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Punita Batra

Publications and source records attributed to Punita Batra.

16 recordsLinked to original sources
arXiv2026

Integrable Representations for Toroidal Lie Algebras Co-ordinated by Rational Quantum Torus

We classify irreducible integrable modules with finite-dimensional weight spaces for toroidal Lie algebras coordinated by rational quantum torus with trivial central action. Let $\mathbb{C}_q$ denote the rational quantum torus associated with a rational quantum matrix $q$, and let $\hatτ(d,q)$ be the toroidal Lie algebra coordinated by rational quantum torus obtained by adjoining the derivation space $D$ to the universal central extension $\tildeτ(d,q)=\mathfrak{sl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$ of $\mathfrak{sl}_d(\mathbb{C}_q)$. The case of nontrivial central action was previously classified by S. Eswara Rao and K. Zhao. The present work completes the classification by describing all irreducible integrable $\hatτ(d,q)$-modules with finite-dimensional weight spaces in the case where the $n$-dimensional center $C$ acts trivially on the modules.

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arXiv2024

Integrable Modules of Map full Toroidal Lie Algebras

In this paper, we study the irreducible objects of the category Cf in of integrable representations for Map full Toroidal Lie algebras with finite dimensional weight spaces. These representations turn out to be single point evaluation modules and hence are irreducible-integrable modules for the underlying full toroidal algebras.

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arXiv2024

Simple Modules For Twisted Hamiltonian Extended Affine Lie Algebras

In this paper, we consider the twisted Hamiltonian extended affine Lie algebra (THEALA). We classify the irreducible integrable modules for these Lie algebras with finite-dimensional weight spaces when the finite-dimensional center acts non-trivially. This Lie algebra has a triangular decomposition, which is different from the natural triangular decomposition of twisted full toroidal Lie algebra. Any irreducible integrable module of it is the highest weight module with respect to the given triangular decomposition. In this paper, we describe the highest-weight space in detail.

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arXiv2022

Irreducible Integrable Modules for the full Toroidal Lie Algebras co-ordinated by Rational Quantum Torus

Let $\mathbb{C}_q$ be a non-commutative Laurent polynomial ring associated with a $(n+1)\times (n+1)$ rational quantum matrix $q$. Let $\mathfrak{sl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$ be the universal central extension of Lie subalgebra $\mathfrak{sl}_d(\mathbb{C}_q)$ of $\mathfrak{gl}_d(\mathbb{C}_q)$. Now let us take the Lie algebra $τ=\mathfrak{gl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$. Let $Der(\mathbb{C}_q)$ be the Lie algebra of all derivations of $\mathbb{C}_q$. Now we consider the Lie algebra $\tildeτ=τ\rtimes Der(\mathbb{C}_q)$, called as full toroidal Lie algebra co-ordinated by rational quantum tori. In this paper we get a classification of irreducible integrable modules with finite dimensional weight spaces for $\tildeτ$ with nonzero central action on the modules.

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arXiv2021

A class of irreducible modules for loop-Virasoro algebras

Tensor product of highest weight modules and intermediate modules for Virasoro algebra have been studied around 1997. Since then the irreducibility problem for tensor product of modules is open. We consider the loop-Virasoro algebra $Vir \otimes B$, where $Vir$ is the Virasoro algebra and $B$ a commutative associative unital algebra over $\mathbb C$. In this paper we study the irreducibility problem for the tensor product of highest weight modules and intermediate modules for $Vir\otimes B$. Finally we find out a necessary and sufficient conditions for such modules to be isomorphic.

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arXiv2019

Natural Elements of Center of Generalized Quantum Groups

This paper gives elements in the (skew) center of the generalized quantum group corresponding to its irreducible finite dimensional modules. Finally we give a conjecture stating that those must form a basis of the center.

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arXiv2017

Centers of Generalized Quantum Groups

This paper treats the generalized quantum group $U=U(χ,π)$ with a bi-homomorphism $χ$ for which the corresponding generalized root system is a finite set. We establish a Harish-Chandra type theorem describing the (skew) center of $U$.

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arXiv2015

On Integrable modules for the twisted full toroidal Lie algebra

The paper is to classify irreducible integrable modules for the twisted full toroidal Lie algebra with some technical conditions. The twisted full toroidal Lie algebra are extensions of multiloop algebra twisted by sevaral finite order automorphisms. The result genaralizes a result by Fu Jiayuan and Cuipo Jiang, where they consider only one automorphism.

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arXiv2015

The irreducible modules for the derivations of the rational quantum torus

Let $\bbcq$ be the quantum torus associated with the $d \times d$ matrix $q = (q_{ij})$, $q_{ii} = 1$, $q_{ij}^{-1} = q_{ji}$, $q_{ij}$ are roots of unity, for all $1 \leq i, j \leq d.$ Let $\Der(\bbcq)$ be the Lie algebra of all the derivations of $\bbcq$. In this paper we define the Lie algebra $\Der(\bbcq) \ltimes \bbcq$ and classify its modules which are irreducible and have finite dimensional weight spaces. These modules under certain conditions turn out to be of the form $V \otimes \bbcq$, where $V$ is a finite dimensional irreducible $gl_d$-module.

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arXiv2009

Blocks and modules for Whittaker pairs

Inspired by recent activities on Whittaker modules over various (Lie) algebras we describe some general framework for the study of Lie algebra modules locally finite over a subalgebra. As a special case we obtain a very general setup for the study of Whittaker modules, which includes, in particular, Lie algebras with triangular decomposition and simple Lie algebras of Cartan type. We describe some basic properties of Whittaker modules, including a block decomposition of the category of Whittaker modules and certain properties of simple Whittaker modules under some rather mild assumptions. We establish a connection between our general setup and the general setup of Harish-Chandra subalgebras in the sense of Drozd, Futorny and Ovsienko. For Lie algebras with triangular decomposition we construct a family of simple Whittaker modules (roughly depending on the choice of a pair of weights in the dual of the Cartan subalgebra), describe their annihilators and formulate several classification conjectures. In particular, we construct some new simple Whittaker modules for the Virasoro algebra. Finally, we construct a series of simple Whittaker modules for the Lie algebra of derivations of the polynomial algebra, and consider several finite dimensional examples, where we study the category of Whittaker modules over solvable Lie algebras and their relation to Koszul algebras.

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arXiv2008

Representations of Graded Multi-Loop Lie Algebras

Let g_A (respectively, g_A(μ)) be the graded multi-loop Lie algebra (respectively graded twisted multi-loop Lie algebra)" associated with the simple finite dimensional Lie algebra g over the complex field C. In this paper, we prove that irreducible integrable g_A(μ)-modules with finite dimensional weight spaces are either highest weight modules or their duals and classify the isomorphism classes of irreducible integrable g_A-modules and g_A(μ)-modules with finite dimensional weight spaces.

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