Reflection Equation, Twist, and Equivariant Quantization

Twisted symplectic reflection algebras

In this paper we introduce the notion of twisted symplectic reflection algebras and describe the category of representations of such an algebra associated to a non-faithful G-action in terms of those for faithful actions of G.

$(G,χ_ϕ)$-equivariant $ϕ$-coordinated quasi modules for nonlocal vertex algebras

In this paper, we study $(G,χ_ϕ)$-equivariant $ϕ$-coordinated quasi modules for nonlocal vertex algebras. Among the main results, we establish several conceptual results, including a generalized commutator formula and a general construction of weak quantum vertex algebras and their $(G,χ_ϕ)$-equivariant $ϕ$-coordinated quasi modules. As an application, we also construct (equivariant) $ϕ$-coordinated quasi modules for lattice vertex algebras by using Lepowsky's work on twisted vertex operators.

Equivariant K-theory, generalized symmetric products, and twisted Heisenberg algebra

For a space X acted by a finite group $\G$, the product space $X^n$ affords a natural action of the wreath product $\Gn$. In this paper we study the K-groups $K_{\tG_n}(X^n)$ of $\Gn$-equivariant Clifford supermodules on $X^n$. We show that $\tFG =\bigoplus_{n\ge 0}K_{\tG_n}(X^n) \otimes \C$ is a Hopf algebra and it is isomorphic to the Fock space of a twisted Heisenberg algebra. Twisted vertex operators make a natural appearance. The algebraic structures on $\tFG$, when $\G$ is trivial and X is a point, specialize to those on a ring of symmetric functions with the Schur Q-functions as a linea…

Multiplicative Structures on the Twisted Equivariant K-theory of Finite Groups

Let $K$ be a finite group and let $G$ be a finite group acting on $K$ by automorphisms. In this paper we study two different but intimately related subjects: on the one side we classify all possible multiplicative and associative structures with which one can endow the twisted $G$-equivariant K-theory of $K$, and on the other, we classify all possible monoidal structures with which one can endow the category of twisted and $G$-equivariant bundles over $K$. We achieve this classification by encoding the relevant information in the cochains of a sub double complex of the double bar resolution as…

On universal solution to reflection equation

For a given quasitriangular Hopf algebra $\Ha$ we study relations between the braided group $\tilde \Ha^*$ and Drinfeld's twist. We show that the braided bialgebra structure of $\tilde \Ha^*$ is naturally described by means of twisted tensor powers of $\Ha$ and their module algebras. We introduce universal solution to the reflection equation (RE) and deduce a fusion prescription for RE-matrices

复反射群与有理Cherednik代数的表示的扭

Twists of representations of complex reflection groups and rational Cherednik algebras

该研究证明代数分解与诱导表示的上循环扭的一般性结果，将其应用于反射群与有理Cherednik代数，刻画相关模的扭并证明非交换代数的Chevalley定理类似结论。

Local Weyl modules for equivariant map algebras with free abelian group actions

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras. Local Weyl modules play an important role in the theory of finite-dimensional representations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras wh…

Braided Commutative Geometry and Drinfel'd Twist Deformations

In this thesis we give obstructions for Drinfel'd twist deformation quantization on several classes of symplectic manifolds. Motivated from this quantization procedure, we further construct a noncommutative Cartan calculus on any braided commutative algebra, as well as an equivariant Levi-Civita covariant derivative for any non-degenerate equivariant metric. This generalizes and unifies the Cartan calculus on a smooth manifold and the Cartan calculus on twist star product algebras. We prove that the Drinfel'd functor leads to equivalence classes in braided commutative geometry and commutes wit…

扭转、Gorm 与反场，天哪！

Twists and Gorms and Antifields, oh my!

本文利用 BV 场-反场形式化与微分 gorm 几何，系统构造并推广了开代数超荷及多拓扑超荷情形下的超对称场论拓扑扭转，并给出等变拓扑 B 模型等实例。

Braided Cartan Calculi and Submanifold Algebras

We construct a noncommutative Cartan calculus on any braided commutative algebra and study its applications in noncommutative geometry. The braided Lie derivative, insertion and de Rham differential are introduced and related via graded braided commutators, also incorporating the braided Schouten-Nijenhuis bracket. The resulting braided Cartan calculus generalizes the Cartan calculus on smooth manifolds and the twisted Cartan calculus. While it is a necessity of derivation based Cartan calculi on noncommutative algebras to employ central bimodules our approach allows to consider bimodules over…