Leibniz algebras are non-antisymmetric generalizations of Lie algebras. In this paper, we investigate the properties of complete Leibniz algebras under certain conditions on their extensions. Additionally, we explore the properties of derivations and direct sums of Leibniz algebras, proving several results analogous to those in Lie algebras.

}, number={7}, journal={ELECTRONIC RESEARCH ARCHIVE}, author={Misra, Kailash C. and Patlertsin, Sutida and Pongprasert, Suchada and Rungratgasame, Thitarie}, year={2024}, pages={4715–4722} } @article{groves_misra_stitzinger_2024, title={Subinvariance and ascendancy in Leibniz Algebras}, volume={8}, ISSN={["1532-4125"]}, url={https://doi.org/10.1080/00927872.2024.2387050}, DOI={10.1080/00927872.2024.2387050}, abstractNote={Leibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance and ascendancy in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. Kawamoto studied ascendency in Lie algebras. In this paper we define subinvariance and asendency in Leibniz algebras and study their properties. It is shown that the signature results on subinvariance and ascendency in Lie algebras have analogs for Leibniz algebras.}, journal={COMMUNICATIONS IN ALGEBRA}, author={Groves, Emma and Misra, Kailash C. and Stitzinger, Ernie}, year={2024}, month={Aug} } @article{cao_jing_misra_wang_2024, title={Tighter parameterized monogamy relations}, volume={23}, ISSN={["1573-1332"]}, DOI={10.1007/s11128-024-04495-7}, number={8}, journal={QUANTUM INFORMATION PROCESSING}, author={Cao, Yue and Jing, Naihuan and Misra, Kailash and Wang, Yiling}, year={2024}, month={Jul} } @article{jayne_misra_2023, title={Multiplicities of maximal weights of the sℓ (n)-module V (kΛ0)}, ISSN={["1793-6829"]}, DOI={10.1142/S0219498825500987}, abstractNote={ Consider the affine Lie algebra [Formula: see text] with null root [Formula: see text], weight lattice [Formula: see text] and set of dominant weights [Formula: see text]. Let [Formula: see text] [Formula: see text] denote the integrable highest weight [Formula: see text]-module with level [Formula: see text] highest weight [Formula: see text]. Let [Formula: see text] denote the set of weights of [Formula: see text]. A weight [Formula: see text] is a maximal weight if [Formula: see text]. Let [Formula: see text] denote the set of maximal dominant weights which is known to be a finite set. The explicit description of the weights in the set [Formula: see text] is known [R. L. Jayne and K. C. Misra, On multiplicities of maximal dominant weights of [Formula: see text]-modules, Algebr. Represent. Theory 17 (2014) 1303–1321]. In papers [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb. 22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the [Formula: see text]-modules [Formula: see text], Algebr. Represent. Theory 25 (2022) 477–490], the multiplicities of certain subsets of [Formula: see text] 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 [Formula: see text]-module [Formula: see text] are given generalizing the results in [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb. 22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the [Formula: see text]-modules [Formula: see text], Algebr. Represent. Theory 25 (2022) 477–490]. }, journal={JOURNAL OF ALGEBRA AND ITS APPLICATIONS}, author={Jayne, Rebecca L. and Misra, Kailash C.}, year={2023}, month={Nov} } @article{boulware_jing_misra_2022, title={On Smith normal forms of q-Varchenko matrices}, volume={34}, ISSN={["2415-721X"]}, DOI={10.12958/adm2006}, abstractNote={In this paper, we investigate q-Varchenko matrices for some hyperplane arrangements with symmetry in two andthree dimensions, and prove that they have a Smith normal formover Z[q]. In particular, we examine the hyperplane arrangement forthe regular n-gon in the plane and the dihedral model in the spaceand Platonic polyhedra. In each case, we prove that the q-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over Z[q] and realize their congruent transformation matrices over Z[q] as well.}, number={2}, journal={ALGEBRA AND DISCRETE MATHEMATICS}, author={Boulware, N. and Jing, N. and Misra, K. C.}, year={2022}, pages={187–222} } @article{jayne_misra_2022, title={Multiplicities of Some Maximal Dominant Weights of the (sl)over-cap(n)-Modules V (k Lambda(0))}, volume={25}, ISSN={["1572-9079"]}, DOI={10.1007/s10468-021-10031-3}, number={2}, journal={ALGEBRAS AND REPRESENTATION THEORY}, author={Jayne, Rebecca L. and Misra, Kailash C.}, year={2022}, month={Apr}, pages={477–490} } @article{misra_stitzinger_yu_2021, title={Subinvariance in Leibniz algebras}, volume={567}, ISSN={["1090-266X"]}, url={https://doi.org/10.1016/j.jalgebra.2020.09.031}, DOI={10.1016/j.jalgebra.2020.09.031}, abstractNote={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.}, journal={JOURNAL OF ALGEBRA}, publisher={Elsevier BV}, author={Misra, Kailash C. and Stitzinger, Ernie and Yu, Xingjian}, year={2021}, month={Feb}, pages={128–138} } @article{misra_pongprasert_2020, title={-geometric crystal at the spin node}, volume={48}, ISSN={["1532-4125"]}, DOI={10.1080/00927872.2020.1737872}, abstractNote={Abstract Let be an affine Lie algebra with index set It is conjectured that for each Dynkin node the affine Lie algebra has a positive geometric crystal. In this paper we construct a positive geometric crystal for the affine Lie algebra corresponding to the Dynkin spin node k = 6. Communicated by Sarah Witherspoon}, number={8}, journal={COMMUNICATIONS IN ALGEBRA}, author={Misra, Kailash C. and Pongprasert, Suchada}, year={2020}, month={Aug}, pages={3382–3397} } @article{boyle_misra_stitzinger_2020, title={Complete Leibniz algebras}, volume={557}, ISSN={["1090-266X"]}, url={https://doi.org/10.1016/j.jalgebra.2020.04.016}, DOI={10.1016/j.jalgebra.2020.04.016}, abstractNote={Leibniz algebras are certain generalizations 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 a generalization of complete Lie algebras. Then we study properties of complete Leibniz algebras and their holomorphs.}, journal={JOURNAL OF ALGEBRA}, publisher={Elsevier BV}, author={Boyle, Kristen and Misra, Kailash C. and Stitzinger, Ernest}, year={2020}, month={Sep}, pages={172–180} } @article{jing_mangum_misra_2021, title={Fermionic realization of twisted toroidal Lie algebras}, volume={20}, ISSN={["1793-6829"]}, url={https://doi.org/10.1142/S0219498821501437}, DOI={10.1142/S0219498821501437}, abstractNote={ In this paper, we construct a fermionic realization of the twisted toroidal Lie algebra of type [Formula: see text] and [Formula: see text] based on the newly found Moody–Rao–Yokonuma-like presentation. }, number={08}, journal={JOURNAL OF ALGEBRA AND ITS APPLICATIONS}, publisher={World Scientific Pub Co Pte Lt}, author={Jing, Naihuan and Mangum, Chad R. and Misra, Kailash C.}, year={2021}, month={Aug} } @article{igarashi_misra_pongprasert_2019, title={D-5((1))-Geometric crystal corresponding to the Dynkin spin node i=5 and its ultra-discretization}, volume={18}, ISSN={["1793-6829"]}, DOI={10.1142/S021949881950227X}, abstractNote={ Let [Formula: see text] be an affine Lie algebra with index set [Formula: see text] and [Formula: see text] be its Langlands dual. It is conjectured that for each Dynkin node [Formula: see text] the affine Lie algebra [Formula: see text] has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for [Formula: see text]. In this paper, we construct a positive geometric crystal [Formula: see text] in the level zero fundamental spin [Formula: see text]-module [Formula: see text]. Then we define explicit [Formula: see text]-action on the level [Formula: see text] known [Formula: see text]-perfect crystal [Formula: see text] and show that [Formula: see text] is a coherent family of perfect crystals with limit [Formula: see text]. Finally, we show that the ultra-discretization of [Formula: see text] is isomorphic to [Formula: see text] as crystals which proves the conjecture in this case. }, number={12}, journal={JOURNAL OF ALGEBRA AND ITS APPLICATIONS}, author={Igarashi, Mana and Misra, Kailash C. and Pongprasert, Suchada}, year={2019}, month={Dec} } @article{misra_nakashima_2018, title={Affine geometric crystal of An(1) and limit of Kirillov–Reshetikhin perfect crystals}, volume={507}, ISSN={0021-8693}, url={http://dx.doi.org/10.1016/J.JALGEBRA.2018.03.041}, DOI={10.1016/J.JALGEBRA.2018.03.041}, abstractNote={Let g be an affine Lie algebra with index set I={0,1,2,⋯,n} and gL be its Langlands dual. It is conjectured in [16] that for each k∈I∖{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 gL. Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra g=An(1) for each Dynkin index k∈I∖{0} and show that its ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for An(1) given in [24]. In the process we develop and use some lattice-path combinatorics.}, journal={Journal of Algebra}, publisher={Elsevier BV}, author={Misra, Kailash C. and Nakashima, Toshiki}, year={2018}, month={Aug}, pages={249–291} } @article{jayne_misra_2018, title={Lattice Paths, Young Tableaux, and Weight Multiplicities}, volume={22}, ISSN={0218-0006 0219-3094}, url={http://dx.doi.org/10.1007/S00026-018-0374-4}, DOI={10.1007/S00026-018-0374-4}, abstractNote={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 shape $${\lambda \vdash \ell}$$ with $${l(\lambda) \leq k}$$ , which is also the number of (k + 1)k··· 21-avoiding permutations in $${S_\ell}$$ . Finally, we apply this result to the representation theory of the affine Lie algebra $${\widehat{sl}(n)}$$ and show that this gives the multiplicity of certain maximal dominant weights in the irreducible highest weight $${\widehat{sl}(n)}$$ -module $${V(k \Lambda_0)}$$ .}, number={1}, journal={Annals of Combinatorics}, publisher={Springer Science and Business Media LLC}, author={Jayne, Rebecca L. and Misra, Kailash C.}, year={2018}, month={Feb}, pages={147–156} } @article{misra_stitzinger_turner_2018, title={Criteria for solvability and supersolvability in Leibniz algebras}, volume={46}, ISSN={["1532-4125"]}, DOI={10.1080/00927872.2017.1372455}, abstractNote={ABSTRACT Leibniz algebras are certain generalization of Lie algebras. Recently, analyzing the structure of subalgebras, David Towers gave some criteria for the solvability and supersolvability of Lie algebras. In this paper we define analogues concepts for Leibniz algebras and extend some of these results on solvability and supersolvability to that of Leibniz algebras.}, number={5}, journal={COMMUNICATIONS IN ALGEBRA}, author={Misra, Kailash C. and Stitzinger, Ernie and Turner, Bethany}, year={2018}, pages={2083–2088} } @article{cox_futorny_misra_2017, title={Imaginary Verma modules for U-q<((sl(2)))over cap> and crystal-like bases}, volume={481}, ISSN={["1090-266X"]}, DOI={10.1016/j.jalgebra.2017.02.017}, abstractNote={We consider imaginary Verma modules for quantum affine algebra Uq(sl(2)ˆ) and define a crystal-like base which we call an imaginary crystal basis using the Kashiwara algebra Kq constructed in earlier work of the authors. In particular, we prove the existence of imaginary crystal-like bases for a suitable category of reduced imaginary Verma modules for Uq(sl(2)ˆ).}, journal={JOURNAL OF ALGEBRA}, author={Cox, Ben and Futorny, Vyacheslav and Misra, Kailash C.}, year={2017}, month={Jul}, pages={12–35} } @article{demir_misra_stitzinger_2017, title={On classification of four-dimensional nilpotent Leibniz algebras}, volume={45}, ISSN={["1532-4125"]}, DOI={10.1080/00927872.2016.1172626}, abstractNote={ABSTRACT 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.}, number={3}, journal={COMMUNICATIONS IN ALGEBRA}, author={Demir, Ismail and Misra, Kailash C. and Stitzinger, Ernie}, year={2017}, pages={1012–1018} } @article{misra_wilson_2016, title={Root Multiplicities of the Indefinite Kac-Moody Algebra HDn(1)}, volume={44}, ISSN={["1532-4125"]}, DOI={10.1080/00927872.2015.1027369}, abstractNote={In this article, we study the root multiplicities of the indefinite type Kac–Moody algebra . We obtain the multiplicities of infinite family of roots and conjecture that they are polynomials in n of certain degree. Our approach is to view the roots of as weights of certain integrable -modules and then use the path realization of crystal bases for these modules and Kang's multiplicity formula to find their multiplicities. In particular, we prove that mult(−2α−1 − kδ) is a polynomial in n of degree k. We observe that Frenkel's conjectured root multiplicity bound holds in this case.}, number={4}, journal={COMMUNICATIONS IN ALGEBRA}, author={Misra, Kailash C. and Wilson, Evan A.}, year={2016}, pages={1599–1614} } @article{cox_futorny_misra_2015, title={An imaginary PBW basis for quantum affine algebras of type 1}, volume={219}, ISSN={["1873-1376"]}, DOI={10.1016/j.jpaa.2014.04.011}, abstractNote={Let gˆ be an affine Lie algebra of type 1. We give a PBW basis for the quantum affine algebra Uq(gˆ) with respect to the triangular decomposition of gˆ associated with the imaginary positive root system.}, number={1}, journal={JOURNAL OF PURE AND APPLIED ALGEBRA}, author={Cox, Ben and Futorny, Vyacheslav and Misra, Kailash C.}, year={2015}, month={Jan}, pages={83–100} } @article{demir_misra_stitzinger_2016, title={Classification of Some Solvable Leibniz Algebras}, volume={19}, ISSN={["1572-9079"]}, DOI={10.1007/s10468-015-9580-5}, abstractNote={Leibniz algebras are certain generalization of Lie algebras. In this paper we give classification of non-Lie solvable (left) Leibniz algebras of dimension ≤ 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.}, number={2}, journal={ALGEBRAS AND REPRESENTATION THEORY}, author={Demir, Ismail and Misra, Kailash C. and Stitzinger, Ernie}, year={2016}, month={Apr}, pages={405–417} } @article{cox_futorny_misra_2015, title={Imaginary Verma modules and Kashiwara algebras for U-q((g)over-cap)}, volume={424}, journal={Journal of Algebra}, author={Cox, B. and Futorny, V. and Misra, K. C.}, year={2015}, pages={390–415} } @article{cox_futorny_misra_2015, title={Imaginary Verma modules and Kashiwara algebras for Uq(gˆ)}, volume={424}, ISSN={0021-8693}, url={http://dx.doi.org/10.1016/J.JALGEBRA.2014.09.025}, DOI={10.1016/J.JALGEBRA.2014.09.025}, abstractNote={We consider imaginary Verma modules for quantum affine algebra Uq(gˆ), where gˆ has Coxeter–Dynkin diagram of ADE type, and construct Kashiwara type operators and the Kashiwara algebra Kq. We show that a certain quotient Nq− of Uq(gˆ) is a simple Kq-module.}, journal={Journal of Algebra}, publisher={Elsevier BV}, author={Cox, Ben and Futorny, Vyacheslav and Misra, Kailash C.}, year={2015}, month={Feb}, pages={390–415} } @article{khuhirun_misra_stitzinger_2015, title={On nilpotent Lie algebras of small breadth}, volume={444}, ISSN={["1090-266X"]}, DOI={10.1016/j.jalgebra.2015.07.036}, abstractNote={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.}, journal={JOURNAL OF ALGEBRA}, author={Khuhirun, Borworn and Misra, Kailash C. and Stitzinger, Ernie}, year={2015}, month={Dec}, pages={328–338} } @article{jayne_misra_2014, title={On Multiplicities of Maximal Weights of -Modules}, volume={17}, ISSN={["1572-9079"]}, DOI={10.1007/s10468-014-9470-2}, number={4}, journal={ALGEBRAS AND REPRESENTATION THEORY}, author={Jayne, Rebecca L. and Misra, Kailash C.}, year={2014}, month={Aug}, pages={1303–1321} } @article{dunbar_jing_misra_2014, title={REALIZATION OF (sl)over-cap(2)(C) AT THE CRITICAL LEVEL}, volume={16}, ISSN={["1793-6683"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84899490482&partnerID=MN8TOARS}, DOI={10.1142/s0219199714500060}, abstractNote={ An explicit realization of the affine Lie algebra [Formula: see text] 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. }, number={2}, journal={COMMUNICATIONS IN CONTEMPORARY MATHEMATICS}, author={Dunbar, Jonathan and Jing, Naihuan and Misra, Kailash C.}, year={2014}, month={Apr} } @inproceedings{misra_wilson_2014, title={Tensor product decomposition of (sl)over-cap(n) modules and identities}, volume={627}, booktitle={Ramanujan 125}, author={Misra, K. C. and Wilson, E. A.}, year={2014}, pages={131–144} } @inbook{misra_nakashima_2013, place={London}, series={Springer Proceedings in Mathematics & Statistics}, title={An(1)-Geometric Crystal Corresponding to Dynkin Index i=2 and Its Ultra-Discretization}, ISBN={9781447148623 9781447148630}, ISSN={2194-1009 2194-1017}, url={http://dx.doi.org/10.1007/978-1-4471-4863-0_12}, DOI={10.1007/978-1-4471-4863-0_12}, abstractNote={Let $\mathfrak{g}$ be an affine Lie algebra with index set I={0,1,2,…,n} and $\mathfrak{g}^{L}$ be its Langlands dual. It is conjectured in Kashiwara et al. (Trans. Am. Math. Soc. 360(7):3645–3686, 2008) that for each i∈I∖{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 $\mathfrak{g}^{L}$ . We prove this conjecture for i=2 and $\mathfrak{g} = A_{n}^{(1)}$ .}, booktitle={Symmetries, Integrable Systems and Representations}, publisher={Springer}, author={Misra, Kailash C. and Nakashima, Toshiki}, editor={Iohara, K. and Morier-Genoud, S. and Remy, B.Editors}, year={2013}, pages={297–318}, collection={Springer Proceedings in Mathematics & Statistics} } @article{misra_wilson_2013, title={ON TENSOR PRODUCT DECOMPOSITION OF (sl)over-cap(n) MODULES}, volume={12}, ISSN={["1793-6829"]}, DOI={10.1142/s0219498813500540}, abstractNote={ We decompose the [Formula: see text]-module V(Λ0) ⊗ V(Λ0) and give generating function identities for the outer multiplicities. In the process we discover an infinite family of partition identities, which are seemingly new even in the n = 3 case. }, number={8}, journal={JOURNAL OF ALGEBRA AND ITS APPLICATIONS}, author={Misra, Kailash C. and Wilson, Evan A.}, year={2013}, month={Dec} } @article{misra_wilson_2013, title={Soliton cellular automaton associated with D-n((1))-crystal B-2,B-s}, volume={54}, ISSN={["1089-7658"]}, DOI={10.1063/1.4801448}, abstractNote={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 \documentclass[12pt]{minimal}\begin{document}$U_q(D_n^{(1)})$\end{document}Uq(Dn(1))-perfect crystal B2, s. We calculate the combinatorial R matrix for all elements of B2, s ⊗ B2, 1. In particular, we show that the scattering rule for our soliton cellular automaton can be identified with the combinatorial R matrix for \documentclass[12pt]{minimal}\begin{document}$U_q(A_1^{(1)}) \oplus U_q(D_{n-2}^{(1)})$\end{document}Uq(A1(1))⊕Uq(Dn−2(1))-crystals.}, number={4}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Misra, Kailash C. and Wilson, Evan A.}, year={2013}, month={Apr} } @article{beier_misra_2012, title={Demazure Crystals and Extended Young Diagrams}, volume={19}, ISSN={["1005-3867"]}, DOI={10.1142/s1005386712000193}, abstractNote={ For a suitable sequence of Weyl group elements {w(l)}l≥0 we give explicit combinatorial descriptions of the crystals Bw(l)(kΛ0) for the Demazure modules Vw(l)(kΛ0) of the quantum affine Lie algebra [Formula: see text] in terms of extended Young diagrams. }, number={2}, journal={ALGEBRA COLLOQUIUM}, author={Beier, Julie C. and Misra, Kailash C.}, year={2012}, month={Jun}, pages={293–304} } @article{misra_okado_wilson_2012, title={Soliton cellular automaton associated with G(2)((1)) crystal base}, volume={53}, ISSN={["1089-7658"]}, DOI={10.1063/1.3673541}, abstractNote={We calculate the combinatorial R matrix for all elements of $\mathcal {B}_l\otimes \mathcal {B}_1$Bl⊗B1 where $\mathcal {B}_l$Bl denotes the $G_2^{(1)}$G2(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 $A_1^{(1)}$A1(1)-crystal $\tilde{\mathcal {B}}_{3l}$B̃3l. The scattering rule for our soliton cellular automaton is identified with the combinatorial R matrix for $A_1^{(1)}$A1(1)-crystals.}, number={1}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Misra, Kailash C. and Okado, Masato and Wilson, Evan A.}, year={2012}, month={Jan} } @article{igarashi_misra_nakashima_2012, title={ULTRA-DISCRETIZATION OF THE D(4)((3)-)GEOMETRIC CRYSTAL TO THE G(2)((1))-PERFECT CRYSTALS}, volume={255}, ISSN={["0030-8730"]}, DOI={10.2140/pjm.2012.255.117}, abstractNote={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.}, number={1}, journal={PACIFIC JOURNAL OF MATHEMATICS}, author={Igarashi, Mana and Misra, Kailash C. and Nakashima, Toshiki}, year={2012}, month={Jan}, pages={117–142} } @article{jing_misra_2010, title={Fermionic realization of toroidal Lie algebras of classical types}, volume={324}, ISSN={["1090-266X"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77952959663&partnerID=MN8TOARS}, DOI={10.1016/j.jalgebra.2010.03.021}, abstractNote={We use fermionic operators to construct toroidal Lie algebras of classical types, including in particular that of symplectic affine algebras, which is first realized by fermions.}, number={2}, journal={JOURNAL OF ALGEBRA}, author={Jing, Naihuan and Misra, Kailash C.}, year={2010}, month={Jul}, pages={183–194} } @article{jayne_misra_2011, title={ON DEMAZURE CRYSTALS FOR Uq(G(2)((1)))}, volume={139}, ISSN={["1088-6826"]}, DOI={10.1090/s0002-9939-2010-10663-9}, abstractNote={We show that there exist suitable sequences $\{w^{(k)}\}_{k \ge 0}$ and $\{w'^{(k)}\}_{k \ge 0}$ of Weyl group elements for a given perfect crystal of level $l\ge 1$ such that the path realizations of the Demazure crystals $B_{w^{(k)}}(l\Lambda _0)$ and $B_{w'^{(k)}}(l\Lambda _2)$ for the quantum affine algebra $U_q(G_2^{(1)})$ have tensor-product-like structures with mixing index $\kappa =1$.}, number={7}, journal={PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Jayne, Rebecca L. and Misra, Kailash C.}, year={2011}, month={Jul}, pages={2343–2356} } @article{misra_mohamad_okado_2010, title={Zero Action on Perfect Crystals for Uq(G(2)((1)))}, volume={6}, ISSN={["1815-0659"]}, DOI={10.3842/sigma.2010.022}, abstractNote={The actions of 0-Kashiwara operators on the U1q(G(1)2)-crystal Bl in [Yamane S., J. Algebra 210 (1998), 440–486] are made explicit by using a similarity technique from that of a U1q(D(3)4)-crystal. It is shown that {Bl}l≥1 forms a coherent family of perfect crystals.}, journal={SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS}, author={Misra, Kailash C. and Mohamad, Mahathir and Okado, Masato}, year={2010} } @article{jing_misra_xu_2009, title={BOSONIC REALIZATION OF TOROIDAL LIE ALGEBRAS OF CLASSICAL TYPES}, volume={137}, ISSN={["1088-6826"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77950857029&partnerID=MN8TOARS}, DOI={10.1090/S0002-9939-09-09942-0}, abstractNote={We use fermionic operators to construct toroidal Lie algebras of classical types, including in particular that of symplectic affine algebras, which is first realized by fermions.}, number={11}, journal={PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Jing, Naihuan and Misra, Kailash C. and Xu, Chongbin}, year={2009}, month={Nov}, pages={3609–3618} } @article{klima_misra_2008, title={Root multiplicities of the indefinite Kac-Moody algebras of symplectic type}, volume={36}, ISSN={["0092-7872"]}, DOI={10.1080/00927870701724367}, abstractNote={We study the root multiplicities of the indefinite Kac–Moody algebras by viewing them as weight multiplicities of certain integrable -modules. Then using Kang's root multiplicity formula and the path crystal for integrable -modules we calculate the multiplicities of a family of roots for . In particular, we show that for any positive integer k the multiplicity of − 2α−1 − kδ as a root of is a polynomial in n of degree at most k. Furthermore, we observe that Frenkel's conjectured root multiplicity bound does not hold for roots of .}, number={2}, journal={COMMUNICATIONS IN ALGEBRA}, author={Klima, Vicky W. and Misra, Kaillash C.}, year={2008}, month={Feb}, pages={764–782} } @article{cook_li_misra_2007, title={A recurrence relation for characters of highest weight integrable modules for affine Lie algebras}, volume={9}, ISSN={["1793-6683"]}, DOI={10.1142/S0219199707002368}, abstractNote={ 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. }, number={2}, journal={COMMUNICATIONS IN CONTEMPORARY MATHEMATICS}, author={Cook, William J. and Li, Haisheng and Misra, Kailash C.}, year={2007}, month={Apr}, pages={121–133} } @article{kashiwara_misra_okado_yamada_2007, title={Perfect crystals for U-q (D-4((3)))}, volume={317}, ISSN={["1090-266X"]}, DOI={10.1016/j.jalgebra.2007.02.021}, abstractNote={A perfect crystal of any level is constructed for the Kirillov–Reshetikhin module of [Math Processing Error] corresponding to the middle vertex of the Dynkin diagram. The actions of Kashiwara operators are given explicitly. It is also shown that this family of perfect crystals is coherent. A uniqueness problem solved in this paper can be applied to other quantum affine algebras.}, number={1}, journal={JOURNAL OF ALGEBRA}, author={Kashiwara, M. and Misra, K. C. and Okado, M. and Yamada, D.}, year={2007}, month={Nov}, pages={392–423} } @article{kim_misra_stitzinger_2004, title={On the nilpotency of certain subalgebras of Kac-Moody Lie algebras}, volume={14}, number={1}, journal={Journal of Lie Theory}, author={Kim, Y. and Misra, K. C. and Stitzinger, E.}, year={2004}, pages={23-} } @article{hontz_misra_2002, title={On root multiplicities of HA(n)((1))}, volume={12}, ISSN={["0218-1967"]}, DOI={10.1142/S0218196702000730}, abstractNote={ We determine the root multiplicities of the Kac–Moody Lie algebra [Formula: see text] of indefinite type using a recursive root multiplicity formula due to Kang. We view [Formula: see text] as a representation of its subalgebra [Formula: see text] and then use the combinatorics of the irreducible representations of [Formula: see text] to determine the root multiplicities. }, number={3}, journal={INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION}, author={Hontz, J and Misra, KC}, year={2002}, month={Jun}, pages={477–508} } @article{hontz_misra_2002, title={Root multiplicities of the indefinite Kac-Moody Lie algebras HD4(3) and HG(2)((1))}, volume={30}, ISSN={["0092-7872"]}, DOI={10.1081/AGB-120004000}, abstractNote={ABSTRACT We determine the root multiplicities of the Kac-Moody Lie algebras and of indefinite type using a recursive root multiplicity formula due to Kang. We view both algebras as representations of its subalgebra and then use the combinatorics of the irreducible representations of to determine the root multiplicities.}, number={6}, journal={COMMUNICATIONS IN ALGEBRA}, author={Hontz, J and Misra, KC}, year={2002}, pages={2941–2959} } @article{hara_jing_misra_2001, title={BRST resolution for the principally graded Wakimoto module of (sl)over-cap(2)}, volume={58}, ISSN={["0377-9017"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0041814659&partnerID=MN8TOARS}, DOI={10.1023/A:1014559525117}, number={3}, journal={LETTERS IN MATHEMATICAL PHYSICS}, author={Hara, Y and Jing, NH and Misra, K}, year={2001}, month={Dec}, pages={181–188} } @article{jing_misra_savage_2001, title={On multi-color partitions and the generalized Rogers-Ramanujan identities}, volume={3}, ISSN={["0219-1997"]}, DOI={10.1142/S0219199701000482}, abstractNote={ Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers–Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers–Ramanujan identities in terms of multi-color partitions. }, number={4}, journal={COMMUNICATIONS IN CONTEMPORARY MATHEMATICS}, author={Jing, NH and Misra, KC and Savage, CD}, year={2001}, month={Nov}, pages={533–548} } @article{benkart_kang_lee_misra_shin_2001, title={The polynomial behavior of weight multiplicities for the affine Kac-Moody algebras A(r)((1))}, volume={126}, ISSN={["0010-437X"]}, DOI={10.1023/A:1017584131106}, abstractNote={We prove that the multiplicity of an arbitrary dominant weight for an irreducible highest weight representation of the affine Kac–Moody algebra A(1)r is a polynomial in the rank r. In the process we show that the degree of this polynomial is less than or equal to the depth of the weight with respect to the highest weight. These results allow weight multiplicity information for small ranks to be transferred to arbitrary ranks.}, number={1}, journal={COMPOSITIO MATHEMATICA}, author={Benkart, G and Kang, SJ and Lee, H and Misra, KC and Shin, DU}, year={2001}, month={Mar}, pages={91–111} } @article{kuniba_misra_okado_takagi_uchiyama_2000, title={Paths, Demazure crystals, and symmetric functions}, volume={41}, ISSN={["1089-7658"]}, DOI={10.1063/1.1286284}, abstractNote={The path realization of Demazure crystals is reviewed and Demazure characters in the light of symmetric functions are discussed.}, number={9}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Kuniba, A and Misra, KC and Okado, M and Takagi, T and Uchiyama, J}, year={2000}, month={Sep}, pages={6477–6486} } @article{jing_misra_okado_2000, title={q-wedge modules for quantized enveloping algebras of classical type}, volume={230}, ISSN={["0021-8693"]}, DOI={10.1006/jabr.2000.8325}, abstractNote={Abstract We use the fusion construction in twisted quantum affine algebras to obtain a unified method to deform the wedge product for classical Lie algebras. As a by-product we uniformly realize all non-spin fundamental modules for quantized enveloping algebras of classical types, and show that they admit natural crystal bases as modules for the (derived) twisted quantum affine algebra. These crystal bases are parametrized in terms of the q -wedge products.}, number={2}, journal={JOURNAL OF ALGEBRA}, author={Jing, NH and Misra, KC and Okado, M}, year={2000}, month={Aug}, pages={518–539} } @book{jing_misra_1999, title={Recent developments in quantum affine algebras and related topics: Representations of affine and quantum affine algebras and their applications, North Carolina State University, May 21-24, 1998 / Naihuan Jing, Kailash C. Misra, editors}, ISBN={0821811991}, DOI={10.1090/conm/248}, abstractNote={The polynomial behavior of weight multiplicities for classical simple Lie algebras and classical affine Kac-Moody algebras by G. Benkart, S.-J. Kang, H. Lee, and D.-U. Shin A note on embeddings of some Lie algebras defined by matrices by S. Berman and S. Tan Principal realization for the extended affine Lie algebra of type $sl_2$ with coordinates in a simple quantum torus with two generators by S. Berman and J. Szmigielski Monomial bases of quantized enveloping algebras by V. Chari and N. Xi Quantized W-algebra of ${\mathfrak sl}(2,1)$: a construction from the quantization of screening operators by J. Ding and B. Feigin Affine algebras and non-perturbative symmetries in superstring theory by L. Dolan Automorphism groups and twisted modules for lattice vertex operator algebras by C. Dong and K. Nagatomo Truncated meanders by P. Di Francesco The $q$-characters of representations of quantum affine algebras and deformations of $\mathcal W$-algebras by E. Frenkel and N. Reshetikhin Melzer's identities revisited by O. Foda and T. A. Welsh Automorphisms of lattice type vertex operator algebras and variations, a survey by R. L. Griess, Jr. Remarks on fermionic formula by G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada $q$-vertex operators for quantum affine algebras by N. Jing and K. C. Misra Homology of certain truncated Lie algebras by S. Kumar Vertex operator algebras and the zeta function by J. Lepowsky On $\mathbb Z$-graded associative algebras and their $\mathbb N$-graded modules by H. Li and S. Wang An $\mathbb A$-form technique of quantum deformations by D. J. Melville Determinant formula for the solutions of the quantum Knizhnik-Zamolodchikov equation with $q=1$ by T. Miwa and Y. Takeyama Functorial properties of the hypergeometric map by E. Mukhin and A. Varchenko Polyhedral realizations of crystal bases and braid-type isomorphisms by T. Nakashima Meromorphic tensor categories, quantum affine and chiral algebras I by Y. Soibelman Dual pairs and infinite dimensional Lie algebras by W. Wang.}, publisher={Providence, RI: American Mathematical Society}, author={Jing, Naihuan and Misra, K. C.}, year={1999} } @article{jing_misra_1999, title={Vertex operators for twisted quantum affine algebras}, volume={351}, ISSN={["0002-9947"]}, DOI={10.1090/S0002-9947-99-02098-X}, abstractNote={We construct explicitly the q-vertex operators (intertwining operators) for the level one modules V (�i) of the classical quantum affine algebras of twisted types using interacting bosons, where i = 0,1 for A (2)n−1, i = 0 for D (3) , i = 0, n for D (2)+1, and i = n for A (2)n . A perfect crystal graph for D (3) 4 is constructed as a by-product.}, number={4}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Jing, NH and Misra, KC}, year={1999}, month={Apr}, pages={1663–1690} } @article{jing_koyama_misra_1998, title={Bosonic realizations of U-q(C-n((1)))}, volume={200}, ISSN={["0021-8693"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0039993885&partnerID=MN8TOARS}, DOI={10.1006/jabr.1997.7227}, abstractNote={Abstract We construct explicitly the quantum symplectic affine algebraUq( sp 2n) using bosonic fields. The Fock space decomposes into irreducible modules of level −1/2, quantizing the Feingold–Frenkel construction forq = 1.}, number={1}, journal={JOURNAL OF ALGEBRA}, author={Jing, NH and Koyama, Y and Misra, KC}, year={1998}, month={Feb}, pages={155–172} } @article{kuniba_misra_okado_takagi_uchiyama_1998, title={Characters of Demazure modules and solvable lattice models}, volume={510}, DOI={10.1016/s0550-3213(97)00685-8}, abstractNote={We study the path realization of Demazure crystals related to solvable lattice models in statistical mechanics. Various characters are represented in a unified way as the sums over one-dimensional configurations which we call unrestricted, classically restricted and restricted paths. As an application, characters of Demazure modules are obtained in terms of q-multinomial coefficients for several level-1 modules of classical affine algebras.}, number={3}, journal={Nuclear Physics. B}, author={Kuniba, A. and Misra, K. C. and Okado, M. and Takagi, T. and Uchiyama, J.}, year={1998}, pages={555–576} } @article{kaniba_misra_okado_takagi_uchiyama_1998, title={Crystals for Demazure modules of classical affine Lie algebras}, volume={208}, DOI={10.1006/jabr.1998.7503}, abstractNote={Abstract We study, in the path realization, crystals for Demazure modules of affine Lie algebras of types A (1) n , B (1) n , C (1) n , D (1) n , A (2) 2 n − 1 , A (2) 2 n , and D (2) n + 1 . We find a special sequence of affine Weyl group elements for the selected perfect crystal, and show that if the highest weight is l Λ 0 , the Demazure crystal has a remarkably simple structure.}, number={1}, journal={Journal of Algebra}, author={Kaniba, A. and Misra, K. C. and Okado, M. and Takagi, T. and Uchiyama, J.}, year={1998}, pages={185–215} } @article{kuniba_misra_okado_uchiyama_1998, title={Demazure modules and perfect crystals}, volume={192}, ISSN={["0010-3616"]}, DOI={10.1007/s002200050309}, number={3}, journal={COMMUNICATIONS IN MATHEMATICAL PHYSICS}, author={Kuniba, A and Misra, KC and Okado, M and Uchiyama, J}, year={1998}, month={Apr}, pages={555–567} } @article{foda_misra_okado_1998, title={Demazure modules and vertex models: The (sl)over-cap(2) case}, volume={39}, ISSN={["0022-2488"]}, DOI={10.1063/1.532402}, abstractNote={We characterize, in the case of sl∧(2),the crystal base of the Demazure module Ew(Λ) in terms of extended Young diagrams or paths for any dominant integral weight Λ and Weyl group element w. Its character is evaluated via two expressions, “bosonic” and “fermionic.”}, number={3}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Foda, O and Misra, KC and Okado, M}, year={1998}, month={Mar}, pages={1601–1622} } @article{misra_1987, title={Basic representations of some affine Lie algebras and generalized Euler identities}, volume={42}, ISSN={0263-6115}, url={http://dx.doi.org/10.1017/S1446788700028585}, DOI={10.1017/S1446788700028585}, abstractNote={AbstractWe consider certain affine Kac-Moody Lie algebras. We give a Lie theoretic interpretation of the generalized Euler identities by showing that they are associated with certain filtrations of the basic representations of these algebras. In the case when the algebras have prime rank, we also give algebraic proofs of the corresponding identities.}, number={3}, journal={Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics}, publisher={Cambridge University Press (CUP)}, author={Misra, Kailash C.}, year={1987}, month={Jun}, pages={296–311} } @article{adamovic_jing_misra_2017, title={ON PRINCIPAL REALIZATION OF MODULES FOR THE AFFINE LIE ALGEBRA A(1)((1)) AT THE CRITICAL LEVEL}, volume={369}, ISSN={["1088-6850"]}, DOI={10.1090/tran/7009}, abstractNote={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 ${\Bbb C}[\alpha(-1/2), \alpha(-3/2), ...]$. Explicit combinatorial bases for these modules are also given.}, number={7}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Adamovic, Drazen and Jing, Naihuan and Misra, Kailash C.}, year={2017}, month={Jul}, pages={5113–5136} }