multiplicities of all simple $\frak k$-modules in that direct sum are universally bounded. The goal of this article is to show that the "boundedness" property for a simple $(\frak g, \frak k)$-module $M$ is equivalent to a property of the associated variety of the annihilator of $M$ (this is the closure of a nilpotent coadjoint orbit inside $\frak g^*$) under the assumption that the main field is algebraically closed and of characteristic 0. In particular this implies that if $M_1, M_2$ are simple $(\frak g, \frak k)$-modules such that $M_1$ is bounded and the associated varieties of the annihilators of $M_1$ and $M_2$ coincide then $M_2$ is also bounded. This statement is a geometric analogue of a purely algebraic fact due to I. Penkov and V. Serganova and it was posed as a conjecture in my Ph.D. thesis.

math.RT↗
arXiv2017

Primitive ideals of $\operatorname{U}(\frak{sl}(\infty))$

We provide an explicit description of the primitive ideals of the enveloping algebra $\operatorname{U}(\frak{sl}(\infty))$ of the infinite-dimensional finitary Lie algebra $\frak{sl}(\infty)$ over an uncountable algebraically closed field of characteristic 0. Our main new result is that any primitive ideal of $\operatorname{U}(\frak{sl}(\infty))$ is integrable. A classification of integrable primitive ideals of $\operatorname{U}(\frak{sl}(\infty))$ has been known previously, and relies on the pioneering work of A. Zhilinskii.

math.RT↗
arXiv2016

Finite-dimensional representations of minimal nilpotent W-algebras and zigzag algebras

Let $\frak g$ be a simple finite-dimensional Lie algebra over an algebraically closed field $\mathbb F$ of characteristic 0. We denote by $\operatorname{U}(\frak g)$ the universal enveloping algebra of $\frak g$. To any nilpotent element $e\in \frak g$ one can attach an associative (and noncommutative as a general rule) algebra $\operatorname{U}({\frak g},~e)$ which is in a proper sense a "tensor factor" of $\operatorname{U}(\frak g)$. In this article we consider the case in which $\frak g$ is simple and $e$ belongs of the minimal nonzero nilpotent orbit of $\frak g$. Under these assumptions $\operatorname{U}({\frak g}, e)$ was described explicitly in terms of generators and relations. One can expect that the representation theory of $\operatorname{U}({\frak g}, e)$ would be very similar to the representation theory of $\operatorname{U}(\frak g)$. For example one can guess that the category of finite-dimensional $\operatorname{U}({\frak g}, e)$-modules is semisimple. The goal of this article is to show that this is the case if $\frak g$ is not simply-laced. We also show that, if $\frak g$ is simply-laced and is not of type $A_n$, then the regular block of finite-dimensional $\operatorname{U}({\frak g}, e)$-modules is equivalent to the category of finite-dimensional modules of a zigzag algebra.

math.RT↗
arXiv2016

On ideals in U$(\frak{sl}(\infty))$, U$(\frak o(\infty))$, U$(\frak{sp}(\infty))$

We provide a review of results on two-sided ideals in the enveloping algebra U$(\frak g(\infty))$ of a locally simple Lie algebra $\frak g(\infty)$. We pay special attention to the case when $\frak g(\infty)$ is one of the finitary Lie algebras $\frak{sl}(\infty), \frak o(\infty), \frak{sp}(\infty)$. The main results include a description of all integrable ideals in U$(\frak g(\infty))$, as well as a criterion for the annihilator of an arbitrary (not necessarily integrable) simple highest weight module to be nonzero. This criterion is new for $\frak g(\infty)=\frak o(\infty), \frak{sp}(\infty)$. All annihilators of simple highest weight modules are integrable ideals for $\frak g(\infty)=\frak{sl}(\infty), \frak o(\infty)$. Finally, we prove that the lattices of ideals in U$(\frak o(\infty))$ and U$(\frak{sp}(\infty))$ are isomorphic.

math.RT↗
arXiv2015

On Gelfand-Kirillov conjecture for some W-algebras

Consider the W-algebra $W$ attached to the smallest nilpotent orbit in a simple Lie algebra $\frak g$ over an algebraically closed field of characteristic 0. We show that if an analogue of the Gelfand-Kirillov conjecture holds for such a W-algebra then it holds for the universal enveloping algebra $\mathrm U(\frak g)$. This together with a result of A. Premet implies that the analogue of the Gelfand-Kirillov conjecture fails for some $W$-algebras attached to some nilpotent orbits in Lie algebras of types $B_n~(n\ge 3)$, $D_n~(n\ge 4)$, $E_6, E_7, E_8$, $F_4$.

math.RT↗
arXiv2014

Spherical actions on flag varieties

For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X.

math.AG↗
arXiv2012

On ideals in the enveloping algebra of a locally simple Lie algebra

We study (two-sided) ideals $I$ in the enveloping algebra $\U(\frak g_\infty)$ of an infinite-dimensional Lie algebra $\frak g_\infty$ obtained as the union (equivalently, direct limit) of an arbitrary chain of embeddings of simple finite-dimensional Lie algebras $\frak g_1\to\frak g_2\to...\to\frak g_n\to...$ with $\lim\limits_{n\to\infty}\dim\frak g_n=\infty$. Our main result is an explicit description of the zero-sets of the corresponding graded ideals $\gr I$. We use this description and results of A. Zhilinskii to prove Baranov's conjecture that, if $\frak g_\infty$ is not diagonal in the sense of A. Baranov and A. Zhilinskii, then $\U(\frak g_\infty)$ admits a single non-zero proper ideal: the augmentation ideal. Our study is based on a complete description of the radical Poisson ideals in ${\bf S}^\cdot(\frak g_\infty)$ and their zero-sets. We then discuss in detail integrable ideals of $\U(\frak g_\infty)$, i.e. ideals $I\subset\U (\frak g_\infty)$ for which $I\cap\U (\frak g_n)$ is an intersection of ideals of finite-codimension in $\U(\frak g_n)$ for any $n\ge 1$. We present a classification of prime integrable ideals based on work of A. Zhilinskii. For $\frak g_\infty\cong\frak{sl}_\infty, \frak{so}_\infty$, all zero-sets of radical Poisson ideals of ${\bf S}^\cdot(\frak g_\infty)$ arise from prime integrable ideals of $\U(\frak g_\infty)$. For $\frak g_\infty\cong\frak{sp}_\infty$ only "half" of the zero-sets of Poisson ideals ${\bf S}^\cdot(\frak g_\infty)$ arise from integrable ideals of $\U(\frak g_\infty)$.

math.AG↗
arXiv2011

A geometric approach to (g, k)-modules of finite type

Let $g$ be a semisimple Lie algebra over $\mathbb C$ and $k$ be a reductive in $g$ subalgebra. We say that a simple $g$-module $M$ is a $(g; k)$-module if as a $k$-module $M$ is a direct sum of finite-dimensional $k$-modules. We say that a simple $(g; k)$-module $M$ is of finite type if all $k$-isotypic components of $M$ are finite-dimensional. To a simple $g$-module $M$ one assigns interesting invariants V$(M)$, $\EuScript V(M)$ and L$(M)$ reflecting the 'directions of growth of M'. In this work we prove that, for a given pair $(g; k)$, the set of possible such invariants is finite. Let $K$ be a reductive Lie group with Lie algebra $k$. We say that a $K$-variety $X$ is $K$-spherical if $X$ has an open orbit of a Borel subgroup of $K$. Let $W$ be a finite-dimensional $K$-module. The set of flags ($W_1,..., W_s)$ of $W$ with fixed dimensions $(n_1;...; ns)$ is a homogeneous space of the group GL(W). We call such a variety partial $W$-flag variety. In this work we classify all $K$-spherical partial $W$-flag varieties. We say that a simple $(g; k)$-module is bounded if there exists constant C$_M$ such that, for any simple $k$-module $E$, the isotypic component of $E$ in $M$ is a direct sum of not more than C$_M$-copies of $E$. To any simple sl$(W)$-module one assigns a partial $W$-flag variety. In this thesis we prove that a simple (sl$(W); k$)-module is bounded if and only if the corresponding partial $W$-flag variety is $K$-spherical. Moreover, we prove that the pair (sl$(W); k$) admits an infin? ite-dimensional simple bounded module if and only if P$(W)$ is a $K$-spherical variety. For four particular case we say more about category of bounded modules and the set of simple bounded modules.

math.RT↗
arXiv2011

Bounded reductive subalgebras of sl(n)

Let $\mathfrak g$ be a semisimple Lie algebra and $\mathfrak k\subset\mathfrak g$ be a reductive in $\mathfrak g$ subalgebra. A $(\mathfrak g, \mathfrak k)$-module is a $\mathfrak g$-module which after restriction to $\mathfrak k$ becomes a direct sum of finite-dimensional $\mathfrak k$-modules. I.Penkov and V.Serganova introduce definition of bounded $(\mathfrak g, \mathfrak k)$-modules for reductive subalgebras $\mathfrak k\subset\mathfrak g$, i.e. $(\mathfrak g, \mathfrak k)$-modules whose $\mathfrak k$-multiplicities are uniformly bounded. A question arising in this context is, given $\mathfrak g$, to describe all reductive in $\mathfrak g$ bounded subalgebras, i.e. reductive in $\mathfrak g$ subalgebras $\mathfrak k$ for which at least one infinite-dimensional bounded $(\mathfrak g, \mathfrak k)$-module exists. In the present paper we describe explicitly all reductive in $\mathfrak{sl}_n$ bounded subalgebras. Our technique is based on symplectic geometry and the notion of spherical variety. We also characterize the irreducible components of the associated varieties of simple bounded $(\mathfrak g, \mathfrak k)$-modules.

math.RT↗
arXiv2010

Geometric Description of Epimorphic Subgroups

Let $G$ be an affine algebraic group over an algrebraically closed field $\mathbb K$ of characteristic 0 and $H$ be a subgroup of $G$. The stabilizer of all the set of all vector-functions of $\mathbb K[G]^H$ with respect to the right action of $H$ is $\hat H$. $V^H=V^{\hat H}$ for a $G$-module $V$. The subgroup $H$ is called observable if $H=\hat H$ and epimorphic if $G=\hat H$. In this work I show that under some natural restrictions $H$ is observable if and only if some orbit of some group contains 0 in the closure and $H$ is epimorphic if and only the same orbit is closed.

math.GR↗