Introduction to lie algebras and representation theory

In this paper, we study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver Q. Using a geometric R-matrix formalism, we construct a Hopf algebra Y_Q, the Yangian of Q, acting on the cohomology of these varieties, and show several results about their basic structure theory. We prove a formula for quantum multiplication by divisors in terms of this Yangian action. The quantum connection can be identified with the trigonometric Casimir connection for Y_Q/ equivalently, the divisor operators correspond to certain elements of Baxter subalgebras of Y_Q. A key role is played by geometric shift operators which can be identified with the quantum KZ difference connection. In the second part, we give an extended example of the general theory for moduli spaces of sheaves on C^2, framed at infinity. Here, the Yangian action is analyzed explicitly in terms of a free field realization/ the corresponding R-matrix is closely related to the reflection operator in Liouville field theory. We show that divisor operators generate the quantum ring, which is identified with the full Baxter subalgebras. As a corollary of our construction, we obtain an action of the W-algebra W(gl(r)) on the equivariant cohomology of rank $r$ moduli spaces, which implies certain conjectures of Alday, Gaiotto, and Tachikawa.

http://inspirehep.net/record/1201562/files/arXiv%3A1211.1287.pdf

Quantum Groups and Quantum Cohomology

http://eprints.gla.ac.uk/86333/1/86333.pdf

Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions

We study macroscopically two dimensional ${\mathcal{N}=(2,2)}$ supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product ${\mathbb{T}^{d} \times \mathbb{R}^{2\epsilon}}$ of a d-dimensional torus and a two dimensional cigar with ${\Omega}$ -deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories and the Yangian ${\mathbf{Y}_{\epsilon}(\mathfrak{g}_{\Gamma})}$ , quantum affine algebra ${\mathbf{U}^{\mathrm{aff}}_q(\mathfrak{g}_{\Gamma})}$ , or the quantum elliptic algebra ${\mathbf{U}^{\mathrm{ell}}_{q,p}(\mathfrak{g}_{\Gamma})}$ associated to Kac–Moody algebra ${\mathfrak{g}_{\Gamma}}$ for quiver ${\Gamma}$ .

Quantum geometry and quiver gauge theories

We study the PBW filtration on the highest weight representations V(λ) of $$ \mathfrak{s}{\mathfrak{l}_{n + 1}} $$
 . This filtration is induced by the standard degree filtration on $$ {\text{U}}\left( {{\mathfrak{n}^{-} }} \right) $$
 . We give a description of the associated graded $$ S\left( {{\mathfrak{n}^{-} }} \right) $$
 -module gr V(λ) in terms of generators and relations. We also construct a basis of gr V(λ). As an application we derive a graded combinatorial character formula for V(λ), and we obtain a new class of bases of the modules V(λ) conjectured by Vinberg in 2005.

PBW FILTRATION AND BASES FOR IRREDUCIBLE MODULES IN TYPE An

We begin a study of the representation theory of quantum continuous $\mathfrak{gl}_\infty$, which we denote by $\mathcal E$. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of $\mathcal E$ are labeled by a continuous parameter $u\in {\mathbb C}$. The representation theory of $\mathcal E$ has many properties familiar from the representation theory of $\mathfrak{gl}_\infty$: vector representations, Fock modules, semi-infinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from $\mathcal E$ to spherical double affine Hecke algebras $S\ddot H_N$ for all $N$. A key step in this construction is an identification of a natural bases of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the $K$-theory of Hilbert schemes.

Quantum continuous $\mathfrak{gl}_\infty$: Semi-infinite construction of representations

We begin a study of the representation theory of quantum continuous gl∞, which we denote by E. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of E are labeled by a continuous parameter u∈C. The representation theory of E has many properties familiar from the representation theory of gl∞: vector representations, Fock modules, and semiinfinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from E to spherical double affine Hecke algebras SHN for all N. A key step in this construction is an identification of a natural basis of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the K-theory of Hilbert schemes.

/pdf/quantum-continuous-mathfrak-gl-infty-semiinfinite-4390uqmfmu.pdf

Quantum continuous $\mathfrak{gl}_{\infty}$: Semiinfinite construction of representations

Let ${\mathcal{F}_\lambda}$ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module Vλ. We define a flat degeneration ${\mathcal{F}_\lambda^a}$ , which is a ${\mathbb{G}^M_a}$ variety. Moreover, there exists a larger group Ga acting on ${\mathcal{F}_\lambda^a}$ , which is a degeneration of the group G. The group Ga contains ${\mathbb{G}^M_a}$ as a normal subgroup. If G is of type A, then the degenerate flag varieties can be embedde‘d into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of ${\mathcal{F}_\lambda}$ is generated by the set of degenerate Plucker relations. We prove that the coordinate ring of ${\mathcal{F}_\lambda^a}$ is isomorphic to a direct sum of dual PBW-graded ${\mathfrak{g}}$ -modules. We also prove that there exists bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogs of semistandard tableaux.

$${\mathbb{G}_a^ M}$$ degeneration of flag varieties

Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proven that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits, and a cellular decomposition. For type A quivers explicit formulas for the Euler characteristic (the median Genocchi numbers) and the Poincare polynomials are derived.

/pdf/quiver-grassmannians-and-degenerate-flag-varieties-1nc7hq6a2y.pdf

Evgeny Feigin

Papers

PBW FILTRATION AND BASES FOR IRREDUCIBLE MODULES IN TYPE An

Quantum continuous $\mathfrak{gl}_\infty$: Semi-infinite construction of representations

Quantum continuous $\mathfrak{gl}_{\infty}$: Semiinfinite construction of representations

$${\mathbb{G}_a^ M}$$ degeneration of flag varieties

Quiver Grassmannians and degenerate flag varieties