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Kailash C. Misra

Publications and source records attributed to Kailash C. Misra.

At least 19 recordsLinked to original sources
arXiv2026

Representations of twisted quantum affine algebras

We develop the representation theory of imaginary Verma modules for twisted quantum affine algebras and construct the corresponding Kashiwara algebras. The twisted case presents substantial new difficulties compared with the untwisted setting: the PBW root vectors have nontrivial orbit structure, the imaginary root spaces occur with multiplicities, and roots of unity enter essentially into the defining commutation relations. Our first main result is an explicit PBW-type basis for twisted quantum imaginary Verma modules associated with the natural imaginary partition of the affine root system. This provides a precise compatibility between the twisted quantum and classical theories that is not immediate from the standard PBW theory. We then determine the structure and irreducibility of the twisted quantum imaginary Verma modules. We prove that the Heisenberg submodule generated by the imaginary root vectors is irreducible precisely at nonzero central charge, and establish the corresponding irreducibility criterion for the reduced twisted imaginary Verma modules at zero central charge. The second main part of the paper introduces the Kashiwara algebra in the twisted setting. The required current formula for the generators differs essentially from the untwisted formula and is needed to construct the Omega operators. We derive the resulting Omega-operator commutation relations, including the root-of-unity factors specific to the twisted cases. These relations lead to a new presentation of the Kashiwara algebra associated with the reduced twisted imaginary Verma modules. We prove that the negative current algebra is a simple module over this Kashiwara algebra and construct a symmetric non-degenerate bilinear form characterized by the Omega operators.

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arXiv2024

On $C_n^{(1)}$-Geometric Crystal and its Ultradiscretization

Let $\mathfrak{g}$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and $\mathfrak{g}^L$ be its Langlands dual. It is conjectured that for each Dynkin node $i \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak{g}$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a certain coherent family of perfect crystals for the Langland dual $\mathfrak{g}^L$. In this paper we construct positive geometric crystals for $\mathcal{V}(C_n^{(1)})$ in the level zero fundamental spin $C_n^{(1)}$- module $W(\varpi_n)$ for $n = 2, 3,4$ and show that its ultra-discretization is isomorphic to the limit $B^{n, \infty}$ of a coherent family $\{B^{n, l}\}_{l \geq 1}$ of perfect crystals for the Langland dual $D_n^{(2)}$ which proves the conjecture in these cases.

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arXiv2023

Crystal bases for reduced imaginary Verma modules of untwisted quantum affine algebras

We consider reduced imaginary Verma modules for the untwisted quantum affine algebras $U_q(\hat{\g})$ and define a crystal-like base which we call imaginary crystal base using the Kashiwara algebra $\mathcal K_q$ constructed in earlier work by Ben Cox and two of the authors. We prove the existence of the imaginary crystal base for any object in a suitable category $\mc{O}^q_{red,im}$ containing the reduced imaginary Verma modules for $U_q(\hat{\g})$.

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arXiv2022

Multiplicities of maximal weights of the $\hat{s\ell}(n) $-module $V(kΛ_0)$

Consider the affine Lie algebra $\hat{s\ell}(n)$ with null root $δ$, weight lattice $P$ and set of dominant weights $P^+$. Let $V(kΛ_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\hat{s\ell}(n)$-module with level $k \geq 1$ highest weight $kΛ_0$. Let $wt(V)$ denote the set of weights of $V(kΛ_0)$. A weight $μ\in wt(V)$ is a maximal weight if $μ+ δ\not\in wt(V)$. Let $max^+(kΛ_0)= max(kΛ_0)\cap P^+$ denote the set of maximal dominant weights which is known to be a finite set. In 2014, the authors gave the complete description of the set $max^+(kΛ_0)$. In subsequent papers the multiplicities of certain subsets of $max^+(kΛ_0)$ were given in terms of some pattern-avoiding permutations using the associated crystal base theory. In this paper the multiplicity of all the maximal dominant weights of the $\hat{s\ell}(n) $-module $V(kΛ_0)$ are given generalizing the known results.

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arXiv2022

On Smith normal forms of $q$-Varchenko matrices

In this paper, we investigate $q$-Varchenko matrices for some hyperplane arrangements with symmetry in two and three dimensions, and prove that they have a Smith normal form over $\mathbb Z[q]$. In particular, we examine the hyperplane arrangement for the regular $n$-gon in the plane and the dihedral model in the space and Platonic polyhedra. In each case, we prove that the $q$-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over $\mathbb Z[q]$ and realize their congruent transformation matrices over $\mathbb Z[q]$ as well.

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arXiv2020

Subinvariance in Leibniz Algebras

Leibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. In this paper we define subinvariant subalgebras of Leibniz algebras and study their properties. It is shown that the signature results on subinvariance in Lie algebras have analogs for Leibniz algebras.

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arXiv2020

Multiplicities of some maximal dominant weights of the $\widehat{s\ell}(n)$-modules $V(kΛ_0)$

For $n \geq 2$ consider the affine Lie algebra $\widehat{s\ell}(n)$ with simple roots $\{α_i \mid 0 \leq i \leq n-1\}$. Let $V(kΛ_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\widehat{s\ell}(n)$-module with highest weight $kΛ_0$. It is known that there are finitely many maximal dominant weights of $V(kΛ_0)$. Using the crystal base realization of $V(kΛ_0)$ and lattice path combinatorics we determine the multiplicities of a large set of maximal dominant weights of the form $kΛ_0 - λ^\ell_{a,b}$ where $ λ^\ell_{a,b} = \ellα_0 + (\ell-b)α_1 + (\ell-(b+1))α_2 + \cdots + α_{\ell-b} + α_{n-\ell+a} + 2α_{n - \ell+a+1} + \ldots + (\ell-a)α_{n-1}$, and $k \geq a+b$, $a,b \in \mathbb{Z}_{\geq 1}$, $\max\{a,b\} \leq \ell \leq \left \lfloor \frac{n+a+b}{2} \right \rfloor-1 $. We show that these weight multiplicities are given by the number of certain pattern avoiding permutations of $\{1, 2, 3, \ldots \ell\}$.

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arXiv2020

Complete Leibniz Algebras

Leibniz algebras are certain generalization of Lie algebras. It is natural to generalize concepts in Lie algebras to Leibniz algebras and investigate whether the corresponding results still hold. In this paper we introduce the notion of complete Leibniz algebras as generalization of complete Lie algebras. Then we study properties of complete Leibniz algebras and their holomorphs.

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arXiv2020

Ultra-Discretization of $D_6^{(1)}$- Geometric Crystal at the spin node

Let $\mathfrak g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$. It is conjectured in \cite{KNO} that for each Dynkin node $k \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for the Langland dual ${\mathfrak g} ^L$. In this paper we show that at the spin node $k=6$, the family of perfect crystals given in \cite{KMN2} form a coherent family and show that its limit $B^{6,\infty}$ is isomorphic to the ultra-discretization of the positive geometric crystal we constructed in \cite{MP} for the affine Lie algebra $D_6^{(1)}$ which proves the conjecture in this case.

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arXiv2019

$D_6^{(1)}$- Geometric Crystal at the spin node

Let $\mathfrak{g}$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$. It is conjectured that for each Dynkin node $k \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak{g}$ has a positive geometric crystal. In this paper we construct a positive geometric crystal for the affine Lie algebra $D_6^{(1)}$ corresponding to the Dynkin spin node $k= 6$.

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arXiv2018

$D_5^{(1)}$- Geometric Crystal corresponding to the Dynkin spin node $i=5$ and its ultra-discretization

Let $g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and $g^L$ be its Langlands dual. It is conjectured that for each Dynkin node $i \in I \setminus \{0\}$ the affine Lie algebra $g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for $g^L$. In this paper we construct a positive geometric crystal $V(D_5^{(1)})$ in the level zero fundamental spin $D_5^{(1)}$- module $W(\varpi_5)$. Then we define explicit $0$-action on the level $l$ known $D_5^{(1)}$- perfect crystal $B^{5, l}$ and show that $\{B^{5, l}\}_{l \geq 1}$ is a coherent family of perfect crystals with limit $B^{5, \infty}$. Finally we show that the ultra-discretization of $V(D_5^{(1)})$ is isomorphic to $B^{5, \infty}$ as crystals which proves the conjecture in this case.

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arXiv2018

Imaginary crystal bases for $U_q(\widehat{\mathfrak{sl}(2)})$-modules in category $\mathcal O^q_{\text{red,im}}$

Recently we defined imaginary crystal bases for $U_q(\widehat{\mathfrak{sl}(2)})$- modules in category $\mathcal O^q_{\text{red,im}}$ and showed the existence of such bases for reduced quantized imaginary Verma modules for $U_q(\widehat{\mathfrak{sl}(2)})$. In this paper we show the existence of imaginary crystal basis for any object in the category $\mathcal O^q_{\text{red,im}}$.

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arXiv2016

Affine Geometric Crystal of $A^{(1)}_n$ and Limit of Kirillov-Reshetikhin Perfect Crystals

Let $\mathfrak g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and ${\mathfrak g}^L$ be its Langlands dual. It is conjectured by Kashiwara et al.([16]) that for each $k \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for ${\mathfrak g}^L$. Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra ${\mathfrak g}= A^{(1)}_n$ for each Dynkin index $k\in I\setminus\{0\}$ and show that its ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for $A^{(1)}_n$ given by Okado et al.([29]). In the process we develop and use some lattice-path combinatorics.

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arXiv2016

On realization of some twisted toroidal Lie algebras

Toroidal Lie algebras are generalizations of affine Lie algebras. In 1990, Moody, Rao and Yokonuma gave a presentation for untwisted toroidal Lie algebras. In this paper we give a presentation for the twisted toroidal Lie algebras of type $A$ and $D$ constructed by Fu and Jiang.

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arXiv2015

On principal realization of modules for the affine Lie algebra $A_1 ^{(1)}$ at the critical level

We present complete realization of irreducible $A_1 ^{(1)}$-modules at the critical level in the principal gradation. Our construction uses vertex algebraic techniques, the theory of twisted modules and representations of Lie conformal superalgebras. We also provide an alternative Z-algebra approach to this construction. All irreducible highest weight $A_1 ^{(1)}$-modules at the critical level are realized on the vector space $M_{\tfrac{1}{2} + \Bbb Z} (1) ^{\otimes 2}$ where $M_{\tfrac{1}{2} + \Bbb Z} (1) $ is the polynomial ring ${\Bbb C}[α(-1/2), α(-3/2), ...]$. Explicit combinatorial bases for these modules are also given.

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arXiv2015

On Classification of Four Dimensional Nilpotent Leibniz Algebras

Leibniz algebras are certain generalization of Lie algebras. In this paper we give the classification of four dimensional non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear forms and some other techniques to obtain our result.

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arXiv2015

Lattice Paths, Young Tableaux, and Weight Multiplicities

For $\ell \geq 1$ and $k \geq 2$, we consider certain admissible sequences of $k-1$ lattice paths in a colored $\ell \times \ell$ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of partitions of $\ell$ with height $\leq k$, which is also the number of $(k+1)k\cdots21$-avoiding permutations of $\{1, 2, \ldots, \ell\}$. Finally, we apply this result to the representation theory of the affine Lie algebra $\widehat{sl}(n)$ and show that this quantity gives the multiplicity of certain maximal dominant weights in the irreducible module $V(kΛ_0)$.

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