Affine Lie Algebra Of Type Research Articles

Towards geometric Satake correspondence for Kac-Moody algebras --- Cherkis bow varieties and affine Lie algebras of type $A$

Towards geometric Satake correspondence for Kac-Moody algebras --- Cherkis bow varieties and affine Lie algebras of type $A$

On string functions and double-sum formulas

String functions are important building blocks of characters of integrable highest modules over affine Kac--Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type $A_{1}^{(1)}$ in terms of Dedekind eta functions. We produce new relations between string functions by writing them as double-sums and then using certain symmetry relations. We evaluate the series using special double-sum formulas that express Hecke-type double-sums in terms of Appell--Lerch functions and theta functions, where we point out that Appell--Lerch functions are the building blocks of Ramanujan's classical mock theta functions.

New partition identities from \(C^{(1)}_\ell\)-modules

In this paper we conjecture combinatorial Rogers-Ramanu­jan type colored partition identities related to standard representations of the affine Lie algebra of type \(C^{(1)}_\ell\), \(\ell\geq2\), and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras.

NONRECURSIVE CANONICAL BASIS COMPUTATIONS FOR LOW RANK KASHIWARA CRYSTALS OF TYPE A

We give an improved algorithm for the action of the divided power of a Chevalley basis element of an affine Lie algebra of type A on canonical basis elements satisfying an easily checked uniformity condition and compare calculation times for our algorithm against the standard algorithm. For symmetric Kashiwara crystals of affine type A and rank e=2, and for the canonical basis elements that we call external, corresponding to weights on the outer skin of the Kashiwara crystal, we construct the canonical basis elements in a nonrecursive manner. In particular, for a symmetric crystal with Λ=aΛ0+aΛ1, we give formulae for the canonical basis elements for all the e-regular multipartitions with defects either k(a−k) or k(a−k)+2a, for 0≤k≤a.

Quadratic Relations of the Deformed W-Algebra for the Twisted Affine Lie Algebra of Type A<sup>(2)</sup><sub>2N</sub>

We revisit the free field construction of the deformed $W$-algebra by Frenkel and Reshetikhin [Comm. Math. Phys. 197 (1998), 1-32], where the basic $W$-current has been identified. Herein, we establish a free field construction of higher $W$-currents of the deformed $W$-algebra associated with the twisted affine Lie algebra $A_{2N}^{(2)}$. We obtain a closed set of quadratic relations and duality, which allows us to define deformed $W$-algebra ${\mathcal W}_{x,r}\big(A_{2N}^{(2)}\big)$ using generators and relations.

On the R-matrix realization of quantum loop algebras

We consider rr-matrix realization of the quantum deformations of the loop algebras \tilde{\mathfrak{g}}𝔤̃ corresponding to non-exceptional affine Lie algebras of type \widehat{\mathfrak{g}}=A^{(1)}_{N-1}𝔤̂=AN−1(1), B^{(1)}_nBn(1), C^{(1)}_nCn(1), D^{(1)}_nDn(1), A^{(2)}_{N-1}AN−1(2). For each U_q(\tilde{\mathfrak{g}})Uq(𝔤̃) we investigate the commutation relations between Gauss coordinates of the fundamental \mathbb{L}𝕃-operators using embedding of the smaller algebra into bigger one. The new realization of these algebras in terms of the currents is given. The relations between all off-diagonal Gauss coordinates and certain projections from the ordered products of the currents are presented. These relations are important in applications to the quantum integrable models.

Harmonic analysis of boxed hyperoctahedral Hall-Littlewood polynomials

For positive integers n and c with c≥2n (= the Coxeter number of the hyperoctahedral group of signed permutations of degree n), we present a finite-dimensional discrete orthogonality relation for Macdonald's three-parameter hyperoctahedral Hall-Littlewood polynomials of degree at most c in each of the n variables. These polynomials are labeled by partitions λ that fit inside a rectangular box of shape cn, i.e. the partitions in question are of length ≤n and have parts of size ≤c. We employ coordinate patches around the vertices of the alcove of boxed partitions Λ(n,c)={λ⊆cn} to establish the self-adjointness of a one-parameter family of commuting discrete difference operators ▪ acting on functions f:Λ(n,c)→C. By construction, the basis of hyperoctahedral Hall-Littlewood polynomials constitutes a joint eigenbasis for ▪ with simple spectrum, which gives rise to our discrete orthogonality relation. From the point of view of quantum integrable particle dynamics, the present geometric construction establishes the orthogonality of the Bethe Ansatz eigenfunctions for a recently studied q-boson system on a finite lattice with integrable open-end boundary conditions. The Bethe Ansatz equations enter in the geometric picture as compatibility conditions between coordinate patches stemming from distinct vertices. Two applications of the orthogonality relations are highlighted: (i) a cubature rule for the integration of symmetric functions with respect to the Haar measure on the compact symplectic group Sp(n), and (ii) a Verlinde formula for the structure constants of a deformation of the Wess-Zumino-Witten fusion ring associated with the affine Lie algebra of type Cˆn (= Cn(1)).

Affine Pieri rule for periodic Macdonald spherical functions and fusion rings

Let gˆ be an untwisted affine Lie algebra or the twisted counterpart thereof (which excludes the affine Lie algebras of type BCˆn=A2n(2)). We present an affine Pieri rule for a basis of periodic Macdonald spherical functions associated with gˆ. In type Aˆn−1=An−1(1) the formula in question reproduces an affine Pieri rule for cylindric Hall-Littlewood polynomials due to Korff, which at t=0 specializes in turn to a well-known Pieri formula in the fusion ring of genus zero slˆ(n)c-Wess-Zumino-Witten conformal field theories.

A note on principal subspaces of the affine Lie algebras in types and

We construct quasi-particle bases of principal subspaces of standard modules where and Λj denotes the fundamental weight of affine Lie algebras of type or of level one. From the given bases we find characters of principal subspaces.

Modular Representation Theory of Affine and Cyclotomic Yokonuma-Hecke Algebras

We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type A (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic p. Secondly, the modular branching rules for these algebras are obtained; moreover, the resulting modular branching graphs for cyclotomic Yokonuma-Hecke algebras are identified with crystal graphs of irreducible integrable representations of affine Lie algebras of type A.

Frobenius structures and characters of affine Lie algebras

The explicit description of the Frobenius structure for the elliptic root system of type $D_4^{(1,1)}$ in terms of the characters of an affine Lie algebra of type $D_4^{(1)}$ is given.

Leading terms of relations for standard modules of the affine Lie algebras $$C_{n}^{(1)}$$ C n ( 1 )

In this paper, we give a combinatorial parametrization of leading terms of defining relations for the vacuum level k standard modules for the affine Lie algebra of type $$C_{n}^{(1)}$$ . Using this parametrization, we conjecture colored Rogers–Ramanujan type combinatorial identities for $$n\ge 2$$ and $$k\ge 2$$ ; the identity in the case $$n=k=1$$ is equivalent to one of Capparelli’s identities.

Demazure flags, q-Fibonacci polynomials and hypergeometric series

We study a family of finite-dimensional representations of the hyperspecial parabolic subalgebra of the twisted affine Lie algebra of type $$\mathtt A_2^{(2)}$$ . We prove that these modules admit a decreasing filtration whose sections are isomorphic to stable Demazure modules in an integrable highest weight module of sufficiently large level. In particular, we show that any stable level $$m'$$ Demazure module admits a filtration by level m Demazure modules for all $$m\ge m'$$ . We define the graded and weighted generating functions which encode the multiplicity of a given Demazure module and establish a recursive formulae. In the case when $$m'=1,2$$ and $$m=2,3$$ , we determine these generating functions completely and show that they define hypergeometric series and that they are related to the q-Fibonacci polynomials defined by Carlitz.

Bases of standard modules for affine Lie algebras of type

ABSTRACTFeigin-Stoyanovsky’s type subspaces for affine Lie algebras of type have monomial bases with a nice combinatorial description. We describe bases of whole standard modules in terms of semi-infinite monomials obtained as “a limit of translations” of bases for Feigin-Stoyanovsky’s type subspaces.

Quasi-particle bases of principal subspaces of the affine Lie algebra of type G_2^{(1)}

The aim of this work is to construct the quasi-particle basis of principal subspace of standard module of highest weight $k\Lambda_0$ of level $k\geq 1$ of affine Lie algebra of type $G_2^{;(1)};$ by means of which we obtain the basis of principal subspace of generalized Verma module.

The superconformal index of the (2,0) theory with defects

We compute the supersymmetric partition function of the six-dimensional $(2,0)$ theory of type $A_{N-1}$ on $S^1 \times S^5$ in the presence of both codimension two and codimension four defects. We concentrate on a limit of the partition function depending on a single parameter. From the allowed supersymmetric configurations of defects we find a precise match with the characters of irreducible modules of $W_N$ algebras and affine Lie algebras of type $A_{N-1}$.

Twisted Γ-Lie algebras and their vertex operator representations

Let Γ be a generic subgroup of the multiplicative group C⁎ of nonzero complex numbers. We define a class of Lie algebras associated to Γ, called twisted Γ-Lie algebras, which are natural generalizations of the twisted affine Lie algebras. Starting from an arbitrary even sublattice Q of ZN and an arbitrary finite order isometry of ZN preserving Q, we construct a family of twisted Γ-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted Γ-Lie algebras. As an application, this recovers a number of known vertex operator realizations for infinite dimensional Lie algebras, such as twisted affine Lie algebras, extended affine Lie algebras of type A, trigonometric Lie algebras of series A and B, unitary Lie algebras, and BC-graded Lie algebras.

An imaginary PBW basis for quantum affine algebras of type 1

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.

Free Field Realizations of Induced Modules for Affine Lie Algebras

We construct a free field realization of generalized loop modules induced from admissible diagonal ℤ-graded irreducible modules of nonzero level for a Heisenberg subalgebra of untwisted affine Lie algebra of type A. Such generalized loop modules are irreducible as it was shown in Bekkert et al. [2]. We also extend this construction to modules induced from an arbitrary parabolic subalgebra of whose Levi factor is infinite-dimensional and contains .

FUSION RULES AND COMPLETE REDUCIBILITY OF CERTAIN MODULES FOR AFFINE LIE ALGEBRAS

We develop a new method for obtaining branching rules for affine Kac–Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in our previous paper J. Algebra319 (2008) 2434–2450, is closed under fusion. Then, we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type [Formula: see text], obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for ℓ ≥ 3. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type [Formula: see text]. Next, we notice that the category of [Formula: see text] modules at level -2ℓ + 3 has the isomorphic fusion algebra. This enables us to decompose certain [Formula: see text] and [Formula: see text]-modules at negative levels.