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Kailash C. Misra

Publications and source records attributed to Kailash C. Misra.

At least 37 records · Page 2Linked to original sources
arXiv2015

Classification of some Solvable Leibniz Algebras

Leibniz algebras are certain generalization of Lie algebras. In this paper we give classification of non-Lie solvable (left) Leibniz algebras of dimension $\leq 8$ with one dimensional derived subalgebra. We use the canonical forms for the congruence classes of matrices of bilinear forms to obtain our result. Our approach can easily be extended to classify these algebras of higher dimensions. We also revisit the classification of three dimensional non-Lie solvable (left) Leibniz algebras.

math.RA↗
arXiv2014

On nilpotent Lie algebras of small breadth

A Lie algebra $L$ is said to be of breadth $k$ if the maximal dimension of the images of left multiplication by elements of the algebra is $k$. In this paper we give characterization of finite dimensional nilpotent Lie algebras of breadth less than or equal to two. Furthermore, using these characterizations we determined the isomorphism classes of these algebras.

math.RA↗
arXiv2014

An imaginary PBW basis for quantum affine algebras of type 1

Let $\hat{\mathfrak g}$ be an affine Lie algebra of type 1. We give a PBW basis for the quantum affine algebra $U_q(\hat{\mathfrak g})$ with respect to the triangular decomposition of $\hat{\mathfrak g}$ associated with the imaginary positive root system.

math.RT↗
arXiv2013

On multiplicities of maximal weights of $\hat{sl}(n)$-modules

We determine explicitly the maximal dominant weights for the integrable highest weight $\hat{sl}(n)$-modules $V((k-1)Λ_0 + Λ_s)$, $0 \leq s \leq n-1$, $ k \geq 2$. We give a conjecture for the number of maximal dominant weights of $V(kΛ_0)$ and prove it in some low rank cases. We give an explicit formula in terms of lattice paths for the multiplicities of a family of maximal dominant weights of $V(kΛ_0)$. We conjecture that these multiplicities are equal to the number of certain pattern avoiding permutations. We prove that the conjecture holds for $k=2$ and give computational evidence for the validity of this conjecture for $k >2$.

math.RT↗
arXiv2013

Kashiwara Algebras and Imaginary Verma Modules for $U_q(\hat{\mathfrak{g}})$

We consider imaginary Verma modules for quantum affine algebra $U_q(\hat{\mathfrak{g}})$, where $\hat{\mathfrak{g}}$ is of type 1 i.e. of non-twisted type, and construct Kashiwara type operators and the Kashiwara algebra $\mathcal K_q$. We show that a certain quotient $\mathcal N_q^-$ of $U_q(\hat{\mathfrak{g}})$ is a simple $\mathcal K_q$-module.

math.QA↗
arXiv2013

Realization of \hat{sl}_2(C) at the Critical Level

An explicit realization of the affine Lie algebra \hat{sl}_2(C) at the critical level is constructed using a mixture of bosons and parafermions. Subsequently a representation of the associated Lepowsky-Wilson Z-algebra is given on a space of the tensor product of bosonic fields and certain semi-infinite wedge products.

math.QA↗
arXiv2013

On some structures of Leibniz algebras

Leibniz algebras are certain generalization of Lie algebras. In this paper we survey the important results in Leibniz algebras which are analogs of corresponding results in Lie algebras. In particular we highlight the differences between Leibniz algebras and Lie algebras.

math.RA↗
arXiv2013

A note on $U_q(D_4^{(3)})$ - Demazure crystals

We show that there exists a suitable sequence $\{w^{(k)}\}_{k \ge 0}$ of Weyl group elements for the perfect crystal $B = B^{1,3l}$ such that the path realizations of the Demazure crystals $B_{w^{(k)}}(lΛ_2)$ for the quantum affine algebra $U_q(D_4^{(3)})$ have tensor product like structure with mixing index $κ=1$.

math.QA↗
arXiv2012

$A_n^{(1)}$-Geometric Crystal corresponding to Dynkin index $i=2$ and its ultra-discretization

Let $g$ be an affine Lie algebra with index set $I = \{0, 1, 2,..., n\}$ and $g^L$ be its Langlands dual. It is conjectured that for each $i \in I \setminus \{0\}$ 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 $g^L$. We prove this conjecture for $i=2$ and $g = A_n^{(1)}$.

math.QA↗
arXiv2012

Soliton cellular automaton associated with $D_n^{(1)}$-crystal $B^{2,s}$

A solvable vertex model in ferromagnetic regime gives rise to a soliton cellular automaton which is a discrete dynamical system in which site variables take on values in a finite set. We study the scattering of a class of soliton cellular automata associated with the $U_q(D_n^{(1)})$-perfect crystal $B^{2,s}$. We calculate the combinatorial $R$ matrix for all elements of $B^{2,s} \otimes B^{2,1}$. In particular, we show that the scattering rule for our soliton cellular automaton can be identified with the combinatorial $R$ matrix for $U_q(A_1^{(1)}) \oplus U_q(D_{n-2}^{(1)})$-crystals.

math.QA↗
arXiv2011

Soliton cellular automaton associated with $G_2^{(1)}$ crystal base

We calculate the combinatorial $R$ matrix for all elements of $\mathcal{B}_l\otimes \mathcal{B}_1$ where $\mathcal{B}_l$ denotes the $G_2^{(1)}$-perfect crystal of level $l$, and then study the soliton cellular automaton constructed from it. The solitons of length $l$ are identified with elements of the $A_1^{(1)}$-crystal $\tilde{\mathcal{B}}_{3l}$. The scattering rule for our soliton cellular automaton is identified with the combinatorial $R$ matrix for $A_1^{(1)}$-crystals.

math.QA↗
arXiv2010

Zero Action on Perfect Crystals for U_q(G_2^{(1)})

The actions of 0-Kashiwara operators on the $U'_q(G_2^{(1)})$-crystal $B_l$ in [Yamane S., J. Algebra 210 (1998), 440-486] are made explicit by using a similarity technique from that of a $U'_q(D_4^{(3)})$-crystal. It is shown that $\{B_l\}_{l\ge1}$ forms a coherent family of perfect crystals.

math.QA↗
arXiv2010

Ultra-discretization of the D_4^3-Geometric Crystals to the G_2^1-Perfect Crystals

Let g be an affine Lie algebra and g^L be its Langlands dual. It is conjectured that g has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for g^L. We prove that the ultra-discretization of the positive geometric crystal for g = D_4^3 given by Igarashi and Nakashima is isomorphic to the limit of the coherent family of perfect crystals for g^L= G_2^1 constructed recently by Misra, Mohamad and Okado.

math.QA↗
arXiv2006

A recurrence relation for characters of highest weight integrable modules for affine Lie algebras

Using certain results for the vertex operator algebras associated with affine Lie algebras we obtain recurrence relations for the characters of integrable highest weight irreducible modules for an affine Lie algebra. As an application we show that in the simply-laced level 1 case, these recurrence relations give the known characters, whose principal specializations naturally give rise to some multisum Macdonald identities.

math.QA↗