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For n ≥ 2 consider the affine Lie algebra $\widehat {s\ell }(n)$ with simple roots {αi∣0 ≤ i ≤ n − 1}. Let $V(k{\Lambda }_{0}), k \in \mathbb {Z}_{\geq 1}$ denote the integrable highest weight $\widehat {s\ell }(n)$ -module with highest weight kΛ0. It is known that there are finitely many maximal dominant weights of V (kΛ0). Using the crystal base realization of V (kΛ0) and lattice path combinatorics we examine the multiplicities of a large set of maximal dominant weights of the form $k{\Lambda }_{0} - \lambda ^{\ell }_{a,b}$ where $ \lambda ^{\ell }_{a,b} = \ell \alpha _{0} + (\ell -b)\alpha _{1} + (\ell -(b+1))\alpha _{2} + {\cdots } + \alpha _{\ell -b} + \alpha _{n-\ell +a} + 2\alpha _{n - \ell +a+1} + {\ldots } + (\ell -a)\alpha _{n-1}$ , and k ≥ a + b, $a,b \in \mathbb {Z}_{\geq 1}$ , $\max \limits \{a,b\} \leq \ell \leq \left \lfloor \frac {n+a+b}{2} \right \rfloor -1 $ . We obtain two formulae to obtain these weight multiplicities - one in terms of certain standard Young tableaux and the other in terms of certain pattern-avoiding permutations.

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.

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.

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 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 g L . Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra g = A n ( 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 A n ( 1 ) given in [24] . In the process we develop and use some lattice-path combinatorics.

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 $V(k\Lambda_0)$., Comment: 11 pages

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 Li...

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., Comment: 7 pages

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.

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 $k= 6$., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:1812.01651

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., 9 pages

We consider imaginary Verma modules for quantum affine algebra U q ( g ˆ ) , where g ˆ has Coxeter–Dynkin diagram of ADE type, and construct Kashiwara type operators and the Kashiwara algebra K q . We show that a certain quotient N q − of U q ( g ˆ ) is a simple K q -module.

In this paper, we construct a fermionic realization of the twisted toroidal Lie algebra of type $A_{2n-1}, D_{n+1}, A_{2n}$ and $D_4$ based on the newly found Moody-Rao-Yokonuma-like presentation., Comment: 22 pages. Final version. Proof of the MRY presentation upgrades that of arXiv:1605.05300

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., Comment: Introduction rewritten plus some typos corrected

We determine explicitly the maximal dominant weights for the integrable highest weight $\widehat {sl}(n)$ -modules V((k − 1)Λ 0 + Λ s ), 0 ≤ s ≤ n − 1, k ≥ 2. We give a conjecture for the number of maximal dominant weights of V(k Λ 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(k Λ 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.

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.

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.

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., Comment: Updated version to include new fermionic construction for symplectic types

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 .

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.

A perfect crystal of any level is constructed for the Kirillov–Reshetikhin module of U q ( D 4 ( 3 ) ) 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.

We construct realizations for the 2-toroidal Lie algebra associated with the Lie algebra A1 using vertex operators based on bosonic elds. In particular our construction realizes higher-level representations of the 2-toroidal algebra for any given pair of levels (k0;k1) with k00. We also construct a smaller module of level (k0;0) for the toroidal algebra from the Fock space using certain screening vertex operator, and this later representation generalizes the higher-level construction of the ane Lie algebra 2.

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., Comment: 25 pages

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., Comment: 10 pages

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.

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.

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.

We use the fusion construction in the twisted quantum affine algebras to obtain a unified method to deform the wedge product for classical Lie algebras. As a byproduct 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., 20 pages, Latex file

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.”

We construct explicitly the level-one vertex operators for the fundamental modulesV(Δ1) (i=0, 1,n) of the quantum affine algebra of typeB using free boson and fermion fields.

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 \(\mathfrak{g} = A_{n}^{(1)}\).