arXiv subjects
Publications and source records attributed to Apoorva Khare.
Motivated by the orthogonality relations for irreducible characters of a finite group, we evaluate the sum of a finite group of linear characters of a Hopf algebra, at all grouplike and skew-primitive elements. We then discuss results for products of skew-primitive elements. Examples include groups, (quantum groups over) Lie algebras, the small quantum groups of Lusztig, and their variations (by Andruskiewitsch and Schneider).
This article is an exposition of the 1967 Annals paper by Parthasarathy, Ranga Rao, and Varadarajan, on irreducible admissible Harish-Chandra modules over complex semisimple Lie groups and Lie algebras. It was written in Winter 2012 to be part of a special collection organized to mark 10 years and 25 volumes of the series "Texts and Readings in Mathematics (TRIM)". Each article in this collection is intended to give nonspecialists in its field, an appreciation for the impact and contributions of the paper being surveyed. Thus, the author has kept the prerequisites for this article down to a basic course on complex semisimple Lie algebras. While it arose out of the grand program of Harish-Chandra on admissible representations of semisimple Lie groups, the work by Parthasarathy et al also provided several new insights on highest weight modules and related areas, and these results have been the subject of extensive research over the last four decades. Thus, we also discuss its results and follow-up works on the classification of irreducible Harish-Chandra modules; on the PRV conjecture and tensor product multiplicities; and on PRV determinants for (quantized) affine and semisimple Lie algebras.
Coupled discrete models abound in several areas of physics. Here we provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lamé polynomials of arbitrary order. The models discussed are (i) coupled Salerno model, (ii) coupled Ablowitz-Ladik model, (iii) coupled $ϕ^4$ model, and (iv) coupled $ϕ^6$ model. In all these cases we show that the coefficients of the Lamé polynomials are such that the Lamé polynomials can be reexpressed in terms of Chebyshev polynomials of the relevant Jacobi elliptic function.
Let $\g$ be a reductive Lie algebra and $V$ a $\g$-semisimple module. In this article, we study the category $\G$ of graded finite-dimensional representations of $\g \ltimes V$. We produce a large class of truncated subcategories, which are directed and highest weight. Suppose $V$ is finite-dimensional with weights $\wt(V)$. Let $Ψ\subset \wt(V)$ be the set of weights contained in a face $\F$ of the polytope that is the convex hull of $\wt(V)$. For each such $Ψ$, we produce quasi-hereditary Koszul algebras. We use these Koszul algebras to construct an infinite-dimensional graded subalgebra $\spg$ of the locally finite part of the algebra of invariants $(END{\C} (\V) \otimes \Sym V)^{\g}$, where $\V$ is the direct sum of all simple finite-dimensional $\g$-modules. We prove that $\spg$ is Koszul of finite global dimension.
The paper is motivated by the study of graded representations of Takiff algebras, cominuscule parabolics, and their generalizations. We study certain special subsets of the set of weights (and of their convex hull) of the generalized Verma modules (or GVM's) of a semisimple Lie algebra $\lie g$. In particular, we extend a result of Vinberg and classify the faces of the convex hull of the weights of a GVM. When the GVM is finite-dimensional, we ask a natural question that arises out of Vinberg's result: when are two faces the same? We also extend the notion of interiors and faces to an arbitrary subfield $\F$ of the real numbers, and introduce the idea of a weak $\F$-face of any subset of Euclidean space. We classify the weak $\F$-faces of all lattice polytopes, as well as of the set of lattice points in them. We show that a weak $\F$-face of the weights of a finite-dimensional $\lie g$-module is precisely the set of weights lying on a face of the convex hull.
In this paper, we compute the center of the infinitesimal Hecke algebras Hz associated to sl_2 ; then using nontriviality of the center, we study representations of these algebras in the framework of the BGG category O. We also discuss central elements in infinitesimal Hecke algebras over gl(n) and sp(2n) for all n. We end by proving an analogue of the theorem of Duflo for Hz.
This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category O. We study functorial properties of O across various setups. The first setup is over a skew group ring, involving a finite group $Γ$ acting on a regular triangular algebra $A$. We develop Clifford theory for $A \rtimes Γ$, and obtain results on block decomposition, complete reducibility, and enough projectives. O is shown to be a highest weight category when $A$ satisfies one of the "Conditions (S)"; the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module. Next, we turn to tensor products of such skew group rings; such a product is also a skew group ring. We are thus able to relate four different types of Categories O; more precisely, we list several conditions, each of which is equivalent in any one setup, to any other setup - and which yield information about O.
In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of partitioning V into subspaces.
This paper has been withdrawn by the author; see the much expanded, improved, and generalized version at arXiv:0811.2080.
This paper has been withdrawn by the author; see the much expanded, improved, and generalized version at arXiv:0811.2073.
A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with U_q(sl_2). We study its representation theory, and in particular, its category O.
In this article, we try to explain and unify standard divisibility tests found in various books. We then look at recurring decimals, and list a few of their properties. We show how to compute the number of digits in the recurring part of any fraction. Most of these results are accompanied by a proof (along with the assumptions needed), that works in a Euclidean domain. We then ask some obvious questions related to the results, and mention some similar questions that have been answered. In the final section (written jointly with P. Moree), some quantitative statements regarding the asymptotic behaviour of various sets of primes satisfying related properties, are considered.
We discuss the representation theory of $H_f$, which is a deformation of the symplectic oscillator algebra $sp(2n) \ltimes h_n$, where $h_n$ is the ((2n+1)-dimensional) Heisenberg algebra. We first look at a more general setup, involving an algebra with a triangular decomposition. Assuming the PBW theorem, and one other hypothesis, we show that the BGG category $\mathcal{O}$ is abelian, finite length, and self-dual. We decompose $\mathcal{O}$ as a direct sum of blocks $\calo(\la)$, and show that each block is a highest weight category. In the second part, we focus on the case $H_f$ for $n=1$, where we prove all these assumptions, as well as the PBW theorem.