Masaki Kashiwara, Toshiyuki Tanisaki
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.
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In this paper, we classify the finite dimensional irreducible modules for affine BMW algebra over an algebraically closed field with arbitrary characteristic.
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 $U_q(g)^{\geq 0}$.
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 introduce and study an affine analogue of skew Young diagrams and tableaux on them. It turns out that the double affine Hecke algebra of type $A$ acts on the space spanned by standard tableaux on each diagram. It is shown that the modules obtained this way are irreducible, and they exhaust all irreducible modules of a certain class over the double affine Hecke algebra. In particular, the classification of irreducible modules of this class, announced by Cherednik, is recovered.
By using methods developed in arXiv:math/0602181 we study the irreducibility of certain Wakimoto modules for $\widehat{sl_2}$ at the critical level. We classify all $χ\in {\Bbb C}((z))$ such that the corresponding Wakimoto module $W_χ$ is irreducible. It turns out that zeros of Schur polynomials play important rule in the classification result.
We classify the finite dimensional irreducible representations of affine Hecke algebras of type B_2 with unequal parameters.
We construct a new family of irreducible modules over any basic classical affine Kac-Moody Lie superalgebra which are induced from modules over the Heisenberg subalgebra. We also obtain irreducible deformations of these modules for the quantum affine superalgebra of type A.
We show that ${\rm End}_{\bf U}(V_λ\otimes V^{\otimes n})$ is generated by the affine braid group $AB_n$ where ${\bf U}=U_q\mathfrak g(G_2)$, $V$ is its 7-dimensional irreducible representation and $V_λ$ is an arbitrary irreducible representation.
Let $U_q(\hat{sl}_2)^{\geq 0}$ denote the Borel subalgebra of the quantum affine algebra $U_q(\hat{sl}_2)$. We show that the following hold for any choice of scalars $ε_0, ε_1$ from the set ${1,-1}$. (i) Let $V$ be a finite-dimensional irreducible $U_q(\hat{sl}_2)^{\geq 0}$-module of type $(ε_0,ε_1)$. Then the action of $U_q(\hat{sl}_2)^{\geq 0}$ on $V$ extends uniquely to an action of $U_q(\hat{sl}_2)$ on $V$. The resulting $U_q(\hat{sl}_2)$-module structure on $V$ is irreducible and of type $(ε_0,ε_1)$. (ii) Let $V$ be a finite-dimensional irreducible $U_q(\hat{sl}_2)$-module of type $(ε_0,ε…
For a twisted affine Lie superalgebra with nonzero odd part, we study {tight irreducible weight modules} with bounded weight multiplicities and show that if the action of nonzero real vectors of each affine component of the zero part is neither completely injective nor completely locally nilpotent, then these modules are parabolically induced.
Young wall realizations of level 1 irreducible highest weight and Fock space crystals of quantum affine algebras in type E
本文为$E_{6}^{(1)}$、$E_{7}^{(1)}$和$E_{8}^{(1)}$型量子仿射代数的1级不可约最高权与Fock空间表示晶体基构造Young wall模型，通过彩色块的堆叠与增减描述晶体结构。
Oscillator representations of quantum affine orthosymplectic superalgebras
本文引入D型量子仿射超代数的q-振子表示范畴并构造新族不可约表示，基于超对偶理论证明其插值两类不可约表示，建立了Howe对偶对应的量子仿射类比。
We classify the irreducible integrable modules for the twisted toroidal extended affine Lie algebras with fnite diemnsional weight spaces where the fnite dimensional center acts trivially. We have proved that the entire central extension part acts trivially on the modules and our modules turn out to be highest weight modules.
We study $\mathbb Z$-graded modules of nonzero level with arbitrary weight multiplicities over Heisenberg Lie algebras and the associated generalized loop modules over affine Kac-Moody Lie algebras. We construct new families of such irreducible modules over Heisenberg Lie algebras. Our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine Lie algebras. In particular, to any function $ϕ:\mathbb N\rightarrow \{\pm\}$ we associate a $ϕ$-highest weight module over the Heisen…
We study representations of the double affine Lie algebra associated to a simple Lie algebra. We construct a family of indecomposable integrable representations and identify their irreducible quotients. We also give a condition for the indecomposable modules to be irreducible, this is analogous to a result in the representation theory of quantum affine algebras. Finally, in the last section of the paper, we show, by using the notion of fusion product, that our modules are generically reducible.
Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1 irreducible integrable highest weight representations and classify the finite dimensional irreducible representations of Uq(g). This makes essential use of the Drinfeld realisation for Uq(g), and quantum correspondences between affine Kac-Moody superalgebras, developed in earlier papers.
We construct irreducible modules for twisted toroidal Lie algebras and extended affine Lie algebras. This is done by combining the representation theory of untwisted toroidal algebras with the technique of thin coverings of modules. We illustrate our method with examples of extended affine Lie algebras of Clifford type.
We prove a character formula for the irreducible modules from the category $\mathcal{O}$ over the simple affine vertex algebra of type $A_n$ and $C_n$ $(n \geq 2)$ of level $k=-1$. We also give a conjectured character formula for types $D_4$, $E_6$, $E_7$, $E_8$ and levels $k=-1, \cdots, -b$, where $b=2,3,4,6$ respectively.
This paper classifies irreducible, integrable highest weight modules for "current Kac-Moody Algebras" with finite dimensional weight spaces. We prove that these modules turn out to be modules of appropriate direct sums of finitely many copies of Kac-Moody Lie algebras.
In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra of type $A_1$ by taking tensor products of a finite number of irreducible modules $M(λ, α, β, γ)$ with irreducible highest weight modules $V(η, ε, θ)$. We obtain the necessary and sufficient conditions for such tensor product modules to be irreducible, and determine the necessary and sufficient conditions for such two modules to be isomorphic. We also compare these modules with other known non-weight modules, showing that these irreducible modules are new.
Koenig and Xi introduced {\em affine cellular algebras}. Kleshchev and Loubert showed that an important class of {\em infinite dimensional} algebras, the KLR algebras $R(Γ)$ of finite Lie type $Γ$, are (graded) affine cellular; in fact, the corresponding affine cell ideals are idempotent. This additional property is reminiscent of the properties of {\em quasihereditary algebras} of Cline-Parshall-Scott in a {\em finite dimensional} situation. A fundamental result of Cline-Parshall-Scott says that a finite dimensional algebra $A$ is quasihereditary if and only if the category of finite dimensio…
Affine highest weight structures on module categories over quiver Hecke algebras
本文证明任意类型箭图Hecke代数的有限生成分次模范畴具有多种Kleshchev分层，从而得到与Weyl群元素相关的量子幺正子群范畴为仿射最高权范畴，推广了多项早期工作，关键利用行列式模和R-矩阵技术。
In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra of type $A_1$ by taking tensor products of irreducibles defined in [Q. Chen, J. Han, Non-weight modules over the affine-Virasoro algebra of type $A_1$, J. Math. Phys. 60, 071707 (2019)] with irreducible highest weight modules. The irreducibility and the isomorphism classes of these modules are determined. Moreover, we show that these tensor product modules are different from the known non-weight modules. Finally, we realize some tensor product modules as induced modules from modules over certain subalgeb…
For a Dynkin quiver $Q$ (of type ADE), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc's monoidal category $\mathcal{C}_Q$ of modules over the quantum loop algebra $U_q(L\mathfrak{g})$ via Nakajima's homomorphism. As an application, we show that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives an equivalence between the cate…
We give a full classification, in terms of periodic skew diagrams, of irreducible semisimple modules in category O for the degenerate double affine Hecke algebra of type A which can be realized as submodules of Verma modules.
Let $G$ be the Weil restriction of a general linear group. By extending the method of semi-modules developed by de Jong, Oort, Viehmann and Hamacher, we obtain a stratification of the affine Deligne-Lusztig varieties for $G$ (in the affine Grassmannian) attached to a minuscule coweight and a basic element. As an application, we verify a conjecture by Chen and Zhu on irreducible components of affine Deligne-Lusztig varieties for $G$.
We give a detailed account of a combinatorial construction, due to Cherednik, of cyclic generators for irreducible modules of the affine Hecke algebra of the general linear group with generic parameter q.
The Lascoux-Leclerc-Thibon conjecture, reformulated and solved by S. Ariki, asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with the Lie algebra $gl_\infty$ or the affine Lie algebra $A^{(1)}_\ell$, and the irreducible representations correspond to the upper global bases. In this note, we formulate analogous conjectures for certain classes of irreducible representations of affine Hecke algebras of type B. We corrected typos.
This paper introduces calibrated representations for affine Hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations. The primary technique is to provide indexing sets for controlling the weight space structure of finite dimensional modules for the affine Hecke algebra. Using these indexing sets we show that (1) irreducible calibrated representations are indexed by skew local regions, (2) the dimension of an irreducible calibrated representation is the number of chambers in the local region, (3) each irreducible calibrated representation is con…
In this paper, we explore natural connections among trigonometric Lie algebras, (general) affine Lie algebras, and vertex algebras. Among the main results, we obtain a realization of trigonometric Lie algebras as what were called the covariant algebras of the affine Lie algebra $\widehat{\mathcal{A}}$ of Lie algebra $\mathcal{A}=\frak{gl}_{\infty}\oplus\frak{gl}_{\infty}$ with respect to certain automorphism groups. We then prove that restricted modules of level $\ell$ for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras $V_{\widehat{\…
In this paper we construct a class of new irreducible modules over untwisted affine Kac-Moody algebras $\widetilde{\mathfrak{g}}$, generalizing and including both highest weight modules and Whittaker modules. These modules allow us to obtain a complete classification of irreducible $\widetilde{\mathfrak{g}}$-modules on which the action of each root vector in $\widetilde{\mathfrak{n}}_+$ is locally finite, where $\widetilde{\mathfrak{n}}_+$ is the locally nilpotent subalgebra (or positive part) of $\widetilde{\mathfrak{g}}$. The necessary and sufficient conditions for two such irreducible $\wid…
We study a class of representations called ``calibrated representations'' of the degenerate double affine Hecke algebra and those of the rational Cherednik algebra of type ${\mathrm{GL}}_n$. We give a realization of calibrated irreducible modules as spaces of coinvariants constructed from integrable modules over the affine Lie algebra $\hat{\mathfrak{gl}}_m$. Moreover, we give a character formula of these irreducible modules in terms of a level-restricted Kostka polynomials. These results were conjectured by Arakawa, Suzuki and Tsuchiya based on the conformal field theory. The proofs using rec…
We describe Borel and parabolic subalgebras of affine Lie superalgebras and study the Verma type modules associated to such subalgebras. We give necessary and sufficient conditions under which these modules are simple.
We give a combinatorial description of all affine spherical varieties with prescribed weight monoid $Γ$. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion's moduli scheme $\mathrm M_Γ$ for such varieties. Moreover, we find several sufficient conditions for $\mathrm M_Γ$ to be irreducible and exhibit several examples where $\mathrm M_Γ$ is reducible. Finally, we provide examples of non-reduced $\mathrm M_Γ$.
Classification of irreducible representations of affine group superschemes and the division superalgebras of their endomorphisms
本文在特征不为2的域上，基于可分闭包上的表示及其Galois扭转，分类了仿射群超概形的不可约表示，计算了其自同态的除超代数，并利用Shibata的Borel--Weil理论给出了特定条件下拟约化代数超群的数值结论。
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.
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.
We define a filtration indexed by the integers on the tensor product of an integrable highest weight module and a loop module for a quantum affine algebra. We prove that the filtration is either trivial or strictly decreasing and give sufficient conditions for this to happen. In the first case we prove that the module is irreducible and in the second case we prove that the intersection of all the modules is zero, thus allowing us to define the completed tensor product. In certain special cases, we identify the subsequent quotients of filtration. These are certain highest weight integrable modu…
Two classes of irreducible highest weight modules of the general linear Lie superalgebra $gl(1/\infty)$ are constructed. Within each module a basis is introduced and the transformation relations of the basis under the action of the algebra generators are written down.
Quasi-finite modules over affine and extended affine Lie algebras
本文针对无中心核心为多圈代数的扩张仿射李代数，分类了其核心作用非平凡的不可约拟有限模，同时推广了仿射Kac-Moody代数上的相关经典分类结果。
Gelfand-Tsetlin bases of representations for super Yangian and quantum affine superalgebra
本文通过构造超 Yangian 模的 Gelfand-Tsetlin 基并给出 Drinfeld 生成元的显式作用，重新证明其不可约性，同时刻画了有限维不可约驯顺 Y(gl(1|1))-模恰为薄模，并将相应结论推广到 A 型量子仿射超代数。
This paper introduces the notion of calibrated representations for affine Hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations. The main results are that (1) irreducible calibrated representations are indexed by placed skew shapes, (2) the dimension of an irreducible calibrated representation is the number of standard Young tableaux corresponding to the placed skew shape and (3) each irreducible calibrated representation is constructed explicitly by formulas which describe the action of each generator of the affine Hecke algebra on a specif…
We will classify finite dimensional irreducible modules for affine quantum Schur algebras at roots of unity and generalize \cite[(6.5f) and (6.5g)]{Gr80} to the affine case in this paper.
In 2006, Gao and Zeng \cite{GZ} gave the free field realizations of highest weight modules over a class of extended affine Lie algebras. In the present paper, applying the technique of localization to those free field realizations, we construct a class of new weight modules over the extended affine Lie algebras. We give necessary and sufficient conditions for these modules to be irreducible. In this way, we construct free field realizations for a class of simple weight modules with infinite weight multiplicities over the extended affine Lie algebras.
We study the crystal graphs of irreducible $U_{v}(\hat{sl}}_{e})$-modules of higher level l. Generalizing works of the first author, we obtain a simple description of the bijections between the classes of multipartitions which naturally label these graphs: the Uglov multipartitions. This is achieved by expliciting an embedding of the $U_{v}(\hat{sl}}_{e})$-crystals of level l into $U_{v}(\hat{sl}_{\infty})$-crystals associated to highest weight modules.
In this paper we give a new proof for the classification of irreducible modules of an affine Hecke algebra of type $A_n$, which was obtained by G. E. Murphy in 1995.
Let L(n-l+1/2,0) be the vertex operator algebra associated to an affine Lie algebra of type B_l^(1) at level n-l+1/2, for a positive integer n. We classify irreducible L(n-l+1/2,0)-modules and show that every L(n-l+1/2,0)-module is completely reducible. In the special case n=1, we study a category of weak L(-l+3/2,0)-modules which are in the category $\cal{O}$ as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.
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…
The Lascoux-Leclerc-Thibon-Ariki theory asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra $g$ where $g$ is $gl_\infty$ or the affine Lie algebra $A^{(1)}_\ell$, and the irreducible representations correspond to the upper global bases. Recently, N. Enomoto and the first author presented the notion of symmetric crystals and formulated analogous conjectures for the affine Hecke algebras of type B. In this note, we present similar conjectures f…
We show that the representation theory for the toroidal extended affine Lie algebras is controlled by a vertex operator algebra which is a tensor product of four VOAs: a sub-VOA of the hyperbolic lattice VOA, two affine VOAs and a Virasoro VOA. A tensor product of irreducible modules for these VOAs admits the structure of an irreducible module for the toroidal extended affine Lie algebra. We also show that for N=12, the sub-VOA of the hyperbolic lattice VOA becomes an exceptional irreducible module for the extended affine Lie algebra of rank 0.
In this paper we study the representations of loop Affine-Virasoro Algebras. As they have canonical triangular decomposition, we define Verma modules and its irreducible quotients. We give necessary and sufficient condition for an irreducible highest weight module to have finite dimensional weight spaces. We prove that an irreducible integrable module is either an highest weight module or a lowest weight module whenever the canonical central element acts non trivially. At the end we construct Affine central operators for each integer and they commute with the action of the Affine Lie Algebra.
We describe the category of integrable sl(1|n)^ -modules with the positive central charge and show that the irreducible modules provide the full set of irreducible representations for the corresponding simple vertex algebra.
We introduce the notion of standard multipartitions and establish a one-to-one correspondence between standard multipartitions and irreducible representations with integral weights for the affine Hecke algebra of type A with a parameter q which is not a root of unity. We then extend the correspondence to all Kleshchev multipartitions for Ariki-Koike algebras of integral type. By the affine Schur--Weyl duality, we further extend this to a correspondence between standard multipartitions and Drinfeld multipolynomials of integral type whose associated irreducible polynomial representations complet…
A cuspidal system for an affine Khovanov-Lauda-Rouquier algerba $R_\al$ yields a theory of standard modules. This allows us to classify the irreducible modules over $R_\al$ up to the so-called imaginary modules. We make a conjecture on reductions modulo $p$ of irreducible $R_\al$-modules, which generalizes James Conjecture. We also describe minuscule imaginary modules, laying the groundwork for future study of imaginary Schur-Weyl duality. We introduce colored imaginary tensor spaces and reduce a classification of imaginary modules to one color. We study the characters of cuspidal modules. We …
On the computation of the canonical basis for irreducible highest weight $U_q (\mathfrak{gl}_{\infty})$-modules
本文提出了一种新的方法，用于计算不可约最高权 $U_q (\mathfrak{gl}_{\infty})$-模的 canonical basis，并将其应用于 Calogero-Moser 细胞特征标和 Ariki-Koike 代数的分解矩阵。
We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and $q$-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the $σ$-centralizer group to the Mirkovic-Vilonen basis of a certain weight space of a representation of the Langlands dual group.
In this paper, we characterize quasi-integrable modules, of nonzero level, over twisted affine Lie superalgebras. We show that quasi-integrable modules are not necessarily highest weight modules. We prove that each quasi-integrable module is parabolically induced from a cuspidal module, over a finite dimensional Lie superalgebra having a Cartan subalgebra whose corresponding root system just contain real roots; in particular, the classification of quasi-integrable modules is reduced to the known classification of cuspidal modules over such Lie superalgebras.
Let ${\mathsf F}$ be the Schur functor from the category of finite dimensional ${\mathcal H}_{\vartriangle}(r)_\mathbb C$-modules to the category of finite dimensional ${\mathcal S}_{\vartriangle}(n,r)_{\mathbb{C}}$-modules, where ${\mathcal H}_{\vartriangle}(r)_\mathbb C$ is the extended affine Hecke algebra of type $A$ over ${\mathbb C}$ and ${\mathcal S}_{\vartriangle}(n,r)_{\mathbb{C}}$ is the affine quantum Schur algebras over $\mathbb{C}$. The Drinfeld polynomials associated with ${\mathsf F}(V)$ were determined in \cite[7.6]{CP96} and \cite[4.4.2]{DDF} in the case of $n>r$, where $V$ is…
On multiplicity-free weight modules over quantum affine algebras
本文旨在构造并研究量子仿射代数的无重数权模，通过引入关于可对称化广义 Cartan 矩阵的可移性条件，探讨了其在代数结构与无穷维模实现中的应用，并计算了相关最高 $\ell$-权。
By extending the method of semi-modules developed by de Jong, Oort, Viehmann and Hamacher, we introduce a stratification for the affne Deligne-Lusztig variety (in the affne Grassmannian) attached to attached to a minuscule cocharacter and a basic element. As an application, we complete the proof of a conjecture on the irreducible components of affne Deligne-Lusztig varieties, due to Miaofen Chen and Xinwen Zhu.
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