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Ivan Penkov

Publications and source records attributed to Ivan Penkov.

At least 55 records · Page 3Linked to original sources
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 $\U(\frak g_\infty)$.

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arXiv2012

A Koszul category of representations of finitary Lie algebras

We find for each simple finitary Lie algebra $\mathfrak{g}$ a category $\mathbb{T}_\mathfrak{g}$ of integrable modules in which the tensor product of copies of the natural and conatural modules are injective. The objects in $\mathbb{T}_\mathfrak{g}$ can be defined as the finite length absolute weight modules, where by absolute weight module we mean a module which is a weight module for every splitting Cartan subalgebra of $\mathfrak{g}$. The category $\mathbb{T}_\mathfrak{g}$ is Koszul in the sense that it is antiequivalent to the category of locally unitary finite-dimensional modules over a certain direct limit of finite-dimensional Koszul algebras. We describe these finite-dimensional algebras explicitly. We also prove an equivalence of the categories $\mathbb{T}_{o(\infty)}$ and $\mathbb{T}_{sp(\infty)}$ corresponding respectively to the orthogonal and symplectic finitary Lie algebras $o(\infty)$, $sp(\infty)$.

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arXiv2011

On the structure of the fundamental series of generalized Harish-Chandra modules

We continue the study of the fundamental series of generalized Harish-Chandra modules initiated in [PZ2]. Generalized Harish-Chandra modules are (g,k)-modules of finite type where g is a semisimple Lie algebra and k \subset g is a reductive in g subalgebra. A first result of the present paper is that a fundamental series module is a g-module of finite length. We then define the notions of strongly and weakly reconstructible simple (g,k)-modules M which reflect to what extent M can be determined via its appearance in the socle of a fundamental series module. In the second part of the paper we concentrate on the case k \simeq sl(2) and prove a sufficient condition for strong reconstructibility. This strengthens our main result from [PZ2] for the case k = sl(2). We also compute the sl(2)-characters of all simple strongly reconstructible (and some weakly reconstructible) (g,sl(2))-modules. We conclude the paper by discussing a functor between a generalization of the category O and a category of (g,sl(2))-modules, and we conjecture that this functor is an equivalence of categories.

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arXiv2011

Levi components of parabolic subalgebras of finitary Lie algebras

We characterize locally semisimple subalgebras $ł$ of $\sl_\infty$, $\so_\infty$, and $\sp_\infty$ which are Levi components of parabolic subalgebras. Given $ł$, we characterize the parabolic subalgebras $\p$ such that $ł$ is a Levi component of $\p$. When the set of such self-normalizing parabolic subalgebras $\p$ is finite, we prove an estimate on its cardinality. We consider various examples which highlight the differences from the case of parabolic subalgebras of finite-dimensional simple Lie algebras.

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arXiv2010

Categories of integrable $sl(\infty)$-, $o(\infty)$-, $sp(\infty)$-modules

We investigate several categories of integrable $sl(\infty)$-, $o(\infty)$-, $sp(\infty)$-modules. In particular, we prove that the category of integrable $sl(\infty)$-, $o(\infty)$-, $sp(\infty)$-modules with finite-dimensional weight spaces is semisimple. The most interesting category we study is the category $\widetilde{\mathrm{Tens}}_{\mathfrak{g}}$ of tensor modules. Its objects $M$ are defined as integrable modules of finite Loewy length such that the algebraic dual $M^*$ is also integrable and of finite Loewy length. We prove that the simple objects of $\widetilde{\mathrm{Tens}}_{\mathfrak{g}}$ are precisely the simple tensor modules, i.e. the simple subquotients of the tensor algebra of the direct sum of the natural and conatural representations. We also study injectives in $\widetilde{\mathrm{Tens}}_{\mathfrak{g}}$ and compute the Ext$^1$'s between simple modules. Finally, we characterize a certain subcategory $\mathrm{Tens}_{\mathfrak{g}}$ of $\widetilde{\mathrm{Tens}}_{\mathfrak{g}}$ as the unique minimal abelian full subcategory of the category of integrable modules which contains a non-trivial module and is closed under tensor product and algebraic dualization.

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arXiv2009

A Bott-Borel-Weil theorem for diagonal ind-groups

We establish a theorem computing the cohomology groups of line bundles on homogeneous ind-varieties $G/B$ for diagonal ind-groups $G$. The main difficulty in proving this analog of the classical Bott-Borel-Weil theorem is in defining an appropriate analog $W_B$ of the Weyl group so that the action of $W_B$ on weights of $G$ is compatible with the analog of the Demazure "action" of the Weyl group on the cohomology of line bundles.

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arXiv2009

Parabolic and Levi subalgebras of finitary Lie algebras

Let $\g$ be a locally reductive complex Lie algebra which admits a faithful countable-dimensional finitary representation $V$. Such a Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of $\sl_\infty$, $\so_\infty$, $\sp_\infty$, and finite-dimensional simple Lie algebras. A parabolic subalgebra of $\g$ is any subalgebra which contains a maximal locally solvable (that is, Borel) subalgebra. Building upon work by Dimitrov and the authors of the present paper, we give a general description of parabolic subalgebras of $\g$ in terms of joint stabilizers of taut couples of generalized flags. The main differences with the Borel subalgebra case are that the description of general parabolic subalgebras has to use both the natural and conatural modules, and that the parabolic subalgebras are singled out by further "trace conditions" in the suitable joint stabilizer. The technique of taut couples can also be used to prove the existence of a Levi component of an arbitrary subalgebra $\k$ of $\gl_\infty$. If $\k$ is splittable, we show that the linear nilradical admits a locally reductive complement in $\k$. We conclude the paper with descriptions of Cartan, Borel, and parabolic subalgebras of arbitrary splittable subalgebras of $\gl_\infty$.

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arXiv2009

Parabolic Subgroups of Real Direct Limit Groups

Let $G_R$ be a classical real direct limit Lie group and $g_R$ its Lie algebra. The parabolic subalgebras of the complexification $g_C$ were described by the first two authors. In the present paper we extend these results to $g_R$. This also gives a description of the parabolic subgroups of $G_R$. Furthermore, we give a geometric criterion for a parabolic subgroup $P_C$ of $G_C$ to intersect $G_R$ in a parabolic subgroup. This criterion involves the $G_R$-orbit structure of the flag ind-manifold $G_C/P_C$.

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arXiv2007

Rank 2 vector bundles on ind-Grassmannians

The simplest example of an ind-Grassmannian is the infinite projective space $\mathbf P^\infty$. The Barth-Van de Ven-Tyurin (BVT) Theorem, proved more than 30 years ago \cite{BV}, \cite{T}, \cite{Sa} (see also a recent proof by A. Coandă and G. Trautmann, \cite{CT}), claims that any vector bundle of finite rank on $\mathbf P^\infty$ is isomorphic to a direct sum of line bundles. In the last decade natural examples of infinite flag varieties (or flag ind-varieties) have arisen as homogeneous spaces of locally linear ind-groups, \cite{DPW}, \cite{DiP}. In the present paper we concentrate our attention to the special case of ind-Grassmannians, i.e. to inductive limits of Grassmannians of growing dimension.

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arXiv2007

Bounded generalized Harish-Chandra modules

Let $\gg$ be a complex reductive Lie algebra and $\kk\subset\gg$ be any reductive in $\gg$ subalgebra. We call a $(\gg,\kk)$-module $M$ bounded if the $\kk$-multiplicities of $M$ are uniformly bounded. In this paper we initiate a general study of simple bounded $(\gg,\kk)$-modules. We prove a strong necessary condition for a subalgebra $\kk$ to be bounded (Corollary \ref{cor1.6}), i.e. to admit an infinite-dimensional simple bounded $(\gg,\kk)$-module, and then establish a sufficient condition for a subalgebra $\kk$ to be bounded (Theorem \ref{thGroups2}). As a result we are able to classify all maximal bounded reductive subalgebras of $\gg=\sl(n)$. In the second half of the paper we describe in detail simple bounded infinite-dimensional $(\gg,\sl(2))$-modules, and in particular compute their characters and minimal $\sl(2)$-types. We show that if $\sl(2)$ is a bounded subalgebra of $\gg$ which is not contained in a proper ideal of $\gg$, then $\gg\simeq \sl(2)\oplus \sl(2), \sl(3),\sp(4)$; alltogether, up to conjugation there are five possible embeddings of $\sl(2)$ as a bounded subalgebra into $\gg$ as above. In two of these cases $\sl(2)$ is a symmetric subalgebra, and many results about simple bounded $(\gg,\sl(2))$-modules are known. A case where our results are entirely new is the case of a principal $\sl(2)$-subalgebra in $\sp(4)$.

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arXiv2007

Triviality of vector bundles on sufficiently twisted ind-Grassmannians

Twisted ind-Grassmannians are ind-varieties $\GG$ obtained as direct limits of Grassmannians $G(r_m,V^{r_m})$, for $m\in\ZZ_{>0}$, under embeddings $ϕ_m:G(r_m,V^{r_m})\to G(r_{m+1}, V^{r_{m+1}})$ of degree greater than one. It has been conjectured in \cite{PT} and \cite{DP} that any vector bundle of finite rank on a twisted ind-Grassmannian is trivial. We prove this conjecture under the assumption that the ind-Grassmannian $\GG$ is sufficiently twisted, i.e. that $\lim_{m\to\infty}\frac{r_m}{°ϕ_1...\degϕ_m}=0$.

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arXiv2007

A construction of generalized Harish-Chandra modules for locally reductive Lie algebras

We study cohomological induction for a pair $(\frak g,\frak k)$, $\frak g$ being an infinite dimensional locally reductive Lie algebra and $\frak k \subset\frak g$ being of the form $\frak k_0 + C_\gg(\frak k_0)$, where $\frak k_0\subset\frak g$ is a finite dimensional reductive in $\frak g$ subalgebra and $C_{\gg} (\frak k_0)$ is the centralizer of $\frak k_0$ in $\frak g$. We prove a general non-vanishing and $\frak k$-finiteness theorem for the output. This yields in particular simple $(\frak g,\frak k)$-modules of finite type over $\frak k$ which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in \cite{PZ1} and \cite{PZ2}. We study explicit versions of the construction when $\frak g$ is a root-reductive or diagonal locally simple Lie algebra.

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arXiv2006

A Construction of Generalized Harish-Chandra Modules with Arbitrary Minimal k-Type

Let g be a semisimple complex Lie algebra and k in g be any algebraic subalgebra reductive in g. For any simple finite dimensional k-module V, we construct simple (g; k)-modules M with finite dimensional k-isotypic components such that V is a k-submodule of M and the Vogan norm of any simple k-submodule V' of M; V' not isomorphic to V, is greater than the Vogan norm of V . The (g; k)-modules M are subquotients of the fundamental series of (g; k)-modules introduced in [PZ2].

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arXiv2006

Cartan subalgebras of root-reductive Lie algebras

Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras under injections which preserve the root spaces. It is known that a root-reductive Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of finite-dimensional simple Lie algebras as well as copies of the three simple infinite-dimensional root-reductive Lie algebras sl_infty, so_infty, and sp_infty. As part of a structure theory program for root-reductive Lie algebras, Cartan subalgebras of the Lie algebra gl_infty were introduced and studied in a paper of Neeb and Penkov. In the present paper we refine and extend the results of [N-P] to the case of a general root-reductive Lie algebra g. We prove that the Cartan subalgebras of g are the centralizers of maximal toral subalgebras and that they are nilpotent and self-normalizing. We also give an explicit description of all Cartan subalgebras of the simple Lie algebras sl_infty, so_infty, and sp_infty. We conclude the paper with a characterization of the set of conjugacy classes of Cartan subalgebras of the Lie algebras gl_infty, sl_infty, so_infty, and sp_infty with respect to the group of automorphisms of the natural representation which preserve the Lie algebra.

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arXiv2004

Generalized Harish-Chandra modules with generic minimal $\frak k$-type

We make a first step towards a classification of simple generalized Harish-Chandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras $(\g,\k)$, we construct, via cohomological induction, the fundamental series $F^\cdot (\p,E)$ of generalized Harish-Chandra modules. We then use $F^\cdot (\p,E)$ to characterize any simple generalized Harish-Chandra module with generic minimal $\k$-type. More precisely, we prove that any such simple $(\g,\k)$-module of finite type arises as the unique simple submodule of an appropriate fundamental series module $F^s(\p,E)$ in the middle dimension $s$. Under the stronger assumption that $\k$ contains a semisimple regular element of $\g$, we prove that any simple $(\g,\k)$-module with generic minimal $\k$-type is necessarily of finite type, and hence obtain a reconstruction theorem for a class of simple $(\g,\k)$-modules which can a priori have infinite type. We also obtain generic general versions of some classical theorems of Harish-Chandra, such as the Harish-Chandra admissibility theorem. The paper is concluded by examples, in particular we compute the genericity condition on a $\k$-type for any pair $(\g,\k)$ with $\k\simeq s\ell (2)$.

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arXiv2004

Ind--varieties of generalized flags as homogeneous spaces for classical ind--groups

The purpose of the present paper is twofold: to introduce the notion of a generalized flag in an infinite dimensional vector space $V$ (extending the notion of a flag of subspaces in a vector space), and to give a geometric realization of homogeneous spaces of the ind--groups $SL(\infty)$, $SO(\infty)$ and $Sp(\infty)$ in terms of generalized flags. Generalized flags in $V$ are chains of subspaces which in general cannot be enumerated by integers. Given a basis $E$ of $V$, we define a notion of $E$--commensurability for generalized flags, and prove that the set $\cFl (\cF, E)$ of generalized flags E$--commensurable with a fixed generalized flag $\cF$ in $V$ has a natural structure of an ind--variety. In the case when $V$ is the standard representation of $G = SL(\infty)$, all homogeneous ind--spaces $G/P$ for parabolic subgroups $P$ containing a fixed splitting Cartan subgroup of $G$, are of the form $\cFl (\cF, E)$. We also consider isotropic generalized flags. The corresponding ind--spaces are homogeneous spaces for $SO(\infty)$ and $Sp(\infty)$. As an application of the construction, we compute the Picard group of $\cFl (\cF, E)$ (and of its isotropic analogs) and show that $\cFl (\cF, E)$ is a projective ind--variety if and only if $\cF$ is a usual, possibly infinite, flag of subspaces in $V$.

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arXiv2003

Generalized Harish-Chandra Modules: A New Direction

Let $\frak g$ be a reductive Lie algebra over $\bold C$. We say that a $\frak g$-module $M$ is a generalized Harish-Chandra module if, for some subalgebra $\frak k \subset\frak g$, $M$ is locally $\frak k$-finite and has finite $\frak k$-multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when $\frak k$ is a Cartan subalgebra. We also review the recent determination of which reductive in $\frak g$ subalgebras $\frak k$ are essential to a classification. Finally, we present in detail the emerging picture for the case when $\frak k$ is a principal 3-dimensional subalgebra.

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