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Let g be a complex simple Lie algebra with a Borel subalgebra b. Consider the semidirect product I b = b ⋉ b ⁎ , where the dual b ⁎ of b is equipped with the coadjoint action of b and is considered as an abelian ideal of I b. We describe the automorphism group Aut (I b) of the Lie algebra I b. In particular we prove that it contains the automorphism group of the extended Dynkin diagram of g. In type A n , the dihedral subgroup was recently proved to be contained in Aut (I b) by Dror Bar-Natan and Roland van der Veen in [1] (where I b is denoted by I u n). Their construction is ad hoc and they asked for an explanation which is provided by this note. Let n denote the nilpotent radical of b. We obtain similar results for I b ‾ = b ⋉ n ⁎ that is both an Inönü-Wigner contraction of g and the quotient of I b by its center. [ABSTRACT FROM AUTHOR]

In this research, we studied the affine Lie algebra aff(2,R). The aim of this research is to determine the 1-form in affine Lie algebra aff(2,R) which is associated with its symplectic structure so that affine Lie algebra aff(2,R) is a Frobenius Lie algebra. Realized the elements of the affine Lie algebra aff(2,R) in matrix form, then calculated the Lie brackets and formed the structure matrix of the affine Lie algebra aff(2,R). 1-form of the affine Lie algebra aff(2,R) is obtained from the determinant of the structure matrix of the affine Lie algebra aff(2,R). Furthermore, proved that the 2-form is symplectic and related to the 1-form. The result obtained is that the affine Lie algebra aff(2,R) has 1-form α=ε_12^*+ε_23^* on aff(2,R)^* which is related to its symplectic structure, β=ε_11^*∧ε_12^*+ε_12^*∧ε_22^*+ε_21^*∧ε_13^*+ε_22^*∧ε_23^* such that the affine Lie algebra aff(2,R) is a Frobenius Lie algebra. For further research, it can be developed into an affine Lie algebra with dimensions n(n+1).

We demonstrate that for a fixed dominant integral weight and fixed defect d, there are only a finite number of Morita equivalence classes of blocks of cyclotomic Hecke algebras, by combining some combinatorics with the Chuang-Rouquier categorification of integrable highest weight modules over Kac-Moody algebras of affine type A. This is an extension of a proof for symmetric groups of a conjecture known as Donovan's conjecture. We fix a dominant integral weight Λ. The blocks of cyclotomic Hecke algebras H n Λ for the given Λ correspond to the weights P(Λ) of a highest weight representation with highest weight Λ. We connect these weights into a graph we call the reduced crystal P ̂ (Λ) , in which vertices are connected by i-strings. We define the hub of a weight and show that a vertex is i-external for a residue i if the defect is less than the absolute value of the i-component of the hub. We demonstrate the existence of a bound on the degree after which all vertices of a given defect d are i-external in at least one i-string, lying at the high degree end of the i-string. For e = 2, we calculate an approximation to this bound. [ABSTRACT FROM AUTHOR]

Consider the affine Lie algebra sℓ ̂(n) with null root δ, weight lattice P and set of dominant weights P+. Let V (kΛ0), k ∈ ℤ≥1 denote the integrable highest weight sℓ ̂(n)-module with level k ≥ 1 highest weight kΛ0. Let wt(V ) denote the set of weights of V (kΛ0). A weight μ ∈wt(V ) is a maximal weight if μ + δ∉wt(V ). Let max+(kΛ 0) = max(kΛ0) ∩ P+ denote the set of maximal dominant weights which is known to be a finite set. The explicit description of the weights in the set max+(kΛ 0) is known [R. L. Jayne and K. C. Misra, On multiplicities of maximal dominant weights of sl ̂(n)-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 sℓ ̂(n)-modules V (kΛ0), Algebr. Represent. Theory 25 (2022) 477–490], 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 sℓ ̂(n)-module V (kΛ0) 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 sℓ ̂(n)-modules V (kΛ0), Algebr. Represent. Theory 25 (2022) 477–490]. [ABSTRACT FROM AUTHOR]

Using the bases of principal subspaces for twisted affine Lie algebras except A 2 l (2) by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight kΛ0 and parafermionic spases for the same affine Lie algebras. As a result, we obtain their character formulas conjectured in Hatayama et al. (2001). [ABSTRACT FROM AUTHOR]

Chess tableaux are a special kind of standard Young tableaux where, in the chessboard coloring of the Young diagram, even numbers always appear in white cells and odd numbers in black cells. If, for λ a partition of n, Chess (λ) denotes the number of chess tableaux of shape λ , then Chow, Eriksson and Fan observed that ∑ λ ⊢ n Chess (λ) 2 is divisible by unusually large powers of 2. In this paper, we give an explanation for this phenomenon, proving a lower bound of n - O (n) for the 2-adic valuation of this sum and a generalization of it. We do this by exploiting a connection with a certain representation of the affine Lie algebra sl 2 ^ on the vector space with basis indexed by partitions. Our result about chess tableaux then follows from a study of the basic representation of sl 2 ^ with coefficients taken from the ring of rational numbers with odd denominators. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that the diagonal coset vertex operator algebra C (L g (k + 2 , 0) , L g (k , 0) ⊗ L g (2 , 0)) is rational and C 2 -cofinite in case g = s o (2 n) , n ≥ 3 and k is an admissible number for g ˆ. It is also shown that the diagonal coset vertex operator algebra C (L s l 2 (k + 4 , 0) , L s l 2 (k , 0) ⊗ L s l 2 (4 , 0)) is rational and C 2 -cofinite in case k is an admissible number for s l 2 ˆ. Furthermore, irreducible modules of C (L s l 2 (k + 4 , 0) , L s l 2 (k , 0) ⊗ L s l 2 (4 , 0)) are classified in case k is a positive odd integer. [ABSTRACT FROM AUTHOR]

For the Kashiwara crystal of a highest weight representation of an affine Lie algebra of type A and rank e, with highest weight Λ, there is a labeling by multipartitions and by piecewise-linear paths in the real weight space called Littelmann paths. Both labelings are constructed recursively, but the crystals are isomorphic, so there is a bijection between the labels. We choose a multicharge (k 1 , ... , k r) , with 0 ≤ k1 ≤ k2.... ≤ kr ≤ e − 1. We put ki in the node at the upper left corner of partition i and let the residues from ℤ / e ℤ increase across rows and decrease down columns. For e = 2, we call a multipartition residue-homogeneous if all nonzero rows end in nodes of the same residue and if partitions with the same corner residue have first rows of the same parity. The multipartition is called strongly residue-homogeneous if each partition ends in a right triangle of whose side has length one less than the first row of the next partition. We give explicit examples, including many of the important case of symmetric groups. We show that such a multipartition corresponds to a Littelmann path which is unidirectional in the sense that the projection of the the main part of the path to the coordinates of the fundamental weights consists of long paths all lying in either the second or fourth quadrant, separated by short paths between fixed integers encoding the number of rows and addable nodes in the multipartition. The path corresponding to a strongly residue-homogeneous multipartition can be constructed non-recursively using only integers describing the multipartition. [ABSTRACT FROM AUTHOR]

This article, titled "Coupled Free Fermion Conformal Field Theory and Representations," explores a specific subclass of conformal field theories (CFTs) that involve either uncoupled or coupled free fermions. The study analyzes the representation spaces of these CFTs and reveals the exclusion statistics of coupled free fermions with universal chiral partition functions under specific bases. The article also discusses the connection between the coset construction, lattice construction, and orbifold construction within the context of coupled free fermions. Additionally, the author provides explicit expressions of certain string functions in terms of Dedekind eta functions. The article was written by Bolin Han from the Mathematical Sciences Institute at the Australian National University. [Extracted from the article]

This article, titled "Coupled Free Fermion Conformal Field Theory and Representations," explores a specific subclass of conformal field theories (CFTs) that involve either uncoupled or coupled free fermions. The study analyzes the representation spaces of these CFTs and reveals the exclusion statistics of coupled free fermions with universal chiral partition functions under specific bases. The article also discusses the connection between the coset construction, lattice construction, and orbifold construction within the context of coupled free fermions. Additionally, the author provides explicit expressions of certain string functions in terms of Dedekind eta functions. The article was written by Bolin Han from the Mathematical Sciences Institute at the Australian National University. [Extracted from the article]

We consider a Demazure slice of type A 2 l (2) , that is an associated graded piece of an infinite-dimensional version of a Demazure module. We show that a global Weyl module of a hyperspecial current algebra of type A 2 l (2) is filtered by Demazure slices. We calculate extensions between a Demazure slice and a usual Demazure module and prove that a graded character of a Demazure slice is equal to a nonsymmetric Macdonald-Koornwinder polynomial divided by its square norm. In the last section, we prove that a global Weyl module of the special current algebra of type A 2 l (2) is a free module over the polynomial ring arising as the endomorphism ring of itself. [ABSTRACT FROM AUTHOR]

For n ≥ 2 consider the affine Lie algebra s ℓ ̂ (n) with simple roots {αi∣0 ≤ i ≤ n − 1}. Let V (k Λ 0) , k ∈ ℤ ≥ 1 denote the integrable highest weight s ℓ ̂ (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 Λ 0 − λ a , b ℓ where λ a , b ℓ = ℓ α 0 + (ℓ − b) α 1 + (ℓ − (b + 1)) α 2 + ⋯ + α ℓ − b + α n − ℓ + a + 2 α n − ℓ + a + 1 + ... + (ℓ − a) α n − 1 , and k ≥ a + b, a , b ∈ ℤ ≥ 1 , max { a , b } ≤ ℓ ≤ n + a + b 2 − 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. [ABSTRACT FROM AUTHOR]

We study the representation theory of the Kazama–Suzuki coset vertex operator superalgebra associated to the pair of a complex simple Lie algebra and its Cartan subalgebra. In the case of type A 1 , B.L. Feigin, A.M. Semikhatov, and I.Yu. Tipunin introduced another coset construction, which is "inverse" of the Kazama–Suzuki coset construction. In this paper we generalize the latter coset construction to arbitrary type and establish a categorical equivalence between the categories of certain modules over an affine vertex operator algebra and the corresponding Kazama–Suzuki coset vertex operator superalgebra. Moreover, when the affine vertex operator algebra is regular, we prove that the corresponding Kazama–Suzuki coset vertex operator superalgebra is also regular and the category of its ordinary modules carries a braided monoidal category structure by the theory of vertex tensor categories. [ABSTRACT FROM AUTHOR]

We construct a combinatorial crystal structure on the Kirillov–Reshetikhin crystal B 7 , s in type E 7 (1) , where 7 is the unique node in the orbit of 0 in the affine Dynkin diagram. We then describe the combinatorial R-matrix R : B 7 , s ⊗ B 7 , s ′ → B 7 , s ′ ⊗ B 7 , s . [ABSTRACT FROM AUTHOR]

We study the characters of simple modules in the parabolic BGG category of the affine Lie algebra in Deligne's category. More specifically, we take the limit of the Weyl-Kac formula to compute the character of the irreducible quotient L (X , k) of the parabolic Verma module M (X , k) of level k , where X is an indecomposable object of Deligne's category Re _ p (G L t) , Re _ p (O t) , or Re _ p (S p t) , under conditions that the highest weight of X plus the level gives a fundamental weight, t is transcendental, and the base field k has characteristic 0. We compare our result to the partial result in [7, Problem 6.2] , and evaluate the characters to the categorical dimensions to get a categorical interpretation of the Nekrasov-Okounkov hook length formula, [12, Formula (6.12)]. [ABSTRACT FROM AUTHOR]

In this paper, under the assumption that the diagonal coset vertex operator algebra C (L g (k + l , 0) , L g (k , 0) ⊗ L g (l , 0)) is rational and C 2 -cofinite, the global dimension of C (L g (k + l , 0) , L g (k , 0) ⊗ L g (l , 0)) is obtained and the quantum dimensions of multiplicity spaces viewed as C (L g (k + l , 0) , L g (k , 0) ⊗ L g (l , 0)) -modules are also obtained. As an application, a method to classify irreducible modules of C (L g (k + l , 0) , L g (k , 0) ⊗ L g (l , 0)) is provided. As an example, we prove that the diagonal coset vertex operator algebra C (L E 8 (k + 2 , 0) , L E 8 (k , 0) ⊗ L E 8 (2 , 0)) is rational, C 2 -cofinite, and classify irreducible modules of C (L E 8 (k + 2 , 0) , L E 8 (k , 0) ⊗ L E 8 (2 , 0)) . [ABSTRACT FROM AUTHOR]

The main result of the paper establishes the irreducibility of a large family of nonzero central charge induced modules over Affine Lie algebras for any non standard parabolic subalgebra. It generalizes all previously known partial results and provides a construction of many new irreducible modules. [ABSTRACT FROM AUTHOR]

For l≥1 and k≥2, we consider certain admissible sequences of k-1 lattice paths in a colored l×l 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 λ+l with l(λ)≤k, which is also the number of (k + 1)k··· 21-avoiding permutations in Sl. Finally, we apply this result to the representation theory of the affine Lie algebra sl(n) and show that this gives the multiplicity of certain maximal dominant weights in the irreducible highest weight sl(n) -module V(kλ0). [ABSTRACT FROM AUTHOR]

We study graded nonlocal $$\underline{\mathsf {q}}$$ -vertex algebras and we prove that they can be generated by certain sets of vertex operators. As an application, we consider the family of graded nonlocal $$\underline{\mathsf {q}}$$ -vertex algebras $$V_{c,1}$$ , $$c\ge 1$$ , associated with the principal subspaces $$W(c\Lambda _0)$$ of the integrable highest weight $$U_q (\widehat{\mathfrak {sl}}_2)$$ -modules $$L(c\Lambda _0)$$ . Using quantum integrability, we derive combinatorial bases for $$V_{c,1}$$ and compute the corresponding character formulae. [ABSTRACT FROM AUTHOR]

Theℤ-grading determined by a long simple root of a rankn+1 affine Lie algebra overℂarises from a representation of a ranknsemi-simple complex Lie algebra. Analysis of the relationship between the grading and the representation yields constructions that generalize the minuscule and adjoint algorithms as well as Kac’s construction of nontwisted affine Lie algebras. [ABSTRACT FROM AUTHOR]

Let be an untwisted affine Kac-Moody Lie algebra. The top of every irreducible highest weight integrable -module is the finite-dimensional irreducible -module, where the action of the simple Lie algebra is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level Frenkel-Jing operators corresponding to Drinfeld realization of the quantum affine algebra . By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator , we obtain an infinite-dimensional vector space . Next, we introduce an associative algebra , a certain quantum analogue of the universal enveloping algebra , and construct some infinite-dimensional -modules corresponding to the finite-dimensional irreducible -modules . We show that the space carries a structure of an -module and, furthermore, we prove that the -module is isomorphic to the -module . [ABSTRACT FROM AUTHOR]

In this paper, irreducible modules of the diagonal coset vertex operator algebra C (L g (k + l , 0) , L g (k , 0) ⊗ L g (l , 0)) are classified under the assumption that C (L g (k + l , 0) , L g (k , 0) ⊗ L g (l , 0)) is rational, C 2 -cofinite and certain additional assumption. An explicit modular transformation formula of traces functions of C (L g (k + l , 0) , L g (k , 0) ⊗ L g (l , 0)) is obtained. As an application, the fusion rules of C (L E 8 (k + 2 , 0) , L E 8 (k , 0) ⊗ L E 8 (2 , 0)) are determined by using the Verlinde formula. [ABSTRACT FROM AUTHOR]

Let g ˜ be the affine Lie algebra of type A 2 l (2). The integrable highest weight g ˜ -module L (k Λ 0) called the standard g ˜ -module is realized by a tensor product of the twisted module V L T for the lattice vertex operator algebra V L. By using such vertex algebraic construction, we construct bases of the standard module, its principal subspace and the parafermionic space. As a consequence, we obtain their character formulas and settle the conjecture for vacuum modules stated in [15]. [ABSTRACT FROM AUTHOR]

In the context of fusion coefficients we construct Markovian processes with values in a fixed level alcove associated to the root system of an affine Lie algebra. In this context we show that for a large class of simple random walks, the discretized characters of irreducible representations of a semi-simple complex Lie algebra appear naturally as the eigenfunctions of the Dirichlet problem on such an alcove. We establish a correspondence between the hypergroup of conjugacy classes of a compact Lie group and the fusion hypergroup. For the stochastic model appearing in the context of fusion coefficients, we prove finally a convergence in distribution towards the radial part of a Brownian motion on a compact Lie group. [ABSTRACT FROM AUTHOR]

We consider two different methods of associating vertex algebraic structures with the level 1 principal subspaces for U q ( sl ˆ 2 ) . In the first approach, we introduce certain commutative operators and study the corresponding vertex algebra and its module. We find combinatorial bases for these objects and show that they coincide with the principal subspace bases found by B.L. Feigin and A.V. Stoyanovsky. In the second approach, we introduce the, so-called nonlocal q _ -vertex algebras, investigate their properties and construct the nonlocal q _ -vertex algebra and its module, generated by Frenkel–Jing operator and Koyama's operator respectively. By finding the combinatorial bases of their suitably defined subspaces, we establish a connection with the sum sides of the Rogers–Ramanujan identities. Finally, we discuss further applications to quantum quasi-particle relations. [ABSTRACT FROM AUTHOR]

The realizations of the basic representation of $$\widehat{\mathfrak {gl}}_r$$ are known to be parametrized by partitions of r and have an explicit description in terms of vertex operators on the bosonic/fermionic Fock space. In this paper, we give a geometric interpretation of these realizations in terms of geometric operators acting on the equivariant cohomology of certain Nakajima quiver varieties. [ABSTRACT FROM AUTHOR]