Ivan Dimitrov

• Queen's University

We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs). Since both Kostant root systems and root systems of Lie superalgebras are examples of GRSs, studying GRSs provides a uniform axiomatic approach to...

We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs for short). Since both Kostant root systems and root systems of Lie superalgebras are examples of GRSs, studying GRSs provides a uniform axiomatic ap...

We use quivers and their representations to bring new perspectives on the subregular J-ring JC of a Coxeter system (W,S), a subring of Lusztig's J-ring. We prove that JC is isomorphic to a suitable quotient of the path algebra of the double quiver of (W,S). Up to Morita equivalence, such quotients include the group algebras of all free products of...

In this paper, we study the existence and classification problems of left-symmetric superalgebras on special linear Lie superalgebras ${\mathfrak{sl}}(m|n)$ with $m\neq n$. The main three results of this paper are: (i) a complete classification of the left-symmetric superalgebras on ${\mathfrak{sl}}(2|1)$, (ii) ${\mathfrak{sl}}(m|1)$ does not admit...

Let $\mathbb{K}$ denote an algebraically closed field and $A$ a free product of finitely many semisimple associative $\mathbb{K}$-algebras. We associate to $A$ a finite acyclic quiver $\Gamma$ and show that the category of finite dimensional $A$-modules is equivalent to a full subcategory of the category ${\rm rep}(\Gamma)$ of finite dimensional re...

We study the subregular $J$-ring $J_C$ of a Coxeter system $(W,S)$, a subring of Lusztig's $J$-ring. We prove that $J_C$ is isomorphic to a quotient of the path algebra of the double quiver of $(W,S)$ by a suitable ideal that we associate to a family of Chebyshev polynomials. As applications, we use quiver representations to study the category mod-...

We study the eigenspace decomposition of a basic classical Lie superalgebra under the adjoint action of a toral subalgebra, thus extending results of Kostant. In recognition of Kostant's contribution we refer to the eigenfunctions appearing in the decomposition as Kostant roots. We then prove that Kostant root systems inherit the main properties of...

In this paper we show that when $\mathrm{G}$ is a classical semi-simple algebraic group, $\mathrm{B}\subset\mathrm{G}$ a Borel subgroup, and $\mathrm{X} = \mathrm{G}/\mathrm{B}$, then the structure coefficients of the Belkale-Kumar product $\odot_{0}$ on $\mathrm{H}^{*}(\mathrm{X}, \mathbf{Z})$ are all either $0$ or $1$.

We study the eigenspace decomposition of a basic classical Lie superalgebra under the adjoint action of a toral subalgebra, thus extending results of Kostant. In recognition of Kostant's contribution we refer to the eigenspaces appearing in the decomposition as Kostant roots. We then prove that Kostant root systems inherit the main properties of cl...

Let \(\mathfrak {g}\) be a simple complex Lie algebra and let \(\mathfrak {t} \subset \mathfrak {g}\) be a toral subalgebra of \(\mathfrak {g}\). As a \(\mathfrak {t}\)-module \(\mathfrak {g}\) decomposes as $$\mathfrak{g} = \mathfrak{s} \oplus \left( \oplus_{\nu \in \mathcal{R}}~ \mathfrak{g}^{\nu}\right)$$ where \(\mathfrak {s} \subset \mathfrak...

In this paper we show that when $\mathrm{G}$ is a classical semi-simple algebraic group, $\mathrm{B}\subset\mathrm{G}$ a Borel subgroup, and $\mathrm{X} = \mathrm{G}/\mathrm{B}$, then the structure coefficients of the Belkale-Kumar product $\odot_{0}$ on $\mathrm{H}^{*}(\mathrm{X}, \mathbf{Z})$ are all either $0$ or $1$.

If $\sigma \in S_n$ is a permutation of $\{1, 2, \ldots, n\}$, the inversion set of $\sigma$ is $\Phi(\sigma) = \{ (i, j) \, | \, 1 \leq i < j \leq n, \sigma(i) > \sigma(j)\}$. We describe all $r$-tuples $\sigma_1, \sigma_2, \ldots, \sigma_r \in S_n$ such that $\Delta_n^+ = \{ (i, j) \, | \, 1 \leq i < j \leq n\}$ is the disjoint union of $\Phi(\si...

A diagonal ind-group is a direct limit of classical affine algebraic groups of growing rank under a class of inclusions that contains the inclusion as a typical special case. If G is a diagonal ind-group and B ⊂ G is a Borel ind-subgroup, we consider the ind-variety G / B and compute the cohomology H 𝓁 ( G / B ,𝒪 −λ ) of any G -equivariant line bu...

All simple weight modules with finite dimensional weight spaces over affine Lie algebras are classified. Comment: Preliminary version, 25 pages

Let X=G/B be a complete flag variety, and L' and L" two line bundles on X. Consider the cup product map H^{d'}(X,L') x H^{d"}(X, L") --> H^{d}(X,L), where L=L' x L" and d=d'+d". We answer two natural questions about the map above: When is it a nonzero map of irreducible G-representations? Conversely, given generic irreducible representations V' and...

This is a companion paper to “Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one” [http://arxiv.org/abs/0909.2280]. It is mostly expository and focuses on the representation-theoretic and combinatorial aspects of the main problems considered in the other article. Let X=G/B and let ℒ 1 and ℒ 2 be...

We describe all locally semisimple subalgebras and all maximal subalgebras of the finitary Lie algebras gl(∞), sl(∞), so(∞), and sp(∞). For simple finite-dimensional Lie algebras these classes of subalgebras have been described in the classical works of A. Malcev and E. Dynkin.

We compare two combinatorial definitions of parabolic sets of roots. We show that these definitions are equivalent for simple finite dimensional Lie algebras, affine Lie algebras, and toroidal Lie algebras. In contrast, these definitions are not always equivalent for simple finite dimensional Lie superalgebras.

We describe all locally semisimple subalgebras and all maximal subalgebras of the finitary Lie algebras $\gl(\infty), \sl(\infty), \so(\infty)$, and $\sp(\infty)$. For simple finite--dimensional Lie algebras these classes of subalgebras have been described in the classical works of A. Malcev and E. Dynkin. Key words (2000 MSC): 17B05 and 17B65.

Let be a toroidal Lie algebra corresponding to a semisimple Lie algebra We describe all Borel subalgebras of which contain the Cartan subalgebra where is a fixed Cartan subalgebra of We show that each such Borel subalgebra determines a parabolic decomposition where is a proper toroidal subalgebra of and Our first main result is that, for any weight...

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(∞), SO(∞), and Sp(∞) in terms of generalized flags. Generalized flags in V...

We develop a Bott-Borel-Weil theory for direct limits of algebraic groups. Some of our results apply to locally reductive ind-groups G in general, i.e., to arbitrary direct limits of connected reductive linear algebraic groups. Our most explicit results concern root-reductive ind-groups G, the locally reductive ind-groups whose Lie algebras admit r...

In this article we initiate a systematic study of irreducible weight modules over direct limits of reductive Lie algebras, and in particular over the simple Lie algebras $A(\infty)$, $B(\infty)$, $C(\infty)$ and $D(\infty)$. Our main tool is the shadow method introduced recently in \cite{DMP}. The integrable irreducible modules are an important par...

We prove a more general version of a result announced without proof in [DP], claiming roughly that in a partially integrable highest weight module over a Kac-Moody algebra the integrable directions from a parabolic subalgebra.

We build higest weight representations of the Virasoro algebra with highest weight vectors tau-functions of Toda deformations of Laguerre polynomials. Then we describe all rationals solutions of the Toda sys- tem and prove that the corresponding tau-functions are highest weight vectors of irreducible degenerate representations of the Virasoro algeb...

The Gelfand-Dikii Hamiltonian H = -q14-q22+3q2q12-q1p22-2p1p2 is studied from a topological point of view. The topology of the real level sets for all values of the constants of motion is described using the complex algebraic structure of the problem.

. Given any simple Lie superalgebra g, we investigate the structure of an arbitrary simple weight g-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsupe...

Partially and Fully Integrable Modules over Lie Superalgebras Partially and Fully Integrable Modules over Lie Superalgebras Abstract . V. Kac and M. Wakimoto have observed in [ KW ] that for certain most natural affine Lie superalgebras g like g =3D sl (m + nε ) (m ≠ 0,1, n ≠ 0,1) or g =3D osp (m + nε ) (m ≠ 1,2 n ≠ 1), and irreducible highest we...