Related papers: Quiver varieties and elliptic quantum groups
We define a class of quantum linear Galois algebras which include the universal enveloping algebra Uq(gln), the quantum Heisenberg Lie algebra and other quantum orthogonal Gelfand-Zetlin algebras of type A, the subalgebras of G-invariants…
We introduce the notion of pure Q-solvable algebra. The quantum matrices, quantum Weyl algebra, U_q(n) are the examples. It is proved that the skew field of fractions of pure Q-solvable algebra is isomorphic to the skew field of twisted…
Quantum general relativity may be considered as generally covariant QFT on differentiable manifolds, without any a priori metric structure. The kinematically covariance group acts by general diffeomorphisms on the manifold and by…
The properties of Hopf star operations and twisted Hopf stars operations on quantum groups are discussed in relation with the theory of representations (star representations). Invariant Hermitian sesquilinear forms (scalar products) on…
We describe the derived category of coherent sheaves on the minimal resolution of the Kleinian singularity associated to a finite subgroup G of SL(2). Then, we give an application to the Euler-characteristic version of the Hall algebra of…
Let $\g$ be an untwisted affine Kac-Moody algebra of type $A^{(1)}_n$ $(n \ge 1)$ or $D^{(1)}_n$ $(n \ge 4)$ and let $\g_0$ be the underlying finite-dimensional simple Lie subalgebra of $\g$. For each Dynkin quiver $Q$ of type $\g_0$,…
Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal…
We define the algebraic elliptic cohomology theory coming from Krichever's elliptic genus as an oriented cohomology theory on smooth varieties over an arbitrary perfect field. We show that in the algebraic cobordism ring with rational…
We prove an identity relating the product of two opposite Schubert varieties in the (equivariant) quantum K-theory ring of a cominuscule flag variety to the minimal degree of a rational curve connecting the Schubert varieties. We deduce…
Given a symmetric quiver with potential, we develop a geometric construction of shifted Yangians acting on the critical cohomologies of antidominantly framed quiver varieties with extended potentials, using the $R$-matrices constructed from…
Two new realizations, denoted $U_{q,x}(\widehat{gl_2})$ and $U(R_{q,x}(\widehat{gl_2}))$ of the trigonometric dynamical quantum affine algebra $U_{q,\lambda}(\widehat{gl_2})$ are proposed, based on Drinfeld-currents and $RLL$ relations…
We find and classify all bialgebras and Hopf algebras or `quantum groups' of dimension $\le 4$ over the field $\Bbb F_2=\{0,1\}$. We summarise our results as a quiver, where the vertices are the inequivalent algebras and there is an arrow…
In this paper we give a geometric construction of the quantum group Ut(G) using Nakajima categories, which were developed in [29]. Our methods allow us to establish a direct connection between the algebraic realization of the quantum group…
We explicitly determine the group of isomorphism classes of equivariant line bundles on the non-archimedean Drinfeld upper half plane for $\mathrm{GL}_2(F)$, for its subgroups of matrices whose determinant has even (respectively trivial)…
We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic…
Adapting a recent work of Brannan et al., on extending graph $C^*$-algebras to Quantum graphs, we introduce "Quantum Quivers" as an analogue of quivers where the edge and vertex set has been replaced by a $C^*$-algebra and the maps between…
We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and…
We study modules and comodules for cohomological Hall algebras equipped with their vertex coproducts arising as objects with classical type stabilizer groups. Specifically we consider how classical type parabolic induction gives rise to…
By using perverse sheaves on representation spaces of quivers over $k[t]/(t^n)$ and jet schemes over flag varieties, we construct a geometric composition algebra $\mathbf K$ under Lusztig's framework on geometric realizations of the…
We study the interaction between geometric operations on stacks and algebraic operations on their categories of sheaves. We work in the general setting of derived algebraic geometry: our basic objects are derived stacks X and their…