We introduce the notion of a spectral character for finite-dimensional representations of affine algebras. These can be viewed as a suitable q=1 limit of the elliptic characters defined by Etingof and Moura for quantum affine algebras. We show that these characters determine blocks of the category of finite-dimensional modules for affine algebras. To do this we use the Weyl modules defined by Chari and Pressley and some indecomposable reducible quotient of the Weyl modules.
Using the orbifold KZ connection we construct a functor from an affine parabolic category O of type A to the category O of a cyclotomic rational double affine Hecke algebra. We give several results concerning this functor.
We give a presentation of localized affine and degenerate affine Hecke algebras of arbitrary type in terms of weights of the polynomial subalgebra and varied Demazure-BGG type operators. We offer a definition of a graded algebra $\mathcal{H}$ whose category of finite-dimensional ungraded nilpotent modules is equivalent to the category of finite-dimensional modules over an associated degenerate affine Hecke algebra. Moreover, unlike the traditional grading on degenerate affine Hecke algebras, this grading factors through central characters, and thus gives a grading to the irreducible representa…
We give a proof of the parabolic/singular Koszul duality for the category O of affine Kac-Moody algebras. The main new tool is a relation between moment graphs and finite codimensional affine Schubert varieties. We apply this duality to q-Schur algebras and to cyclotomic rational double affine Hecke algebras.
We prove that the affine Yokonuma-Hecke algebra defined by Chlouveraki and Poulain d'Andecy is a particular case of the pro-$p$-Iwahori-Hecke algebra defined by Vignéras.
We establish an explicit algebra isomorphism between the affine Yokonuma-Hecke algebra $\widehat{Y}_{r,n}(q)$ and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type $A.$ As an application of this result, we show that $\widehat{Y}_{r,n}(q)$ is affine cellular in the sense of Koenig and Xi, and further prove that it has finite global dimension when the parameter $q$ is not a root of the Poincaré polynomial. As another application, we also recover the modular representation theory of $\widehat{Y}_{r,n}(q)$ previously obtained in [CW].
We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type $A$ (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic $p$. Secondly, the modular branching rules for these alg…
We define an affine partition algebra by generators and relations and prove a variety of basic results regarding this new algebra analogous to those of other affine diagram algebras. In particular we show that it extends the Schur-Weyl duality between the symmetric group and the partition algebra. We also relate it to the affine partition category recently defined by J. Brundan and M. Vargas. Moreover, we show that this affine partition category is a full monoidal subcategory of the Heisenberg category.
After establishing a geometric Schur-Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts have nice ideals coming from geometry, allowing duality for quotients. Some of the quotients of the positive affine Hecke algebra are then identified to some cyclotomic Hecke algebras and the geometric setting allows the construction of canonical bases.
We generalize I. Frenkel's orbital theory for non twisted affine Lie algebras to the case of twisted affine Lie algebras using a character formula for certain non-connected compact Lie groups.
We classify the simple integrable modules of double affine Hecke algebras via perverse sheaves. We get also some estimate for the Jordan-Holder multiplicities of induced modules.
We introduce the notion of spectral transfer morphisms between normalized affine Hecke algebras, and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine Hecke algebras involved. We define a partial ordering on the set of isomorphism classes of normalized affine Hecke algebras, which plays an important role for the Langlands parameters of Lusztig's unipotent representations.
We prove complete reducibility for an integrable module for an affine Lie algebra where the canonical central element acts non-trivially. We further prove that integrable modules does not exists for most of the super affine Lie algebras where the center acts non-trivially.
We use localization technique to construct new families of irreducible modules of affine Kac-Moody algebras. In particular, localization is applied to the first free field realization of the affine Lie algebra A_1^{(1)} or, equivalently, to imaginary Verma modules.
Let $\hat{\mathfrak g}$ be an affine Lie algebra of type 1. We give a PBW basis for the quantum affine algebra $U_q(\hat{\mathfrak g})$ with respect to the triangular decomposition of $\hat{\mathfrak g}$ associated with the imaginary positive root system.
We introduce a new class of extended affine Lie algebras called Hamiltonian Extended Lie Algebras(HEALAs). They are so called because the corresponding derivation algebra is the classical Hamiltonian algebra. We classify the irreducible integrable modules for the HEALA based on the classification of irreducible Jet modules for the Hamiltonian algebra (both zero level and non-zero level.
These are notes for lectures given at MIT in 1999 (Fall). They contain a discussion of affine Hecke algebras with possibly unequal parameters including a theory of cells and a partly conjectural theory of J-rings.
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.
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…
In this paper we prove that the linear Koszul duality equivalence constructed in a previous paper provides a geometric realization of the Iwahori-Matsumoto involution of affine Hecke algebras.
We prove a conjecture of Kashiwara and Miemietz on canonical bases and branching rules of affine Hecke algebras of type D. The proof is similar to the proof of the type B case.
We study Lusztig's t-analog of weight multiplicities associated to level one representations of twisted affine Kac-Moody algebras. An explicit closed form expression is obtained for the corresponding t-string function using constant term identities of Macdonald and Cherednik. The closed form involves the generalized exponents of the graded pieces of the twisted affine algebra, considered as modules for the underlying finite dimensional simple Lie algebra. This extends previous work on level 1 t-string functions for the untwisted simply-laced affine Kac-Moody algebras.
We prove a series of conjectures of Enomoto and Kashiwara on canonical bases and branching rules of affine Hecke algebras of type B. The main ingredient of the proof is a new graded Ext-algebra associated with quiver with involutions that we compute explicitly.
We interpret and develop a theory of loop algebras as torsors (principal homogeneous spaces) over Laurent polynomial rings . As an application, we recover Kac's realization of affine Kac-Moody Lie algebras.
We introduce the spin Hecke algebra, which is a q-deformation of the spin symmetric group algebra, and its affine generalization. We establish an algebra isomorphism which relates our spin (affine) Hecke algebras to the (affine) Hecke-Clifford algebras of Olshanski and Jones-Nazarov. Relation between the spin (affine) Hecke algebra and a nonstandard presentation of the usual (affine) Hecke algebra is displayed, and the notion of covering (affine) Hecke algebra is introduced to provide a link between these algebras. Various algebraic structures for the spin (affine) Hecke algebra are establishe…
The degenerate affine and affine BMW algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper the theory is unified by treating the orthogonal and symplectic cases simultaneously; we make an exact parallel between the degenerate affine and affine cases via a new algebra which takes the role of the affine braid group for the degenerate setting. A main result of this paper is an iden…
The universal enveloping algebra ${\mathcal U}({\widehat{\frak{gl}}_n})$ of ${\widehat{\frak{gl}}_n}$ was realized in \cite[Ch. 6]{DDF} using affine Schur algebras. In particular some explicit multiplication formulas in affine Schur algebras were derived. We use these formulas to study the structure of affine Schur algebras. In particular, we give a presentation of the affine Schur algebra ${\mathcal S}_{\!\vartriangle}(n,r)_{\mathbb Q}$.
We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.
Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the centers. We further develop connections of these algebras with the usual degenerate (i.e. graded) affine Hecke algebras of Lusztig by introducing a notion of degenerate covering affine Hecke algebras.
The affine and degenerate affine Birman-Murakami-Wenzl (BMW) algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce acti…
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.
The affine quantum Schur algebra is a certain important infinite dimensional algebra whose representation theory is closely related to that of quantum affine $\frak{gl}_n$. Finite dimensional irreducible modules for the affine quantum Schur algebra ${\mathcal S}_{\vartriangle}(n,r)_{v}$ were classified in \cite{DDF}, where $v\in{\mathbb C}^*$ is not a root of unity. We will classify blocks of the affine quantum Schur algebra ${\mathcal S}_{\vartriangle}(n,r)_{v}$ in this paper.
In this paper, we classify the finite dimensional irreducible modules for affine BMW algebra over an algebraically closed field with arbitrary characteristic.
Locally affine Lie algebras are generalizations of affine Kac--Moody algebras with Cartan subalgebras of infinite rank whose root system is locally affine. In this note we study a class of representations of locally affine algebras generalizing integrable highest weight modules. In particular, we construct such an integrable representation for each integral weight not vanishing on the center and show that, over the complex numbers, we thus obtain unitary representations w.r.t. a unitary real form. We also use Yoshii's recent classification of locally affine root systems to derive a classificat…
We show that one can construct a classical affine W-algebra via a classical BRST complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional W-algebra has two compatible $λ$-brackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional W-algebra is associated to a minimal nilpotent, we d…
Inspired by the work of Rostam, we establish an explicit categorical equivalence between affine Yokonuma-Hecke algebras and quiver Hecke algebras associated to disjoint copies of quivers of (affine) type $A,$ generalizing Rouquier's categorical equivalence theorem.
We give a description of the center of the affine nilTemperley-Lieb algebra based on a certain grading of the algebra and on a faithful representation of it on fermionic particle configurations. We present a normal form for monomials, hence construct a basis of the algebra, and use this basis to show that the affine nilTemperley-Lieb algebra is finitely generated over its center. As an application, we obtain a natural embedding of the affine nilTemperley-Lieb algebra on N generators into the affine nilTemperley-Lieb algebra on N + 1 generators.
We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.
We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.
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 give a simplified description of quantum affine algebras in their loop presentation. This description is related to Drinfeld's new realization via halves of vertex operators. We also define an idempotent version of the quantum affine algebra which is suitable for categorification.
We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types. Under the specialization of $q$ and coefficients to 1, these bases are generic bases of finite and affine cluster algebras.
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.
We propose a new simplified definition of extended affine Lie algebras (EALAs for short), and also discuss a general version of extended affine Lie algebras, called locally extended affine Lie algebras (LEALAs for short). We prove a conjecture by V. Kac for LEALAs. It turns out that the root system of a LEALA becomes a locally finite version of extended affine root systems. Finally, several examples of new EALAs and LEALAs are introduced, and we classify LEALAs of nullity 0 with connection to locally finite split simple Lie algebras.
We study the structure and representation theory of affine wreath product algebras and their cyclotomic quotients. These algebras, which appear naturally in Heisenberg categorification, simultaneously unify and generalize many important algebras appearing in the literature. In particular, special cases include degenerate affine Hecke algebras, affine Sergeev algebras (degenerate affine Hecke-Clifford algebras), and wreath Hecke algebras. In some cases, specializing the results of the current paper recovers known results, but with unified and simplified proofs. In other cases, we obtain new res…
The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and Dunkl operator representations are established. An algebra isomorphism relating the rational DaHCa to the rational sDaHa is obtained. We further develop a link between the usual rational Cherednik algebra and the rational sDaHa by introducing a notion of rational covering double affine Hecke algebras.
We construct algebra homomorphisms from affine Yangians to the current algebras of rectangular $W$-algebras both in type A. The construction is given via the coproduct and the evaluation map for the affine Yangians. As a consequence, we show that parabolic inductions for representations of the rectangular $W$-algebras can be regarded as tensor product representations of the affine Yangians under the homomorphisms. The same method is applicable also to the super setting.
