In this short note, we show that the Ginzburg-Vasserot map between the quantum affine algebra of type A_(n-1) and the equivariant K-theory group of the Steinberg Variety (of n-step flags in C^d) restricts and remains surjective at the level of the integral forms. In particular, this shows that the parametrization of irreducible, finite-dimensional modules and the Kazhdan-Lusztig multiplicity formulas are valid for the (restricted) specialization at roots of unity.
For any nullity $2$ extended affine Lie algebra $\mathcal{E}$ of maximal type and $\ell\in\mathbb{C}$, we prove that there exist a vertex algebra $V_{\mathcal{E}}(\ell)$ and an automorphism group $G$ of $V_{\mathcal{E}}(\ell)$ equipped with a linear character $χ$, such that the category of restricted $\mathcal{E}$-modules of level $\ell$ is canonically isomorphic to the category of $(G,χ)$-equivariant $ϕ$-coordinated quasi $V_{\mathcal{E}}(\ell)$-modules. Moreover, when $\ell$ is a nonnegative integer, there is a quotient vertex algebra $L_{\mathcal{E}}(\ell)$ of $V_{\mathcal{E}}(\ell)$ modulo…
The q-characters were introduced by Frenkel and mReshetikhin to study finite dimensional representations of the untwisted quantum affine algebra for q generic. The $ε$-characters at roots of unity were constructed by Frenkel and Mukhin to study finite dimensional representations of various specializations of the quantum affine algebra at $q^s=1$. In the finite simply laced case Nakajima defined deformations of q-characters called q,t-characters, for q generic and also at roots of unity. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties…
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 this paper, we study a family of infinite-dimensional Lie algebras $\widehat{X}_{S}$, where $X$ stands for the type: $A,B,C,D$, and $S$ is an abelian group, which generalize the $A,B,C,D$ series of trigonometric Lie algebras. Among the main results, we identify $\widehat{X}_{S}$ with what are called the covariant algebras of the affine Lie algebra $\widehat{\mathcal{L}_{S}}$ with respect to some automorphism groups, where $\mathcal{L}_{S}$ is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted $\widehat{X}_{S}$-modules of level $\ell$ naturally co…
We show that the quantized flag manifold at a root of unity has natural affine open covering parametrized by the elements of the Weyl group. In particular, the quantized flag manifold turns out to be a quasi-scheme in the sense of Rosenberg.
We give complete proofs of the K-theoretic construction of the quantized enveloping algebra of affine gl(n) sketched with V. Ginzburg in Internat. Math. Res. Notices, 3 (1993).
We study the Frobenius-Lusztig kernel for quantum affine algebras at root of unity of small orders that are usually excluded in literature. These cases are somewhat degenerate and we find that the kernel is in fact mostly related to different affine Lie algebras, some even of larger rank, that exceptionally sit inside the quantum affine algebra. This continues the authors study for quantum groups associated to finite-dimensional Lie algebras in [Len14c].
The purpose of this paper is to compute the Drinfel'd polynomials for two types of evaluation representations of quantum affine algebras at roots of unity and construct those representations as the submodules of evaluation Schnizer modules. Moreover, we obtain the necessary and sufficient condition for that the two types of evaluation representations are isomorphic to each other.
We consider various specializations of the non-twisted quantum affine algebras at roots of unity. We define and study the q-characters of their finite-dimensional representations.
We establish several results concerning tensor products, q-characters, and the block decomposition of the category of finite-dimensional representations of quantum affine algebras in the root of unity setting. In the generic case, a Weyl module is isomorphic to a tensor product of fundamental representations and this isomorphism was essential for establishing the block decomposition theorem. This is no longer true in the root of unity setting. We overcome the lack of such a tool by utilizing results on specialization of modules. Furthermore, we establish a sufficient condition for a Weyl modul…
We describe explicitly the canonical map $χ:$ Spec $\ue(\a{g})\ \rightarrow \ $Spec $\ze$, where $\ue(\a{g})$ is a quantum loop algebra at an odd root of unity $\ve$. Here $\ze$ is the center of $\ue(\a{g})$ and Spec $R$ stands for the set of all finite--dimensional irreducible representations of an algebra $R$. We show that Spec $\ze$ is a Poisson proalgebraic group which is essentially the group of points of $G$ over the regular adeles concentrated at $0$ and $\infty$. Our main result is that the image under $χ$ of Spec $\ue(\a{g})$ is the subgroup of principal adeles.
We study the restricted form of the qaunatized enveloping algebra of an untwisted affine Lie algebra and prove a triangular decomposition for it. In proving the decomposition we prove several new identities in the quantized algebra, one of these show a connection between the quantized algebra and Young diagrams. These identities are all invisible in the non-quantum case of the problem which was considered by Garland in 1978. We then study the finite-dimensional irreducible representations and prove a factorization theorem for such representations.
We show that the equivariant hypertoric convolution algebras introduced by Braden-Licata-Proudfoot-Webster are affine quasi hereditary in the sense of Kleshchev and compute the Ext groups between standard modules. Together with the main result of arXiv:2009.03981, this implies a number of new homological results about the bordered Floer algebras of Ozsvath-Szabo, including the existence of standard modules over these algebras. We prove that the Ext groups between standard modules are isomorphic to the homology of a variant of the Lipshitz-Ozsvath-Thurston bordered strands dg algebras.
Here is a list of chapters: 1 Introduction 2 Notation and preliminaries Part I: Finite quantum groups 3 2x2 Matrix quantum groups and the quantum plane 4 Quantum enveloping algebras at a root of unity Part II: q-Oscillators 5 Representations of q-oscillators at a root of unity 6 qr-Oscillator at a root of unity Part III: Infinite quantum groups 7 Quantum affine algebras 8 Quantum affine algebras at a root of unity
We give a combinatorial algorithm for computing Zelevinsky's involution of the set of isomorphism classes of irreducible representations of the affine Hecke algebra $\H_m(t)$ when $t$ is a primitive $n$th root of 1. We show that the same map can also be interpreted in terms of aperiodic nilpotent orbits of $\Zb/n\Zb$-graded vector spaces.
We establish a natural connection of the $q$-Virasoro algebra $D_{q}$ introduced by Belov and Chaltikian with affine Kac-Moody Lie algebras. More specifically, for each abelian group $S$ together with a one-to-one linear character $χ$, we define an infinite-dimensional Lie algebra $D_{S}$ which reduces to $D_{q}$ when $S=\mathbb{Z}$. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra ${\mathfrak{g}}_{S}$ with $S$ as an automorphism group and we prove that $D_{S}$ is isomorphic to the $S$-covariant algebra of the affine Lie algebra $\widehat{…
Inspired by a profound observation on the Racah--Wigner coefficients of $U_q(\mathfrak{sl}_2)$, the Askey--Wilson algebras were introduced in the early 1990s. A universal analog $\triangle_q$ of the Askey--Wilson algebras was recently studied. For $q$ not a root of unity, it is known that $Z(\triangle_q)$ is isomorphic to the polynomial ring of four variables. A presentation for $Z(\triangle_q)$ at $q$ a root of unity is displayed in this paper. As an application, a presentation for the center of the double affine Hecke algebra of type $(C_1^\vee,C_1)$ at roots of unity is obtained.
Using vertex operators, we construct explicitly Lusztig's $\mathbb Z[q, q^{-1}]$-lattice for the level one irreducible representations of quantum affine algebras of ADE type. We then realize the level one irreducible modules at roots of unity and show that the q-dimension is still given by the Weyl-Kac character formula. As a consequence we also answer the corresponding question of realizing the affine Kac-Moody Lie algebras of simply laced type at level one in finite characteristic.
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{\…
We describe vector valued conjugacy equivariant functions on a group K in two cases -- K is a compact simple Lie group, and K is an affine Lie group. We construct such functions as weighted traces of certain intertwining operators between representations of K. For a compact group $K$, Peter-Weyl theorem implies that all equivariant functions can be written as linear combinations of such traces. Next, we compute the radial parts of the Laplace operators of $K$ acting on conjugacy equivariant functions and obtain a comple- tely integrable quantum system with matrix coefficients, which in a speci…
We give shuffle algebra realization of positive part of quantum affine superalgebra $U_{v}(\widehat{\mathfrak{D}}(2,1;θ))$ associated to any simple root systems. We also determine the shuffle algebra associated to $\widehat{\mathfrak{sl}}(2|1)$ with odd root system when $v$ is a primitive root of unity of even order, generalizing results in \cite{FJMMT03}.
We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups had been predicted by certain conformal field theory considerations, but constructions had not appeared until recently. We show that our quantum groups can be identified with those of Creutzig-Gainutdinov-Runkel in type A_1, and Gainutdinov-Lentner-Ohrmann in arbitrary Dynkin type. We discuss conjectural relations with vertex operator algebras at (1,p)-centr…
The natural representation of the quantized affine algebra of type A can be defined via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl modules for a type A quantum group at a root of unity to the action of the quantized affine algebra as the rank tends towards infinity. In this paper we investigate the situation outside of type A. In classical types, we construct embeddings of the Grothendieck group of finite dimensional modules for the corresponding quatum group at a root of unity into Fock spaces of different charges and define an action of an affine quantum symm…
We show that the action of universal $R$-matrix of affine $U_qsl_2$ quantum algebra, when $q$ is a root of unity, can be renormalized by some scalar factor to give a well defined nonsingular expression, satisfying Yang-Baxter equation. It reduced to intertwining operators of all representations, corresponding to Chiral Potts, if the parameters of these representations lie on well known algebraic curve. We also show that affine $U_qsl_2$ for $q$ is a root of unity form the autoquasitriangular Hopf algebra in the sence of Reshetikhin.
Let $\Oq(G)$ be the algebra of quantized functions on an algebraic group $G$ and $\Oq(B)$ its quotient algebra corresponding to a Borel subgroup $B$ of $G$. We define the category of sheaves on the "quantum flag variety of $G$" to be the $\Oq(B)$-equivariant $\Oq(G)$-modules and proves that this is a proj-category. We construct a category of equivariant quantum $\mathcal{D}$-modules on this quantized flag variety and prove the Beilinson-Bernsteins localization theorem for this category in the case when $q$ is not a root of unity.
We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk formula in the torus-equivariant $K$-group of the formal power series model of semi-infinite flag manifolds, both of which are described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or, equivalently, quantum Lakshmibai-Seshadri paths). In view of recent results of Kato, these formulas give an explicit description of the structure constants for the Pontryagin product in the torus-equivariant $K$-group of affine Grassmannians and that for the quantum multiplication of the torus-equivariant (small) qua…
In this paper, we explore natural connections among the representations of the extended affine Lie algebra $\widehat{sl_N}(\mathbb{C}_q)$ with $\mathbb{C}_q=\mathbb{C}_q[t_0^{\pm1},t_1^{\pm1}]$ an irrational quantum 2-torus, the simple affine vertex algebra $L_{\widehat{sl_{\infty}}}(\ell,0)$ with $\ell$ a positive integer, and Levi subgroups $G$ of $GL_\ell(\mathbb{C})$. First, we give a canonical isomorphism between the category of integrable restricted $\widehat{sl_N}(\mathbb{C}_q)$-modules of level $\ell$ and that of equivariant quasi $L_{\widehat{sl_{\infty}}}(\ell,0)$-modules. Second, we…
We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations. The algebra $A$ includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple $(A, H, D)$ with $H=L^2(\mathbb R^n, Λ(\mathbb R^n))$ and the Euler operator $D$, a first order differential operator of index $1$. We show that this spectral triple has simple dimension spectrum: For every operator $B$ in the algebra $Ψ(A,H,D)$ generated by the Shubin type pseudodifferential operators and the elements of $A$, the zeta f…
Let $U_ζ$ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra $\mathfrak g$ and a root of unity $ζ$. When $L,L'$ are irreducible $U_ζ$-modules having regular highest weights, the dimension of $\operatorname{Ext}^n_{U_ζ}(L,L')$ can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of $U_ζ$. This paper shows for $L,L'$ irreducible modules in a singular block that $\dim\operatorname{Ext}^n_{U_ζ}(L,L')$ is explicitly determined using the coefficients of parabolic Kazhdan-Lusztig polynomials. …
Let $G$ be a finite subgroup in SU(2), and $Q$ the corresponding affine Dynkin diagram. In this paper, we review the relation between the categories of $G$-equivariant sheaves on $P^1$ and $Rep Q_h$, where $h$ is an orientation of $Q$, constructing an explicit equivalence of corresponding derived categories.
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.
When the parameter $q\in\mathbb C^*$ is not a root of unity, simple modules of affine $q$-Schur algebras have been classified in terms of Frenkel--Mukhin's dominant Drinfeld polynomials (\cite[4.6.8]{DDF}). We compute these Drinfeld polynomials associated with the simple modules of an affine $q$-Schur algebra which come from the simple modules of the corresponding $q$-Schur algebra via the evaluation maps.
We study the de-equivariantization of a Hopf algebra by an affine group scheme and we apply Tannakian techniques in order to realize it as the tensor category of comodules over a coquasi-bialgebra. As an application we construct a family of coquasi-Hopf algebras $A(H,G,Φ)$ attached to a coradically-graded pointed Hopf algebra $H$ and some extra data.
Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras. Local Weyl modules play an important role in the theory of finite-dimensional representations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras wh…
We establish a q-generalization of Gordon's theorem that the space of diagonal coinvariants has a quotient identified with a perfect representation of the rational double affine Hecke algebra. It leads to a simple proof of his theorem and relates it to the Weyl algebras at roots of unity. The universal double affine Hecke algebra and the corresponding universal double Dunkl operators acting in noncommutative polynomials in terms of two sets of variables are introduced.
We prove that the quotient of the polynomial representation of the double affine Hecke algebra (DAHA) by the radical of the duality pairing is always irreducible assuming that it is finite dimensional (apart from the roots of unity). We also find necessary and sufficient conditions for the radical to be zero, which is a q-generalization of Opdam's formula for the singular k-parameters with the multiple zero-eigenvalues of the corresponding Dunkl operators.
We construct a representation of the affine W-algebra of gl_r on the equivariant homology space of the moduli space of U_r-instantons on A^2, and identify the corresponding module. As a corollary we give a proof of a version of the AGT conjecture concerning pure N=2 gauge theory for the group SU(r). Another proof has been announced by Maulik and Okounkov. Our approach uses a suitable deformation of the universal enveloping algebra of the Witt algebra W_{1+\infty}, which is shown to act on the above homology spaces (for any r) and which specializes to all W(gl_r). This deformation is in turn co…
This paper resolves the unicity conjecture of Bonahon and Wong for the Kauffman bracket skein algebras of all oriented finite type surfaces at all roots of unity. The proof is a consequence of a general unicity theorem that says that the irreducible representations of a prime affine $k$-algebra over an algebraically closed field $k$, that is finitely generated as a module over its center, are generically classified by their central characters. The center of the Kauffman bracket skein algebra of any orientable surface at any root of unity is characterized, and it is proved that the skein algebr…
There are two intriguing statements regarding the quantum cohomology of partial flag varieties. The first one relates quantum cohomology to the affinisation of Lie algebras and the homology of the affine Grassmannian, the second one connects it with the geometry of quiver varieties. The connection with the affine Grassmannian was first discussed in unpublished work of Peterson and subsequently proved by Lam and Shimozono. The second development is based on recent works of Nekrasov, Shatashvili and of Maulik, Okounkov relating the quantum cohomology of Nakajima varieties with integrable systems…
We construct quantizations of multiplicative hypertoric varieties using an algebra of q-difference operators on affine space, where q is a root of unity in C. The quantization defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety and admits an explicit finite étale splitting. The global sections of this Azumaya algebra is a hypertoric quantum group, and we prove a localization theorem. We introduce a general framework of Frobenius quantum moment maps and their Hamiltonian reductions; our results shed light on an instance of this framework.
