arXiv subjects
Publications and source records attributed to Ivan Penkov.
In this note we unveil some subfields of the field of complex numbers which are well known to specialists, but seem to remain hidden for a wider mathematical audience. The existence and construc- tions of these fields follow directly from famous results of the 20-th century, however standard graduate textbooks avoid these topics.
We study the structure of tensor products of $\mathfrak{gl}(\infty) = \varinjlim \mathfrak{gl}(n)$-modules $\mathbf L(\mathbf \lambda) \otimes \mathbf F$ where $\mathbf L(\mathbf \lambda)$ is a simple integrable highest weight module and $\mathbf F$ is a simple integrable weight multiplicity-free module. Both $\mathbf L(\mathbf \lambda)$ and $\mathbf F$ are infinite dimensional, in particular $\mathbf F$ can be a Fock module. Similar tensor products of $\mathfrak{gl}(n)$-modules are semisimple and their simple constituents are described by the classical Pieri rule. We prove that a $\mathfrak{gl}(\infty)$-module $\mathbf M:= \mathbf L(\mathbf \lambda) \otimes \mathbf F$ is semisimple only in relatively trivial cases, and is indecomposable otherwise. Our main results are a description of the simple constituents of $\mathbf M$, and the construction of a linkage filtration on $\mathbf M$ that provides information on when two simple constituents of $\mathbf M$ are linked. Using the linkage filtration, we compute the socle and radical filtrations of $\mathbf M$, and determine when $\mathbf M$ is rigid.
By a grassmannian we understand a usual complex grassmannian or possibly an orthogonal or symplectic grassmannian. We classify, with few exceptions, linear embeddings of grassmannians into larger grassmannians, where the linearity requirement is the condition that the embedding induces an isomorphism on Picard groups. This classification implies that most linear embeddings of grassmannians are equivariant. A linear ind-grassmannian is the direct limit of a chain of linear embeddings of grassmannians. We conclude the paper by classifying linear embeddings of linear ind-grassmannians.
Let $n \in \mathbb{Z}_{\geqslant 2}$. By $P(n)$ we denote the set of all prime divisors of the integers in the sequence $n, n^2-1, (n^2-1)^2-1, \dots$. We ask whether the set $P(n)$ determines $n$ uniquely under the assumption that $n \neq m^2-1$ for $m \in \mathbb{Z}_{\geqslant 2}$. This problem originates in the structure theory of infinite-dimensional Lie algebras. We show that the sets $P(n)$ generate infinitely many equivalence classes of positive integers under the equivalence relation $n_1 \sim n_2 \iff P(n_1) = P(n_2)$. We also prove that the sets $P(n)$ separate all positive integers up to $2^{29}$, and we provide some heuristics on why the answer to our question should be positive.
Ind-varieties of generalized flags have been studied for two decades. However, a precise statement of when two such ind-varieties, one or both being possibly ind-varieties of isotropic generalized flags, are isomorphic, has been missing in the literature. Using some recent results on the structure of ind-varieties of generalized flags, we establish a criterion for the existence of an isomorphism as above. Our result claims that, with only two exceptions, isomorphisms of ind-varieties of generalized flags are induced by isomorphisms of respective generalized flags. The exceptional isomorphisms correlate with a well-known result of A. Onishchik from 1963.
We extend previous work by constructing a universal abelian tensor category ${\bf T}_t$ generated by two objects $X,Y$ equipped with finite filtrations $0\subsetneq X_0\subsetneq ... X_{t+1}\subsetneq X$ and $0\subsetneq Y_0\subsetneq ... Y_{t+1}\subsetneq Y$, and with a pairing $X\otimes Y\to \mathbb{I}$, where $\mathbb{I}$ is the monoidal unit. This category is modeled as a category of representations of a Mackey Lie algebra $\mathfrak{gl}^M(V,V_*)$ of cardinality $2^{\aleph_t}$, associated to a diagonalizable pairing between two complex vector spaces $V,V_*$ of dimension $\aleph_t$. As a preliminary step, we study a tensor category $\mathbb{T}_t$ generated by the algebraic duals $V^*$, $(V_*)^*$. The injective hull of $\mathbb{C}$ in $\mathbb{T}_t$ is a commutative algebra $I$, and the category ${\bf T}_t$ is consists of the free $I$-modules in $\mathbb{T}_t$. An essential novelty in our work is the explicit computation of Ext-groups between simples in both categories ${\bf T}_t$ and $\mathbb{T}_t$, which had been an open problem already for $t=0$. This provides a direct link from the theory of universal tensor categories to Littlewood-Richardson-type combinatorics.
We study the homogeneous ind-spaces $\mathrm{GL}(\mathbf{s})/\mathbf{P}$ where $\mathrm{GL}(\mathbf{s})$ is a strict diagonal ind-group defined by a supernatural number $\mathbf{s}$ and $\mathbf{P}$ is a parabolic ind-subgroup of $\mathrm{GL}(\mathbf{s})$. We construct an explicit exhaustion of $\mathrm{GL}(\mathbf{s})/\mathbf{P}$ by finite-dimensional partial flag varieties. As an application, we characterize all locally projective $\mathrm{GL}(\infty)$-homogeneous spaces, and some direct products of such spaces, which are $\mathrm{GL}(\mathbf{s})$-homogeneous for a fixed $\mathbf{s}$. The very possibility for a $\mathrm{GL}(\infty)$-homogeneous space to be $\mathrm{GL}(\mathbf{s})$-homogeneous for a strict diagonal ind-group $\mathrm{GL}(\mathbf{s})$ arises from the fact that the automorphism group of a $\mathrm{GL}(\infty)$-homogeneous space is much larger than $\mathrm{GL}(\infty)$.
We construct universal monoidal categories of topological tensor supermodules over the Lie superalgebras $\mathfrak{gl}(V\oplus \Pi V)$ and $\mathfrak{osp}(V\oplus \Pi V)$ associated with a Tate space $V$. Here $V\oplus \Pi V$ is a $\mathbb{Z}/2\mathbb{Z}$-graded topological vector space whose even and odd parts are isomorphic to $V$. We discuss the purely even case first, by introducing monoidal categories$\widehat{\mathbf{T}}_{\mathfrak{gl}(V)}$, $\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and $\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$, and show that these categories are anti-equivalent to respective previously studied categories $\mathbb{T}_{\mathfrak{gl}(V)}$, $\mathbb{T}_{\mathfrak{o}(V)}$, $\mathbb{T}_{\mathfrak{sp}(V)}$. These latter categories have certain universality properties as monoidal categories, which consequently carry over to $\widehat{\mathbf{T}}_{\mathfrak{gl}(V)}$, $\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and $\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$. Moreover, the categories $\mathbb{T}_{\mathfrak{o}(V)}$ and $\mathbb{T}_{\mathfrak{sp}(V)}$ are known to be equivalent, and this implies the equivalence of the categories $\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and $\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$. After introducing a supersymmetric setting, we establish the equivalence of the category $\widehat{\mathbf{T}}_{\mathfrak{gl}(V)}$ with the category $\widehat{\mathbf{T}}_{\mathfrak{gl}(V\oplus \Pi V)}$, and the equivalence of both categories $\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and $\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$ with $\widehat{\mathbf{T}}_{\mathfrak{osp}(V\oplus \Pi V)}$.
The Lie algebra $gl(V)$ is the Lie algebra of all endomorphisms of a countable-dimensional complex vector space $V$. We define a tensor category of topological representations of the Lie algebra $gl(V)$, so that $V$, its dual and the adjoint representation $gl(V)$ are objects of this category. This makes it an analogue of the category of finite-dimensional modules over the finite-dimensional Lie algebra $gl(n)$. Our main result is that this category is antiequivalent as a symmetric monoidal category to the category of tensor representations of the Lie algebra of finitary infinite matrices.
We classify simple bounded weight modules over the complex simple Lie superalgebras $\mathfrak{sl}(\infty |\infty)$ and $\mathfrak{osp} (m | 2n)$, when at least one of $m$ and $n$ equals $\infty$. For $\mathfrak{osp} (m | 2n)$ such modules are of spinor-oscillator type, i.e., they combine into one the known classes of spinor $\mathfrak{o} (m)$-modules and oscillator-type $\mathfrak{sp} (2n)$-modules. In addition, we characterize the category of bounded weight modules over $\mathfrak{osp} (m | 2n)$ (under the assumption $\dim \, \mathfrak{osp} (m | 2n) = \infty$) by reducing its study to already known categories of representations of $\mathfrak{sp} (2n)$, where $n$ possibly equals $\infty$. When classifying simple bounded weight $\mathfrak{sl}(\infty |\infty)$-modules, we prove that every such module is integrable over one of the two infinite-dimensional ideals of the Lie algebra $\mathfrak{sl}(\infty |\infty)_{\bar{0}}$. We finish the paper by establishing some first facts about the category of bounded weight $\mathfrak{sl} (\infty |\infty)$-modules.
We compute the group of automorphisms of an arbitrary ind-variety of (possibly isotropic) generalized flags. Such an ind-variety is a homogeneous ind-space for one of the ind-groups $SL(\infty)$, $O(\infty)$ or $Sp(\infty)$. We show that the respective automorphism groups are much larger than $SL(\infty)$, $O(\infty)$ or $Sp(\infty)$, and present the answer in terms of Mackey groups. The latter are groups of automorphisms of nondegenerate pairings of (in general infinite-dimensional) vector spaces. An explicit matrix form of the automorphism group of an arbitrary ind-variety of generalized flags is also given. The case of the Sato grassmannian is considered in detail, and its automorphism group is the projectivization of the connected component of unity in the group Japanese $GL(\infty)$.
Let $V_*\otimes V\rightarrow\mathbb{C}$ be a non-degenerate pairing of countable-dimensional complex vector spaces $V$ and $V_*$. The Mackey Lie algebra $\mathfrak{g}=\mathfrak{gl}^M(V,V_*)$ corresponding to this paring consists of all endomorphisms $\varphi$ of $V$ for which the space $V_*$ is stable under the dual endomorphism $\varphi^*: V^*\rightarrow V^*$. We study the tensor Grothendieck category $\mathbb{T}$ generated by the $\mathfrak{g}$-modules $V$, $V_*$ and their algebraic duals $V^*$ and $V^*_*$. This is an analogue of categories considered in prior literature, the main difference being that the trivial module $\mathbb{C}$ is no longer injective in $\mathbb{T}$. We describe the injective hull $I$ of $\mathbb{C}$ in $\mathbb{T}$, and show that the category $\mathbb{T}$ is Koszul. In addition, we prove that $I$ is endowed with a natural structure of commutative algebra. We then define another category $_I\mathbb{T}$ of objects in $\mathbb{T}$ which are free as $I$-modules. Our main result is that the category ${}_I\mathbb{T}$ is also Koszul, and moreover that ${}_I\mathbb{T}$ is universal among abelian $\mathbb{C}$-linear tensor categories generated by two objects $X$, $Y$ with fixed subobjects $X'\hookrightarrow X$, $Y'\hookrightarrow Y$ and a pairing $X\otimes Y\rightarrow \text{\textbf{1}}$ where \textbf{1} is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $\mathbb{T}$ and ${}_I\mathbb{T}$.
We classify integrable bounded simple weight modules over classical Lie superalgebras at infinity. We also study the categories of such modules, and we prove that for most of the classical Lie superalgebras at infinity the respective category is semisimple.
Let $G$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$, and $P_1,\dots, P_l$ be an arbitrary set of $l$ splitting parabolic subgroups of $G$. We determine all such sets with the property that $G$ acts with finitely many orbits on the ind-variety $X_1\times\dots\times X_l$ where $X_i=G/P_i$. In the case of a finite-dimensional classical linear algebraic group $G$, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar-Weyman-Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for $l=2$, the condition that $G$ acts on $X_1\times X_2$ with finitely many orbits is a rather restrictive condition on the pair $P_1,P_2$. We describe this condition explicitly. Using this result, we tackle the most interesting case where $l=3$, and present the answer in the form of a table. For $l\geq 4$, there always are infinitely many G-orbits on $X_1\times \dots\times X_l$.
We construct a new analogue of the BGG category $\mathcal O$ for the infinite-dimensional Lie algebras $\fg=\mathfrak{sl}(\infty),\mathfrak{o}(\infty), \mathfrak{sp}(\infty)$. A main difference with the categories studied in \cite{Nam} and \cite{CP} is that all objects of our category satisfy the large annihilator condition introduced in \cite{DPS}. Despite the fact that the splitting Borel subalgebras $\fb$ of $\fg$ are not conjugate, one can eliminate the dependency on the choice of $\fb$ and introduce a universal highest weight category $\mathcal {OLA}$ of $\fg$-modules, the letters $\mathcal{LA}$ coming from "large annihilator". The subcategory of integrable objects in $\mathcal {OLA}$ is precisely the category $\mathbb T_{\fg}$ studied in \cite{DPS}. We investigate the structure of $\mathcal {OLA}$, and in particular compute the multiplicities of simple objects in standard objects and the multiplicities of standard objects in indecomposable injectives.
We classify the simple bounded weight modules of ${\mathfrak{sl}(\infty})$, ${\mathfrak{o}(\infty)}$ and ${\mathfrak{sp}(\infty)}$, and compute their annihilators in $U({\mathfrak{sl}(\infty}))$, $U({\mathfrak{o}(\infty))}$, $U({\mathfrak{sp}(\infty))}$, respectively.
We prove a general existence result for infinite-dimensional admissible (g;k)-modules, where g is a reductive finite-dimensional complex Lie algebra and k is a reductive in g algebraic subalgebra.
We study a version of the BGG category O for Dynkin Borel subalgebras of root-reductive Lie algebras g, such as gl(\infty). We prove results about extension fullness and compute the higher extensions of simple modules by Verma modules. In addition, we show that our category O is Ringel self-dual and initiate the study of Koszul duality. An important tool in obtaining these results is an equivalence we establish between appropriate Serre subquotients of category O for g and category O for finite dimensional reductive subalgebras of g.