Realizing Enveloping Algebras via Varieties of Modules

Quantum groups via cyclic quiver varieties I

We construct the quantized enveloping algebra of any simple Lie algebra of type ADE as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig's bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland's recent work for type ADE.

Examples of quantum cluster algebras associated to partial flag varieties

We give several explicit examples of quantum cluster algebra structures, as introduced by Berenstein and Zelevinsky, on quantized coordinate rings of partial flag varieties and their associated unipotent radicals. These structures are shown to be quantizations of the cluster algebra structures found on the corresponding classical objects by Geiss, Leclerc and Schroer, whose work generalizes that of several other authors. We also exhibit quantum cluster algebra structures on the quantized enveloping algebras of the Lie algebras of the unipotent radicals.

Homological description of crystal structures on Lusztig's quiver varieties

Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras. The first realization we consider is a geometric construction in terms of irreducible components of certain quiver varieties established by Kashiwara and Saito. The second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke using Ringel's Hall algebra approach to quantum groups. We show that these two constructions ar…

A geometric construction of crystal graphs using quiver varieties: Extension to the non-simply laced case

We consider a generalization of the quiver varieties of Lusztig and Nakajima to the case of all symmetrizable Kac-Moody Lie algebras. To deal with the non-simply laced case one considers admissible automorphisms of a quiver and the irreducible components of the quiver varieties fixed by this automorphism. We define a crystal structure on these irreducible components and show that the crystals obtained are isomorphic to those associated to the crystal bases of the lower half of the universal enveloping algebra and the irreducible highest weight representations of the non-simply laced Kac-Moody …

$t$--analogs of $q$--characters of quantum affine algebras of type $E_6$, $E_7$, $E_8$

We compute $t$--analogs of $q$--characters of all $l$--fundamental representations of the quantum affine algebras of type $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$ by a supercomputer. In particular, we prove the fermionic formula for Kirillov-Reshetikhin modules conjectured by Hatayama et al. (math.QA/9812022) for these classes of representations. We also give explicitly the monomial realization of the crystal of the corresponding fundamental representations of the qunatum enveloping algebras associated with finite dimensional Lie algebras of types $E_6$, $E_7$, $E_8$. These are computations of Be…

On injective modules and support varieties for the small quantum group

Let $u_ζ(g)$ denote the small quantum group associated to the simple complex Lie algebra $g$, with parameter $q$ specialized to a primitive $\ell$-th root of unity $ζ$ in the field $k$. Generalizing a result of Cline, Parshall and Scott, we show that if $M$ is a finite-dimensional $u_ζ(g)$-module admitting a compatible torus action, then the injectivity of $M$ as a module for $u_ζ(g)$ can be detected by the restriction of $M$ to certain root subalgebras of $u_ζ(g)$. If the characteristic of $k$ is positive, then this injectivity criterion also holds for the higher Frobenius--Lusztig kernels $U…

The structure of smooth algebras in Kapranov's framework for noncommutative geometry

In Kapranov, M. {\it Noncommutative geometry based on commutator expansions,} J. reine angew. Math {\bf 505} (1998) 73-118, a theory of noncommutative algebraic varieties was proposed. Here we prove a structure theorem for the noncommutative coordinate rings of affine open subsets of such of those varieties which are smooth (Theorem 3.4). The theorem describes the local ring of a point as a truncation of a quantization of the enveloping Poisson algebra of a smooth commutative local algebra. An explicit descripition of this quantization is given in Theorem 2.5. A description of the $A$- module …

Representations of tame quivers and affine canonical bases

An integral PBW-basis of type $A_1^{(1)}$ has been constructed by Zhang [Z] and Chen [C] using the Auslander-Reiten quiver of the Kronecker quiver. We associate a geometric order to elements in this basis following an idea of Lusztig [L1] in the case of finite type. This leads to an algebraic realization of a bar-invariant basis of $\uq2$. For any affine symmetric type, we obtain an integral PBW-basis of the generic composition algebra, by using an algebraic construction of the integral basis for a tube in [DDX], an embedding of the module category of the Kronecker quiver into the module categ…

Chiral Koszul duality

We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in \cite{bd}, to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen's homotopy theory of differential graded Lie algebras. We prove the equivalence of higher-dimensional chiral and factorization algebras by embedding factorization algebras into a larger category of chiral commutative coalgebras, then realizing this interrelation as a chiral form of Koszul duality. We apply these…

Finite dimensional Hopf actions on algebraic quantizations

Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra [arxiv.org/abs/1409.1644, arxiv.org/abs/1509.01165], we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z_1,...,z_s] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s=0. We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal envelo…

Gaudin model and opers

This is a review of our previous works (some of them joint with B. Feigin and N. Reshetikhin) on the Gaudin model and opers. We define a commutative subalgebra in the tensor power of the universal enveloping algebra of a simple Lie algebra g. This algebra includes the hamiltonians of the Gaudin model, hence we call it the Gaudin algebra. It is constructed as a quotient of the center of the completed enveloping algebra of the affine Kac-Moody algebra g^ at the critical level. We identify the spectrum of the Gaudin algebra with the space of opers associated to the Langlands dual Lie algebra of g…

双模对偶与斜域扩张的阿廷问题

Bimodule duality and Artin's problem for skew field extensions

该研究解决了阿廷关于斜域扩张左右维数的开放性问题，证明了扭曲卡拉比-丘除代数等类除环为指数对称的，给出了首个满足条件的除环例子。

Opers with irregular singularity and spectra of the shift of argument subalgebra

The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras math.RT/0606380, math.QA/0612798. We prove that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromy-free opers for the Langlands dual group of G on the projective line with regular singularity at one point and irregular singularity of order two at another point. We also prove a multi-point generalization of this result, describing the spectra of commutin…

Symmetries of categorical representations and the quantum Ngô action

We observe that all classical Hamiltonian systems coming from the invariant polynomials on a reductive Lie algebra g can be integrated in a universal way. This is a consequence of Ngô's action of the group scheme J of regular centralizers in G on all centralizers: the Hamiltonian flows associated to invariant polynomials integrate to an action of J as commutative symplectic groupoid. We quantize the Ngô action, providing a universal integration for all quantum Hamiltonian systems coming from the center Z=Z(Ug) of the enveloping algebra. Namely we extend Kostant's Whittaker description of Z to …