Let $\ag$ be an affine Lie algebra, and let $\Ua$ be the quantum affine algebra introduced by Drinfeld and Jimbo. In [Kas94] Kashiwara introduced a $\Ua$-module $V(λ)$, having a global crystal base for an integrable weight $λ$ of level 0. We call it an {\it extremal weight module}.
In [\S13]{Kas00} Kashiwara gave a conjecture on the structure of extremal weight modules. We prove his conjecture when $\ag$ is an untwisted affine Lie algebra of a simple Lie algebra $\g$ of type $ADE$, using a result of Beck-Chari-Pressley [BCP].
We give a path model for a level zero extremal weight module over a quantum affine algebra. By using this result, we prove a branching rule for an extremal weight module with respect to a Levi subalgebra. Furthermore, we also show a decomposition rule of Littelmann type for the concatenation of path models for an integrable highest weight module and a level zero extremal weight module in the case where the extremal weight is minuscule.
We introduce semi-infinite Lakshmibai-Seshadri paths by using the semi-infinite Bruhat order (or equivalently, Lusztig's generic Bruhat order) on affine Weyl groups in place of the usual Bruhat order. These paths enable us to give an explicit realization of the crystal basis of an extremal weight module of an arbitrary level-zero dominant integral extremal weight over a quantum affine algebra. This result can be thought of as a full generalization of our previous result (which uses Littelmann's Lakshmibai-Seshadri paths), in which the level-zero dominant integral weight is assumed to be a posi…
We study the properties of level zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by Akasaka and the present author is given. Several properties of modules generated by extremal vectors are proved. The weights of a module generated by an extremal vector are contained in the convex hull of the Weyl group orbit of the extremalweight. The universal extremal weight module with level zero fundamental weight as an extremal weight is irreducible, and isomorphic to the affinization of an irreducible finite-dimensional module.
Using Lakshmibai-Seshadri paths, we give a combinatorial realization of the crystal basis of an extremal weight module of integral extremal weight over the quantized universal enveloping algebra associated to the infinite rank affine Lie algebra of type $B_{\infty}$, $C_{\infty}$, or $D_{\infty}$. Moreover, via this realization, we obtain an explicit description (in terms of Littlewood-Richardson coefficients) of how tensor products of these crystal bases decompose into connected components when their extremal weights are of nonnegative levels. These results, in types $B_{\infty}$, $C_{\infty}…
We consider the crystal structure of the level zero extremal weight modules $V(λ)$ using the crystal base of the quantum affine algebra constructed by Beck, Chari and Pressley. This approach yields an explicit form for the U^- extremal weight vectors in each connected component of the crystal of $V(λ)$, which are given as Schur functions in the imaginary root vectors. We use this fact to demonstrate Kashiwara's conjectures regarding the crystal structure of $V(λ)$.
We construct by fusion product new irreducible representations of the quantum affinization $U_q(\hat{sl}_\infty)$. The action is defined via the Drinfeld coproduct and is related to the crystal structure of semi-standard tableaux of type $A_\infty$. We call these representations extremal loop weight modules. The main motivations are applications to quantum toroidal algebras $U_q(sl_{n+1}^{tor})$: we prove the conjectural link between $U_q(\hat{sl}_\infty)$ and $U_q(sl_{n+1}^{tor})$ stated in [14] for this family of representations. We recover in this way the extremal loop weight modules obtain…
We construct new irreducible weight modules over quantum affine algebras of type I with all weight spaces infinite-dimensional. These modules are obtained by parabolic induction from irreducible modules over the Heisenberg subalgebra.
For quantized universal enveloping algebras we construct weight modules by inducing representations of the centralizer of the Cartan subalgebra in the quantized universal enveloping algebra. The induced modules arising from finite-dimensional weight modules the centralizer algebra are studied. In particular, we study the induction of one-dimensional modules, and this is related to the study of commutative subalgebras of the centralizer algebra. For the special case of $U_q(\mathfrak{sl}(2,\mathbb{C}))$ we show that we get the admissible unitary representations corresponding to the non-compact …
We show that the finite-dimensional fundamental module over a quantized affine algebra is isomorphic to a Demazure module of a higest weight module of level one as a module over a quantized classical universal enveloping algebra.
We define a family of universal finite-dimensional highest weight modules for affine Lie algebras, we call these Weyl modules. We conjecture that these are the classical limits of the irreducible finite--dimensional representations of the quantum affine algebras. We prove this conjecture in the case of affine sl_2. We establish a criterion for these modules to be irreducible and prove a factorization theorem for them in the general case.
We construct Wakimoto modules for twisted affine Lie algebras, and interpret the construction in terms of vertex algebras and their twisted modules. Using the Wakimoto realization, we prove the Kac-Kazhdan conjecture on the characters of irreducible modules with generic critical highest weights in the twisted case. We provide explicit formulas for the twisted fields in the case of the twisted affine algebra A^{(2)}_{2}.
In this paper, we re-examine certain integrable modules of Chari-Presslely for an (untwisted) affine Lie algebra $\hat{\g}$ by exploiting basic formal variable techniques. We define and study two categories ${\mathcal{E}}$ and ${\mathcal{C}}$ of $\hat{\g}$-modules using generating functions, where ${\mathcal{E}}$ includes evaluation modules and ${\mathcal{C}}$ unifies highest weight modules, evaluation modules and their tensor product modules, and we classify integrable irreducible $\hat{\g}$-modules in categories ${\mathcal{E}}$ and ${\mathcal{C}}$.
In this paper, we give a characterization of the crystal bases $\mathcal{B}_{x}^{+}(λ)$, $x \in W_{\mathrm{af}}$, of Demazure submodules $V_{x}^{+}(λ)$, $x \in W_{\mathrm{af}}$, of a level-zero extremal weight module $V(λ)$ over a quantum affine algebra $U_{q}$, where $λ$ is an arbitrary level-zero dominant integral weight, and $W_{\mathrm{af}}$ denotes the affine Weyl group. This characterization is given in terms of the initial direction of a semi-infinite Lakshmibai-Seshadri path, and is established under a suitably normalized isomorphism between the crystal basis $\mathcal{B}(λ)$ of the le…
We give a proof of the cyclicity conjecture of Akasaka-Kashiwara, for simply laced types, via quiver varieties. We get also an algebraic characterization of the standard modules.
Feigin-Stoyanovsky's type subspaces for affine Lie algebras of type $C_\ell^{(1)}$ 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.
We prove the Kirillov-Reshetikhin conjecture concerning certain finite dimensional representations of a quantum affine algebra $\Ua$ when $\hat\g$ is an untwisted affine Lie algebra of type $ADE$. We use $t$--analog of $q$--characters introduced by the author in an essential way.
Given an affine Kac-Moody Lie algebra $\tilde{\mathfrak{g}}[σ]$ of arbitrary type, we determine certain minimal sets of annihilating fields of standard $\tilde{\mathfrak{g}}[σ]$-modules. We then use these sets in order to obtain a characterization of standard $\tilde{\mathfrak{g}}[σ]$-modules in terms of irreducible loop $\tilde{\mathfrak{g}}[σ]$-modules, which proves to be a useful tool for combinatorial constructions of bases for standard $\tilde{\mathfrak{g}}[σ]$-modules.
Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$. We give a polyhedral realization of the crystal basis for the extremal weight module of extremal weight $λ$, where $λ$ is an integral weight whose Weyl group orbit has neither a dominant integral weight nor an antidominant integral weight.
We study the universal integrable modules W_q(m) of level zero for quantum affine sl_2 and a family of maximal finite--dimensional quotients of these modules. We show that these all have dimension 2^m. Using this, we are able to realize W_q(m) as the space of invariants of a reprsentation of the Hecke algebra H_m. We also give a conjecture in the general case.
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.
In this paper, we study nullity-2 toroidal extended affine Lie algebras in the context of vertex algebras and their $ϕ$-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and $ϕ$-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of irreducible modules for the variant Lie algebras.
We define integrable representations of quantum toroidal algebras of type A by tensor product, using the Drinfeld "coproduct". This allow us to recover the vector representations recently introduced by Feigin-Jimbo-Miwa-Mukhin [6] and constructed by the author [21] as a subfamily of extremal loop weight modules. In addition we get new extremal loop weight modules as subquotients of tensor powers of vector representations. As an application we obtain finite-dimensional representations of quantum toroidal algebras by specializing the quantum parameter at roots of unity.
An action of the Yangian of the general Lie algebra gl(N) is defined on every irreducible integrable highest weight module of affine gl(N) with level greater than 1. This action is derived, by means of the Drinfeld duality and a subsequent semi-infinite limit, from a certain induced representation of the degenerate double affine Hecke algebra H. Each vacuum module of affine gl(N) is decomposed into irreducible Yangian subrepresentations by means of the intertwiners of H. Components of this decomposition are parameterized by semi-infinite skew Young diagrams.
Using the tensor category theory developed by Lepowsky, Zhang and the second author, we construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We conjecture that this braided tensor category is rigid and thus is a ribbon category. We also give conjectures on the modularity of this category and on the equivalence with a suitable quantum group tensor category. In the special case that the affine Lie algebra is $\widehat{\mathfrak{sl}}_2$, we prove the rigidity and modularity conjectures.
We consider a category of $\gl_\infty$-crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight $\gl_\infty$-modules with respect to derivations shifting the characters of fundamental modules. A Littlewood-Richardson rule of extremal weight crystals with non-negative level is described ex…
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.
In this paper we construct a new family of representations for the quantum toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of Hernandez [24]. We construct extremal loop weight modules associated to level 0 fundamental weights $\varpi_\ell$ when $n=2r+1$ is odd and $\ell=1, r+1$ or $n$. To do it, we relate monomial realizations of level 0 extremal fundamental weight crystals with integrable representations of $\mathcal{U}_q(sl_{n+1}^{tor})$, and we introduce promotion operators for the level 0 extremal fundamental weight crystals. By specializing the quantum parameter, w…
Crystal base of the level 0 part of the modified quantum affine algebra $\widetilde U_q(\widehat{sl_2})_0$ is given by path. Weyl group actions, extremal vectors and crystal structure of all irreducible components are described explicitly.
Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.
In this paper, we classify the finite dimensional irreducible modules for affine BMW algebra over an algebraically closed field with arbitrary characteristic.
Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2, and set $λ=Λ_{1} - Λ_{2}$, where $Λ_{1}$, $Λ_{2}$ are the fundamental weights. Denote by $V(λ)$ the extremal weight module of extremal weight $λ$ with $v_λ$ the extremal weight vector, and by $\mathcal{B}(λ)$ the crystal basis of $V(λ)$ with $u_λ$ the element corresponding to $v_λ$. We prove that (i) $\mathcal{B}(λ)$ is connected, (ii) the subset $\mathcal{B}(λ)_μ$ of elements of weight $μ$ in $\mathcal{B}(λ)$ is a finite set for every integral weight $μ$, and $\mathcal{B}(λ)_λ = \{u_λ\}$, (iii) every extremal element in $\mathcal…
Assuming the existence of the perfect crystal bases of Kirillov-Reshetikhin modules over simply-laced quantum affine algebras, we construct certain perfect crystals for twisted quantum affine algebras, and also provide compelling evidence that the constructed crystals are isomorphic to the conjectural crystal bases of Kirillov-Reshetikhin modules over twisted quantum affine algebras.
Kleshchev, Mathas and Ram (2012) gave a presentation for graded Specht modules over Khovanov-Lauda-Rouquier algebras of finite and affine type A. We show that this construction can be applied more generally to skew shapes to give a presentation of graded skew Specht modules, which arise as subquotients of restrictions of Specht modules. As an application, we show that cuspidal modules associated to a balanced convex preorder in affine type A are skew Specht modules for certain hook shapes.
We construct a class of modules for extended affine Lie algebra $\widetilde{\frak{gl}_l({\bc_q})}$ by using the free fields. A necessary and sufficient condition is given for those modules being irreducible.
We prove a bijection between finite-dimensional irreducible modules for an arbitrary quantum affine algebra $U_q(g)$ and finite-dimensional irreducible modules for its Borel subalgebra $U_q(g)^{\geq 0}$.
In recent work, Wang and the third author defined a class of 'extremal' vertex operator algebras (VOAs), consisting of those with at least two simple modules and conformal dimensions as large as possible for the central charge. In this article we show that there are exactly 15 character vectors of extremal VOAs with two simple modules. All but one of the 15 character vectors is realized by a previously known VOA. The last character vector is realized by a new VOA with central charge 33.
We study monoidal categorifications of certain monoidal subcategories $\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over quiver Hecke algebra of type A${}_\infty$. In particular, when the quantum affine algebra is of type A or B, the subcategory coincides with the monoidal category $\mathcal{C}_{\mathfrak{g}}^0$ introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.
We define the extremal projector of the $q$-boson Kashiwara algebra $B_q(\ge)$ and study their basic properties. Applying their proerties to the representation theory of the category ${\cal O}(B_q(\ge))$, whose objects are ''upper bounded '' $B_q(\ge)$-modules, we obtain its semi-simplicity and the classification of simple modules.
We define the categories of weight-finite modules over the type $\mathfrak a_1$ quantum affine algebra $\dot{\mathrm{U}}_q(\mathfrak a_1)$ and over the type $\mathfrak a_1$ double quantum affine algebra $\ddot{\mathrm{U}}_q(\mathfrak a_1)$ that we introduced in a previous paper. In both cases, we classify the simple objects in those categories. In the quantum affine case, we prove that they coincide with the simple finite-dimensional $\dot{\mathrm{U}}_q(\mathfrak a_1)$-modules which were classified by Chari and Pressley in terms of their highest (rational and $\ell$-dominant) $\ell$-weights or…
We shall derive Kazhdan-Lusztig type character formula for the irreducible modules with arbitrary non-critical highest weights over affine Lie algebras from the rational case by using the translation functor, the Enright functor and Jantzen's deformation argument.
We study a class of representations over the degenerate double affine Hecke algebra of gl_n by an algebraic method. As fundamental objects in this class, we introduce certain induced modules and study some of their properties. In particular, it is shown that these induced modules have unique simple quotient modules under certain conditions. Moreover, we show that any simple module in this class is obtained as such a simple quotient, and give a classification of all the simple modules.
We introduce a sequence of $q$-characters of standard modules of a quantum affine algebra and we prove it has a limit as a formal power series. For $\mathfrak{g}=\hat{\mathfrak{sl}_{2}}$, we establish an explicit formula for the limit which enables us to construct corresponding asymptotical standard modules associated to each simple module in the category $\mathcal{O}$ of a Borel subalgebra of the quantum affine algebra. Finally, we prove a decomposition formula for the limit formula into $q$-characters of simple modules in this category $\mathcal{O}$.
We apply the construction of the universal lower-bounded generalized twisted modules by the author to construct universal lower-bounded and grading-restricted generalized twisted modules for affine vertex (operator) algebras. We prove that these universal twisted modules for affine vertex (operator) algebras are equivalent to suitable induced modules of the corresponding twisted affine Lie algebra or quotients of such induced modules by explicitly given submodules.
Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $\mathcal{C}^0_{\mathfrak{g}}$ be Hernandez-Leclerc's category. For a duality datum $\mathcal{D}$ in $\mathcal{C}^0_{\mathfrak{g}}$, we denote by $\mathcal{F}_{\mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give sufficient conditions for a duality datum $\mathcal{D}$ to provide the functor $\mathcal{F}_{\mathcal{D}}$ sending simple modules to simple modules. Then we introduce the notion of cuspidal modules in $\mathcal{C}^0_{\mathfrak{g}}$, and show that all simple modules in $\mathcal{C}…
Using the bases of principal subspaces for twisted affine Lie algebras except $A_{2l}^{(2)}$ by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight $kΛ_0$ and parafermionic spases for the same affine Lie algebras. As a result, we obtain their character formulas conjectured in arXiv:math/0102113.
In this paper, we explore a canonical connection between the algebra of $q$-difference operators $\widetilde{V}_{q}$, affine Lie algebra and affine vertex algebras associated to certain subalgebra $\mathcal{A}$ of the Lie algebra $\mathfrak{gl}_{\infty}$. We also introduce and study a category $\mathcal{O}$ of $\widetilde{V}_{q}$-modules. More precisely, we obtain a realization of $\widetilde{V}_{q}$ as a covariant algebra of the affine Lie algebra $\widehat{\mathcal{A}^{*}}$, where $\mathcal{A}^{*}$ is a 1-dimensional central extension of $\mathcal{A}$. We prove that restricted $\widetilde{V_…
We use combinatorial description of bases of Feigin-Stoyanovsky's type subspaces of standard modules of level 1 for affine Lie algebras of types $A_\ell^{(1)}$ and $D_4^{(1)}$ to obtain character formulas. These descriptions naturally lead to systems of recurrence relations for which we also find solutions.
We define an integral intertwining operator among modules for a vertex operator algebra to be an intertwining operator which respects integral forms in the modules, and we show that an intertwining operator is integral if it is integral when restricted to generators of the integral forms in the modules. We apply this result to classify integral intertwining operators which respect certain natural integral forms in modules for affine Lie algebra and lattice vertex operator algebras.
We construct a family of homomorphisms between Weyl modules for affine Lie algebras in characteristic p, which supports our conjecture on the strong linkage principle in this context. We also exhibit a large class of reducible Weyl modules beyond level one, for p not necessarily small.
Let $L(-{1/2}(l+1),0)$ be the simple vertex operator algebra associated to an affine Lie algebra of type $A_{l}^{(1)}$ with the lowest admissible half-integer level $-{1/2}(l+1)$, for even l. We study the category of weak modules for that vertex operator algebra which are in category $\cal{O}$ as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.
Let $\tilde{\mathfrak{g}}$ be the affine Lie algebra of type $A_{2l}^{(2)}$. The integrable highest weight $\tilde{\mathfrak{g}}$-module $L(kΛ_0)$ called the standard $\tilde{\mathfrak{g}}$-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 arXiv:math/0102113.
We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting), to $\mathfrak{sl}_{n+1}[t]$--stable, prime Demazure modules in level two integrable highest weight rep…
We develop some basic properties such as $p$-centers of affine vertex algebras and free field vertex algebras in prime characteristic. We show that the Wakimoto-Feigin-Frenkel homomorphism preserves the $p$-centers by providing explicit formulas. This allows us to formulate the notion of baby Wakimoto modules, which in particular provides an interpretation in the context of modular vertex algebras for Mathieu's irreducible character formula of modular affine Lie algebras at the critical level.
