209405145
Remarks on the derived center of small quantum groups
Let $u_q(g)$ be the small quantum group associated with a complex semisimple Lie algebra $g$ and a primitive root of unity q, satisfying certain restrictions. We establish the equivalence between three different actions of $g$ on the center of $u_q(g)$ and on the higher derived center of $u_q(g)$. Based on the triviality of this action for $g = sl_2, sl_3, sl_4$, we conjecture that, in finite type A, central elements of the small quantum group $u_q(sl_n)$ arise as the restriction of central elements in the big quantum group $U_q(sl_n)$. 
We also study the role of an ideal $z_{Hig}$ known as the Higman ideal in the center of $u_q(g)$. We show that it coincides with the intersection of the Harish-Chandra center and its Fourier transform, and compute the dimension of zHig in type A. As an illustration we provide a detailed explicit description of the derived center of $u_q(sl_2)$ and its various symmetries.
116931481
Quantum groups, the loop Grassmannian, and the Springer resolution
We establish equivalences of derived categories of the following 3 categories: 
(1) Principal block of representations of the quantum at a root of 1; 
(2) G-equivariant coherent sheaves on the Springer resolution; 
(3) Perverse sheaves on the loop Grassmannian for the Langlands dual group. 
The equivalence (1)-(2) is an `enhancement' of the known expression for quantum group cohomology in terms of nilpotent variety, due to Ginzburg-Kumar. The equivalence (2)-(3) is a step towards resolving an old mystery surrounding the existense of two completely different realizations of the affine Hecke algebra which have played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a p-adic reductive group, while the other is in terms of equivariant K-theory of a complex (Steinberg) variety for the dual group. Our equivalence (2)-(3) may be viewed as a `categorification' of the isomorphism between the corresponding two geometric realizations of the fundamental polynomial representation of the affine Hecke algebra. The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following Lusztig program, from earlier deep results of Kazhdan-Lusztig and Kashiwara-Tanisaki. Our approach is independent of these results and is totally different (it does not rely on representation theory of Kac-Moody algebras). It also gives way to proving Humphreys' conjectures on tilting U_q(g)-modules, as will be explained in a separate paper.
121658727
Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra
0.1. An important role in the theory of modular representations is played by certain finite dimensional Hopf algebras u over Fp (the field with p elements, p = prime). Originally, u was defined (Curtis [3]) as the restricted enveloping algebra of a "simple" Lie algebra over Fp For our purposes, it will be more convenient to define u as follows. Let us fix an indecomposable positive definite symmetric Cartan matrix
11133172
The small quantum group and the Springer resolution
In math.RT/0304173 the derived category of the principal block in modules over the Lusztig quantum algebra at a root of unity is related to the derived category of equivariant coherent sheaves on the Springer resolution. In the present paper we deduce a similar relation between the derived category of the principal block for the {\em small (reduced) quantum algebra} and the derived category of (non-equivariant) coherent sheaves on the Springer resolution. As an application we get a geometric description of Hochschild cohomology (in particular, the center) of the regular block for the small quantum group, and use it to give an explicit description of a certain subalgebra in the center (obtained previously by another method and under more restrictive assumptions in math.QA/0107098. We also briefly explain the relation of our result to the geometric description math.RT/0407048 of the derived category of modules over the De Concini -- Kac quantum algebra.
119628293
The center of small quantum groups II: singular blocks
We generalize to the case of singular blocks the result in \cite{BeLa} that describes the center of the principal block of a small quantum group in terms of sheaf cohomology over the Springer resolution. Then using the method developed in \cite{LQ1}, we present a linear algebro-geometric approach to compute the dimensions of the singular blocks and of the entire center of the small quantum group associated with a complex semisimple Lie algebra. A conjectural formula for the dimension of the center of the small quantum group at an $l$th root of unity is formulated in type A.
201238509
Tensor structures arising from affine Lie algebras. III
This paper is a part of the series [KL]; however, it can be read independently of the first two parts. In [D3], Drinfeld proved the existence of an equivalence between a tensor category of representations of a quantum group over C[[w]] and a tensor category of representations of an undeformed enveloping algebra over C[[w ]]], in which the associativity constraints are given by the Knizhnik-Zamolodchikov equations. However, Drinfeld's proof was nonconstructive, in the sense that he showed that the set of equivalences between the two tensor categories was nonempty, without describing an equivalence explicitly. Our main result is an explicit construction of an equivalence between the tensor categories above. Its main virtue is, as we will show in Part IV, that it "extends analytically" to the case where w is no longer a formal variable
16526954
Conjectures on the Quotient Ring by Diagonal Invariants
AbstractWe formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring 
$$\mathbb{Q}[x_1 , \ldots ,x_n ,y_1 , \ldots ,y_n ]$$
 in two sets of variables by the ideal generated by all Sn invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = {x1, ..., xn} is classical. Introducing the second set of variables leads to a ring about which little is yet understood, but for which there is strong evidence of deep connections with many fundamental results of enumerative combinatorics, as well as with algebraic geometry and Lie theory.
119744431
Examples of tensor categories
Let G be a tensor category over a field k of characteristic 0. In his paper [-i], w 7, Delinge had established rather general conditions for G to be the category of representations of an algebraic group. He has shown that if any object neC satisfies the condition A " n = 0 for a sufficiently large n, then 0 is equivalent to the category Rep(G) of representations of a proalgebraic group scheme G over k. For tensor categories over fields of positive characteristic this result is no longer true. In the first part of this paper we present a construction of a class of tensor categories over fields of positive characteristic for which an analogue of Deligne's theorem does not hold. To describe these categories, denote by Gz an algebraic Chevalley group over Z and by G the corresponding group over Fp (for a sufficiently large p). Our category C will be a certain "subquotient" of the category R(G) of finite-dimensional representations of G. More precisely, let S be the full subcategory of R(G) whose objects (called small representations) are direct sums of the irreducible representations of G with highest weights inside the fundamental alcove. Let also SL 2 c G be the principal SLz-subgroup in G and (SLz)Im c SL2 be the infinitesimal neighbourhood of the Cartan subgroup H in SL2. Denote by O the full subcategory of R(G) consisting of representations M of G (called quasiprojective representations) such that the restriction of M to (SL2)m) is projective. The category 0 is a quotient of the subcategory S+PcR(G). In the last section of the paper we show that the same approach can be applied to representations of quantum groups. Let U be the quantum group at a root of prime order p of 1 (which is a Hopf algebra over the ring of integers in a cyclotomic field, see [2, 3]). Starting from the category of representations of U, we construct a semisimple braided category ~(G, p) with the finite number of simple objects. On the other hand, in Wess-Zumino-Witten conformal field theory one encounters the category O(G, m) of integrable representations of the correspond-
121566811
Braided Groups and Quantum Fourier Transform
Abstract We show that acting on every finite-dimensional factorizable ribbon Hopf algebra H there are invertible operators S , T obeying the modular identities ( S T )3 = λ S 2, where λ is a constant. The class includes the finite-dimensional quantum groups uq(g) associated to complex simple Lie algebras. We give the example of uq(sl(2)) at a root of unity in detail, as well as an example relating to anyons. The operator S plays the role of "quantum Fourier Transform" and acts naturally on H viewed by transmutation as a braided group H (a braided-cocommutative Hopf algebra in a braided category). It obeys S 2 = s −1, where s is the antipode of H . The results follow as an application of previous category-theoretical constructions.
120180108
Characters and a Verlinde-type formula for symmetric Hopf algebras
Abstract We study certain aspects of finite-dimensional non-semisimple symmetric Hopf algebras H and their duals H ∗ . We focus on the set I ( H ) of characters of projective H -modules which is an ideal of the algebra of cocommutative elements of H ∗ . This ideal corresponds via a symmetrizing form to the projective center (Higman ideal) of H which turns out to be Λ H , where Λ is an integral of H and is the left adjoint action of H on itself. We describe Λ H via primitive and central primitive idempotents of H . We also show that it is stable under the quantum Fourier transform. Our best results are obtained when H is a factorizable ribbon Hopf algebra over an algebraically closed field of characteristic 0. In this case Λ H is also the image of I ( H ) under a “translated” Drinfel'd map. We use this fact to prove the existence of a Steinberg-like character. The above ingredients are used to prove a Verlinde-type formula for Λ H .
17450815
A counterpart of the Verlinde algebra for the small quantum group
Let $\bar{Pr}$ denote the ideal spanned by the characters of projective modules in the Grothendieck ring of the category of finite dimensional modules over the small quantum group $u_l$. We show that $\bar{Pr}$ admits a description completely parallel to that of the Verlinde algebra of the fusion category, restricted over $u_l$, with the character of the Steinberg module playing the role of the identity.
14089753
On the center of the small quantum group
Using the quantum Fourier transform F, we describe the block decomposition and multiplicative structure of a subalgebra Z+F(Z) of the center of the small quantum group Uqfin(g) at a root of unity. It contains the known subalgebra Z, which is isomorphic to the algebra of characters of finite-dimensional Uqfin(g)-modules. We prove that the intersection Z∩F(Z) coincides with the annihilator of the radical of Z. Applying representation-theoretical methods, we show that Z surjects onto the algebra of endomorphisms of certain indecomposable projective modules over Uqfin(g). In particular, this leads to the conclusion that the center of Uqfin(g) coincides with Z+F(Z) in the case g=sl2.
16995387
Mapping Class Group Actions on Quantum Doubles December 1993
We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for the S ±1-matrices using the canonical, non degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projective SL(2, Z)-action on the center of U q (sl 2) for q an l = 2m + 1-st root of unity. It appears that the 3m + 1-dimensional representation decomposes into an m + 1-dimensional finite representation and a 2m-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation of SL(2, Z) and the finite, m-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category of U q (sl 2). Since the seminal paper of Atiyah [A] on the abstract definition of a topological quantum field theory (TQFT) much progress has been made in finding non trivial examples and extended structures. The most interesting developments took place in three dimensions where actual models of quantum field theory, like rational conformal field theories and Chern-Simons theory led to the discovery of new invariants. See [Cr] and [Wi]. In an attempt to counterpart these heuristic theories by mathematically rigorous constructions the field theoretical machinery had been replaced by quasitriangular Hopf algebras , or quantum groups. The resulting invariants are described in [TV] and [RT]. From here it is not hard to understand how to associate a TQFT to a rigid, abelian, monoidal category and an extended TQFT to a braided tensor category (BTC). In order for these theories to be well defined one has to make a few more assumptions. One is that the category shall contain only a finite number of inequivalent, simple objects, i.e., it is rational. The other is a technical non degeneracy condition, called " modularity " in [T], which is to assure that elementary cobordisms are associated to identifications rather than projections. Alternatively, if the modularity condition fails to hold, it is standard in the Atiyah [A] description to define a truncated TQFT by reducing …
117258624
Modular representations and quantum groups
The temperature sensors shown utilize a constant frequency fluidic oscillator to excite resonance tubes, whose frequency response is a function of temperature. The outputs of these sensors control a fluidic amplifier, whose output is rectified and filtered to produce a D.C. fluid signal which is a function of the sensed temperature. Different embodiments utilize resonance tubes excited in phase or 180 DEG out of phase to sense the difference between two temperatures or to provide increased sensitivity.
121856083
Quantum groups at roots of 1
We extend to the not necessarily simply laced case the study [8] of quantum groups whose parameter is a root of 1.
14179602
Hochschild cohomology and Atiyah classes
Abstract In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras. In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct. The latter compatibility is an unpublished result by Kontsevich. Our proof is set in the framework of Lie algebroids and so applies without modification in much more general settings as well.
119593371
The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras
Abstract. Let H be a Hopf algebra over the field k which is a finite module over a central affine sub-Hopf algebra R. Examples include enveloping algebras 
$U({\mathfrak g})$ of finite dimensional k-Lie algebras 
${\mathfrak g}$ in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity. The ramification behaviour of the maximal ideals of Z(H) with respect to the subalgebra R is studied, and the conclusions are then applied to the cases of classical and quantised enveloping algebras. In the case of 
$U({\mathfrak g})$ for 
${\mathfrak g}$ semisimple a conjecture of Humphreys [28] on the block structure of 
$U({\mathfrak g})$ is confirmed. In the case of 
$U_{\epsilon}({\mathfrak g})$ for 
${\mathfrak g}$ semisimple and 
$\epsilon$ an odd root of unity we obtain a quantum analogue of a result of Mirković and Rumynin, [35], and we fully describe the factor algebras lying over the regular sheet, [9]. The blocks of 
$U_{\epsilon}({\mathfrak g})$ are determined, and a necessary condition (which may also be sufficient) for a baby Verma 
$U_{\epsilon}({\mathfrak g})$-module to be simple is obtained.