Quantum Groups and Quantum Cohomology

Lecture notes. Introduction to the cohomology of algebras, Lie algebras, Lie bialgebras and quantum groups. Contains a new derivation of the classification of classical r-matrices in terms of deformation cohomology, and a calculation of the esoteric forms of quantum gl(N) by deformation theory.

We study quantum geometry of Nakajima quiver varieties of two different types - framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection between equivariant K-theories thereof with a nontrivial match between their equivariant parameters. In particular, we demonstrate that quantum equivariant K-theory of $A_n$ quiver varieties in a certain $n\to\infty$ limit reproduces equivariant K-theory of the Hilbert scheme of poi...

Beginning with the data of a quiver Q, and its dimension vector d, we construct an algebra D_q=D_q(Mat_d(Q)), which is a flat q-deformation of the algebra of differential operators on the affine space Mat_d(Q). The algebra D_q is equivariant for an action by a product of quantum general linear groups, acting by conjugation at each vertex. We construct a quantum moment map for this action, and subsequently define the Hamiltonian reduction A^lambda_d(Q) of D_q with moment param...

In this note, we study possible $\mathcal{R}$-matrix constructions in the context of quiver Yangians and Yang-Baxter algebras. For generalized conifolds, we also discuss the relations between the quiver Yangians and some other Yangian algebras (and $\mathcal{W}$-algebras) in literature.

The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra, which arises as the K-theoretic Hall algebra of the double cyclic quiver. We prove the isomorphism between the shuffle algebra and the quantum toroidal algebra U_{q,t}(sl_n^^), and identify the quotients of Verma modules for the shuffle algebra wi...

Affine quantum groups are certain pseudo-quasitriangular Hopf algebras that arise in mathematical physics in the context of integrable quantum field theory, integrable quantum spin chains, and solvable lattice models. They provide the algebraic framework behind the spectral parameter dependent Yang-Baxter equation. One can distinguish three classes of affine quantum groups, each leading to a different dependence of the R-matrices on the spectral parameter: Yangians lead to ...

Let $\Psi(\mathbb{z},\mathbb{a},q)$ be the fundamental solution matrix of the quantum difference equation in the equivariant quantum K-theory for Nakajima variety $X$. In this work, we prove that the operator $$ \Psi(\mathbb{z},\mathbb{a},q) \Psi\left(\mathbb{z}^p,\mathbb{a}^p,q^{p^2}\right)^{-1} $$ has no poles at the primitive complex $p$-th roots of unity $\zeta_p$ in the curve counting parameter $q$. As a byproduct, we show that the eigenvalues of the iterated product of ...

The search for elliptic quantum groups leads to a modified quantum Yang-Baxter relation and to a special class of quasi-triangular quasi Hopf algebras. This paper calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only unde...

Framed quiver moduli parametrize stable pairs consisting of a quiver representation and a map to a fixed graded vector space. Geometric properties and explicit realizations of framed quiver moduli for quivers without oriented cycles are derived, with emphasis on their cohomology. Their use for quantum group constructions is discussed.

These lecture notes are devoted to the recent progress in the geometric aspects of quantum integrable systems based on quantum groups solved using the Bethe ansatz technique. One part is devoted to their enumerative geometry realization through the quantum K-theory of Nakajima quiver varieties. The other part describes a recently studied $q$-deformation of the correspondence between oper connections and Gaudin models. The notes are based on a minicourse at C.I.M.E. Summer Sch...