We study finite dimensional representations of current algebras, loop algebras and their quantized versions. For the current algebra of a simple Lie algebra of type {\tt ADE}, we show that Kirillov-Reshetikhin modules and Weyl modules are in fact all Demazure modules. As a consequence one obtains an elementary proof of the dimension formula for Weyl modules for the current and the loop algebra. Further, we show that the crystals of the Weyl and the Demazure module are the same up to some additional label zero arrows for the Weyl module.
For the current algebra $\Lgc$ of an arbitrary simple Lie algebra, the fusion product of Demazure modules of the same level turns out to be again a Demazure module. As an application we construct the $\Lgc$-module structure of the Kac-Moody algebra $\Lhg$-module $V(\ell\Lam_0)$ as a semi-infinite fusion product of finite dimensional $\Lgc$--modules.
We show that every Weyl module for a current algebra has a filtration whose successive quotients are isomorphic to Demazure modules, and that the path model for a tensor product of level zero fundamental representations is isomorphic to a disjoint union of Demazure crystals. Moreover, we show that the Demazure modules appearing in these two objects coincide exactly. Though these results have been previously known in the simply laced case, they are new in the non-simply laced case.
We study a relationship between the graded characters of generalized Weyl modules $W_{w λ}$, $w \in W$, over the positive part of the affine Lie algebra and those of specific quotients $V_{w}^- (λ) / X_{w}^- (λ)$, $w \in W$, of the Demazure submodules $V_{w}^- (λ)$ of the extremal weight modules $V(λ)$ over the quantum affine algebra, where $W$ is the finite Weyl group and $λ$ is a dominant weight. More precisely, we prove that a specific quotient of the Demazure submodule is a quantum analog of a generalized Weyl module.
We construct a Poincare-Birkhoff-Witt type basis for the Weyl modules of the current algebra of $sl_{r+1}$. As a corollary we prove a conjecture made by Chari and Pressley on the dimension of the Weyl modules in this case. Further, we relate the Weyl modules to the fusion modules of the current algebra defined by Feigin and Loktev, and to the Demazure modules in level one representations of the corresponding affine algebra. In particular, this allows us to establish substantial cases of the conjectures of Feigin and Loktev on the structure and graded character of the fusion modules.
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with highest root $θ$ and let $\mathfrak{g}[t]$ be the corresponding current algebra. In this paper, we consider the $\mathfrak{g}[t]$-stable Demazure modules associated to integrable highest weight representations of the affine Lie algebra $\widehat{\mathfrak{g}}$. We prove that the fusion product of Demazure modules of a given level with a single Demazure module of a different level and with highest weight a multiple of $θ$ is a generalized Demazure module, and also give defining relations. This also shows that the fusion …
In this paper, we study local graded Weyl modules and Demazure modules for twisted hypercurrent algebras. We prove that local graded Weyl modules for a twisted hypercurrent algebra are isomorphic to the corresponding level 1 Demazure modules, and moreover, that they are restrictions of corresponding local graded Weyl modules for the untwisted hypercurrent algebra.
We consider a Demazure slice of type $A_{2l}^{(2)}$, 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 $A_{2l}^{(2)}$ 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 norm. In the last section, we prove that a global Weyl module of the special current algebra of type $A_{2l}^{(2)}$ is a free …
We study finite-dimensional respresentations of twisted current algebras and show that any graded twisted Weyl module is isomorphic to level one Demazure module for the twisted affine Kac-Moody algebra. Using the tensor product property of Demazure modules, we obtain, by analyzing the fundamental Weyl modules, dimension and character formulas. Moreover we prove that graded twisted Weyl modules can be obtained by taking the associated graded modules of Weyl modules for the loop algebra, which implies that its dimension and classical character are independent of the support and depend only on it…
The Demazure product gives a natural monoid structure on any Coxeter group. Such structure occurs naturally in many different areas in Lie Theory. This paper studies the Demazure product of an extended affine Weyl group $\tilde W$. The main discovery is a close connection between the Demazure product of an extended affine Weyl group and the quantum Bruhat graph of the finite Weyl group. As applications, we obtain explicit formulas on the generic Newton points and the Demazure products of elements in the lowest two-sided cell/shrunken Weyl chambers of $\tilde W$, and obtain an explicit formula …
We establish the existence of Demazure flags for graded local Weyl modules for hyper current algebras in positive characteristic. If the underlying simple Lie algebra is simply laced, the flag has length one, i.e., the graded local Weyl modules are isomorphic to Demazure modules. This extends to the positive characteristic setting results of Fourier-Littelmann and Naoi for current algebras in characteristic zero. Using this result, we prove that the character of local Weyl modules for hyper loop algebras depend only on the highest weight, but not on the (algebraically closed) ground field, and…
The weights of finite-dimensional representations of simple Lie algebras are naturally associated with Weyl polytopes. Representation characters decompose into multiplicity-free sums over the weights in Weyl polytopes. The Brion formula for these Weyl polytope sums is remarkably similar to the Weyl character formula. Moreover, the same Lie characters are also expressible as Demazure character formulas. This motivates a search for new expressions for Weyl polytope sums, and we prove such a formula involving Demazure operators. It applies to the Weyl polytope sums of the simple Lie algebras $A_n…
In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the twisted current algebras. These modules are indexed by an $|R^+|$-tuple of partitions $\bxi=(ξ^α)_{α\in R^+}$ satisfying a natural compatibility condition. We give three equivalent presentations of these modules and show that for a particular choice of $\bxi$ these modules become isomorphic to Demazure modules in various levels for the twisted affine algebras. As a consequence we see that the defining relations of twisted Demazure modules can be greatly simplified. Furthermore, we investigate the n…
We provide a proof that every Schubert variety of a semi-infinite flag variety is projectively normal. This gives us an interpretation of a Demazure module of a global Weyl module of a current Lie algebra as the (dual) space of the space of global sections of a semi-infinite Schubert variety. Moreover, we give geometric realizations of Feigin-Makedonskyi's generalized Weyl modules, and the $t = \infty$ specialization of non-symmetric Macdonald polynomials.
We consider expansions of products of theta-functions associated with arbitrary root systems in terms of nonsymmetric Macdonald polynomials at $t=\infty$ divided by their norms. The latter are identified with the graded characters of Demazure slices, some canonical quotients of thick (upper) level-one Demazure modules, directly related to recent theory of generalized (nonsymmetric) global Weyl modules. The symmetric Rogers-Ramanujan-type series considered by Cherednik-Feigin were expected to have some interpretation of this kind; the nonsymmetric setting appeared necessary to achieve this. As …
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with highest root $θ$. Given two non-negative integers $m$, $n$, we prove that the fusion product of $m$ copies of the level one Demazure module $D(1,θ)$ with $n$ copies of the adjoint representation ${\rm ev}_0 V(θ)$ is independent of the parameters and we give explicit defining relations. As a consequence, for $\mathfrak{g}$ simply laced, we show that the fusion product of a special family of Chari-Venkatesh modules is again a Chari-Venkatesh module. We also get a description of the truncated Weyl module associated to a mu…
We settle the fusion product decomposition theorem for higher-level affine Demazure modules for the cases $E^{(1)}_{6, 7, 8}, F^{(1)}_4$ and $E^{(2)}_{6}$, thus completing the main theorems of Chari et al. (J. Algebra, 2016) and Kus et al. (Represent. Theory, 2016). We obtain a new combinatorial proof for the key fact, that was used in Chari et al. (op cit.), to prove this decomposition theorem. We give a case free uniform proof for this key fact.
Let g be a finite-dimensional complex simple Lie algebra. Fix a non-negative integer l, we consider the set of dominant weights λ of g such that lΛ_0+λ is a dominant weight for the corresponding untwisted affine Kac-Moody algebra. For these special family of dominant weights, we show that the fusion product of an irreducible g-module V(λ) and a finite number of special family of g-stable Demazure modules of level l (considered in [15] and [16]), for the current algebra g[t] associated to g, again turns out to be a Demazure module. This fact is closely related with several important conjectures…
In this paper, we introduce a family of indecomposable finite--dimensional graded modules for the current algebra associated to a simple Lie algebra. These modules are indexed by a tuple of partitions one for each positive root of the simple Lie algebra. We assume that the partitions satisfy a natural compatibility condition. In the case when the partitions are all rectangular, for instance, we prove that these are precisely the Demazure modules which are stable under the action of the simple Lie algebra. As a consequence we see that the defining relations of these Demazure modules can be grea…
For a classical group $G$ and a Coxeter element $c$ of the Weyl group, it is known that the coordinate ring $\mathbb{C}[G^{e,c^2}]$ of the double Bruhat cell $G^{e,c^2}:=B\cap B_-c^2B_-$ has a structure of cluster algebra of finite type, where $B$ and $B_-$ are opposite Borel subgroups. In this article, we consider the case $G$ is of type ${\rm B}_r$, ${\rm C}_r$ or ${\rm D}_r$ and describe all the cluster variables in $\mathbb{C}[G^{e,c^2}]$ as monomial realizations of certain Demazure crystals.
We prove asymptotic normality of the distributions defined by q-supernomials, which implies asymptotic normality of the distributions given by the central string functions and the basic specialization of fusion modules of the current algebra of sl2. The limit is taken over linearly scaled fusion powers of a fixed collection of irreducible representations. This includes as special instances all Demazure modules of the affine Kac-Moody algebra associated to sl2. Along with an available complementary result on the asymptotic normality of the basic specialization of graded tensors of the type A st…
The classical Pieri formula gives a multiplicity free expansion of an irreducible module with a fundamental one for the complex general linear group. In this article we replace the tensor product by the fusion product and prove an analogue Pieri formula for higher level Demazure modules for the affine Lie algebra $\widehat{\mathfrak{sl}}_{n+1}$. To be more precise, we show that the fusion product of an arbitrary stable Demazure module with a fundamental module admits a multiplicity free excellent filtration and the successive quotients are described explicitly. As a consequence, we derive recu…
In this paper we study the family of prime irreducible representations of quantum affine $\lie{sl}_{n+1}$ which arise from the work of D. Hernandez and B. Leclerc. These representations can also be described as follows: the highest weight is a product of distinct fundamental weights with parameters determined by requiring that the representation be minimal by parts. We show that such representations admit a BGG-type resolution where the role of the Verma module is played by the local Weyl module. This leads to a closed formula (the Weyl character formula) for the character of the irreducible r…
In this paper, we give a representation-theoretic interpretation of the specialization $E_{w_{\circ} λ} (q,\infty)$ of the nonsymmetric Macdonald polynomial $E_{w_{\circ} λ}(q,t)$ at $t=\infty$ in terms of the Demazure submodule $V_{w_\circ}^{-} (λ)$ of the level-zero extremal weight module $V(λ)$ over a quantum affine algebra of an arbitrary untwisted type, here, $λ$ is a dominant integral weight, and $w_{\circ}$ denotes the longest element in the finite Weyl group $W$. Also, for each $x \in W$, we obtain a combinatorial formula for the specialization $E_{x λ} (q, \infty)$ at $t=\infty$ of th…
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ be its two opposite Borel subgroups. For two elements $u$, $v$ of the Weyl group $W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to a cluster algebra $\mathcal{A}(\textbf{i})_{\mathbb C}$ [arXiv:math/0305434, arXiv:1602.00498]. In the case $u=e$, $v=c^2$ ($c$ is a Coxeter element), the algebra ${\mathbb C}[G^{e,c^2}]$ has only finitely many cluster variables. In this article, for $G={\rm SL}_{r+1}(\mathbb{C})$, we obtain explic…
We prove stability of the Chari-Pressley-Loktev bases for natural inclusions of local Weyl modules of the current algebra $sl_2[t]$. These modules being known to be Demazure submodules in the level 1 representations of the affine Lie algebra $\widehat{sl_2}$, we obtain, by passage to the direct limit, bases for the level 1 representations themselves.
We prove stability of the Chari-Loktev bases with respect to the inclusions of local Weyl modules of the current algebra $\mathfrak{sl}_{r+1}[t]$. This is conjectured in \cite{RRV2} and the $r=1$ case is proved in \cite{RRV1}. Local Weyl modules being known to be Demazure submodules in the level one representations of the affine Lie algebra $\widehat{\mathfrak{sl}_{r+1}}$, we obtain, by passage to the direct limit, bases for the level one representations themselves.
We characterise the symplectic Weyl group elements such that the FFLV basis is compatible with the PBW filtration on symplectic Demazure modules, extending type A results by the second author. Surprisingly, the number of such elements depends not on the type A or C of the Lie algebra but on the rank only.
We establish the equality of the specialization $E_{wλ}(x;q,0)$ of the nonsymmetric Macdonald polynomial $E_{wλ}(x;q,t)$ at $t=0$ with the graded character $\mathop{\rm gch} U_{w}^{+}(λ)$ of a certain Demazure-type submodule $U_{w}^{+}(λ)$ of a tensor product of "single-column" Kirillov--Reshetikhin modules for an untwisted affine Lie algebra, where $λ$ is a dominant integral weight and $w$ is a (finite) Weyl group element, this generalizes our previous result, that is, the equality between the specialization $P_λ(x;q,0)$ of the symmetric Macdonald polynomial $P_λ(x;q,t)$ at $t=0$ and the grad…
We study a class of modules, called Chari-Venkatesh modules, for the current superalgebra $\mathfrak{sl}(1|2)[t]$. This class contains other important modules, such as graded local Weyl, truncated local Weyl and Demazure-type modules. We prove that Chari-Venkatesh modules can be realized as fusion products of generalized Kac modules. In particular, this proves Feigin and Loktev's conjecture, that fusion products are independent of their fusion parameters in the case where the fusion factors are generalized Kac modules. As an application of our results, we obtain bases, dimension and character …
Let $G=PSO(2n+1, \mathbb{C}) (n \ge 3)$ and $B$ be the Borel subgroup of $G$ containing maximal torus $T$ of $G.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w.$ In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest element of the Weyl group $W.$ We describe all the reduced express…
We study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for $\mathfrak{sl}_{n+1}$. We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements. These faces are again normal polytopes and their Minkowski sum is compatible with tensor products, which implies that we obtain flat degenerations of the corresponding Schubert vari…
In this paper we study a family of commutative algebras generated by two infinite sets of generators. These algebras are parametrized by Young diagrams. We explain a connection of these algebras with the fusion product of integrable irreducible representations of the affine $sl_2$ Lie algebra. As an application we derive a fermionic formula for the character of the affine fusion product of two modules. These fusion products can be considered as a simplest example of the double affine Demazure modules.
For $G$ a reductive group and $T\subset B$ a maximal torus and Borel subgroup, Demazure modules are certain $B$-submodules, indexed by elements of the Weyl group, of the finite irreducible representations of $G$. In order to describe the $T$-weight spaces that appear in a Demazure module, we study the convex hull of these weights - the Demazure polytope. We characterize these polytopes both by vertices and by inequalities, and we use these results to prove that Demazure characters are saturated, in the case that $G$ is simple of classical Lie type. Specializing to $G=GL_n$, we recover results …
We study a family of posets and the associated chain and order polytopes. We identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple of a fundamental weight. We show that the character of such a Kogan face equals to the character of a Demazure module which occurs in the irreducible representation of $\mathfrak{sl}_n$ having highest weight multiple of fundamental weight and for any such Demazure module there exists a corresponding poset and associated maximal Kogan face. We prove that the chain polytope parametrizes a monomial basis of the associated PBW-…
We develop the theory of global and local Weyl modules for the hyperspecial maximal parabolic subalgebra of type $A_{2n}^{(2)}$. We prove that the dimension of a local Weyl module depends only on its highest weight, thus establishing a freeness result for global Weyl modules. Furthermore, we show that the graded local Weyl modules are level one Demazure modules for the corresponding affine Lie algebra. In the last section we derive the same results for the special maximal parabolic subalgebras of the twisted affine Lie algebras not of type $A_{2n}^{(2)}$.
In this article, we show in the ADE case that the fusion product of Kirillov-Reshetikhin modules for a current algebra, whose character is expressed in terms of fermionic forms, can be constructed from one-dimensional modules by using Joseph functors. As a consequence, we obtain some identity between fermionic forms and Demazure operators. Since the same identity is also known to hold for one-dimensional sums of nonexceptional type, we can show from these results the X = M conjecture for type $A_n^{(1)}$ and $D_n^{(1)}$.
We consider arc spaces for the compositions of Pluecker and Veronese embeddings of the flag varieties for simple Lie groups of types ADE. The arc spaces are not reduced and we consider the homogeneous coordinate rings of the corresponding reduced schemes. We show that each graded component of a homogeneous coordinate ring is a cocyclic module over the current algebra and is acted upon by the algebra of symmetric polynomials. We show that the action of the polynomial algebra is free and that the fiber at the special point of a graded component is isomorphic to an affine Demazure module whose le…
We study Demazure modules which occur in a level $\ell$ irreducible integrable representation of an affine Lie algebra. We also assume that they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie algebra. We prove that such a module is isomorphic to the fusion product of "prime" \ Demazure modules, where the prime factors are indexed by dominant integral weights which are either a multiple of $\ell$ or take value less than $\ell$ on all simple coroots. Our proof depends on a technical result which we prove in all the classical cases and $G_2$. Calculatio…
We introduce the notion of a strong minuscule element, and prove that the dominant integral weight associated to a strong minuscule element is the fundamental weight corresponding to a short simple root. In addition, we enumerate the strong minuscule elements explicitly, and then as an application of this enumeration, determine the dimension of certain Demazure modules in the finite-dimensional irreducible modules whose highest weights are minuscule weights.
The Kashiwara crystal $B(\infty)$ parametrizes a basis for the Verma module of a Kac-Moody algebra. It has a deep combinatorial structure which one seeks to understand. For each sequence $J$ of reduced decompositions of elements of the Weyl group $W$, it has a realization as a subset $B_J(\infty)$ of a crystal $B_J$ which as a set is just $J$ copies of the natural numbers. The goal is to determine $B_J(\infty)$ and in particular to show that it is a polyhedral subset of $B_J$. In earlier work this led to the notion of an $S$-graph associated to a given simple root $α$. Here the notion of a gia…
We study structural properties of truncated Weyl modules. A truncated Weyl module $W_N(λ)$ is a local Weyl module for $\mathfrak g[t]_N = \mathfrak g \otimes \frac{\mathbb C[t]}{t^N\mathbb C[t]}$, where $\mathfrak g$ is a finite-dimensional simple Lie algebra. It has been conjectured that, if $N$ is sufficiently small with respect to $λ$, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when $λ$ is a multiple of certain fundamental weights, including all minuscule ones for simply laced $\mathfrak g$. We also take…
We study the category of finite--dimensional representations for a basic classical Lie superalgebra $\Lg=\Lg_0\oplus \Lg_1$. For the ortho--symplectic Lie superalgebra $\Lg=\mathfrak{osp}(1,2n)$ we show that certain objects in that category admit a fusion flag, i.e. a sequence of graded $\Lg_0[t]$--modules such that the successive quotients are isomorphic to fusion products. Among these objects we find fusion products of finite--dimensional irreducible $\Lg$--modules, truncated Weyl modules and Demazure type modules. Moreover, we establish a presentation for these types of fusion products in t…
In vertex algebra theory, fusion rules are described as the dimension of the vector space of intertwining operators between three irreducible modules. We describe fusion rules in the category of weight modules for the Weyl vertex algebra. This way we confirm the conjecture on fusion rules based on the Verlinde algebra. We explicitly construct intertwining operators appearing in the formula for fusion rules. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules for the affine Lie superalgebra $\widehat{gl(1 \vert 1)}$.
From a certain induced representation $\mathcal{P}_\ell$ of a double affine Weyl group, we construct a ring $\mathcal{F}_\ell$ that is isomorphic to the fusion ring, or Verlinde algebra, associated to affine Lie algebras at fixed positive integer level for both twisted and untwisted type. The induced representation, which also has a natural commutative associative algebra structure and is modular invariant with respect to certain congruence subgroups, contains $\mathcal{F}_\ell$ as an ideal and we show how it naturally inherits the modular invariance property from $\mathcal{P}_\ell.$ This cons…
We use the fusion formulas of the symmetric group and of the Hecke algebra to construct solutions of the Yang-Baxter equation on irreducible representations of $\mathfrak{gl}_N$, $\mathfrak{gl}_{N|M}$, $U_q(\mathfrak{gl}_N)$ and $U_q(\mathfrak{gl}_{N|M})$. The solutions are obtained via the fusion procedure for the Yang--Baxter equation, which is reviewed in a general setting. Distinguished invariant subspaces on which the fused solutions act are also studied in the general setting, and expressed, in general, with the help of a fusion function. Only then, the general construction is specialise…
Let $λ\in P^{+}$ be a level-zero dominant integral weight, and $w$ an arbitrary coset representative of minimal length for the cosets in $W/W_λ$, where $W_λ$ is the stabilizer of $λ$ in a finite Weyl group $W$. In this paper, we give a module $\mathbb{K}_{w}(λ)$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w λ}(q,\,t)$ multiplied by a certain explicit finite product of rational functions of $q$ of the form $(1 - q^{-r})^{-1}$ for a positive integer $r$. This module $\mat…
We study the category of finite--dimensional bi--graded representations of toroidal current algebras associated to finite--dimensional complex simple Lie algebras. Using the theory of graded representations for current algebras, we construct in different ways objects in that category and prove them to be isomorphic. As a consequence we obtain generators and relations for certain types of fusion products including the $N$--fold fusion product of $V(λ)$. This result shows that the fusion product of these types is independent of the chosen parameters, proving a special case of a conjecture by Fei…
Finite dimensional irreducible modules of the two-parameter quantum enveloping algebra $U_{r,s}(\mathfrak{sl}_n)$ are explicitly constructed using the fusion procedure when $rs^{-1}$ is generic. This provides an alternative and combinatorial description of the Schur-Weyl duality for the two-parameter quantum linear algebras of type $A$.
New graded modules for the current algebra of $\mathfrak{sl}_n$ are introduced. Relating these modules to the fusion product of simple $\mathfrak{sl}_n$-modules and local Weyl modules of truncated current algebras shows their expected impact on several outstanding conjectures. We further generalize results on PBW filtrations of simple $\mathfrak{sl}_n$-modules and use them to provide decomposition formulas for these new modules in important cases.
We introduce a new set of $q$-difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebra ${\mathfrak g}[u]$ KR-modules \cite{FL} for ${\mathfrak g}=A_r$. These operators are generalizations of the Kirillov-Noumi \cite{kinoum} Macdonald raising operators, in the dual $q$-Whittaker limit $t\to\infty$. They form a representation of the quantum $Q$-system of type $A$ \cite{qKR}. This system is a subalgebra of a quantum cluster algebra, and is also a discrete integrable system whose conserved quantities, an…
The space spanned by the characters of twisted affine Lie algebras admit the action of certain congruence subgroups of $SL(2,\mathbb{Z})$. By embedding the characters in the space spanned by theta functions, we study an $SL(2,\mathbb{Z})$-closure of the space of characters. Analogous to the untwisted affine Lie algebra case, we construct a commutative associative algebra (fusion algebra) structure on this space through the use of the Verlinde formula and study important quotients. Unlike the untwisted cases, some of these algebras and their quotients, which relate to the trace of diagram autom…
