The defect functor of a homomorphism and direct unions

Grothendieck rings for Lie superalgebras and the Duflo-Serganova functor

We show that the Duflo-Serganova functor on the category of finite-dimensional modules over a finite-dimensional contragredient Lie superalgebra induces a ring homomorphism on a natural quotient of the Grothendieck ring, which is isomorphic to the ring of characters. We realize this homomorphism as a certain evaluation of functions related to the supersymmetry property. We use this realization to describe the kernel and image of the homomorphism induced by the Duflo-Serganova functor.

The rank of a quiver representation

We define a functor which gives the "global rank of a quiver representation" and prove that it has nice properties which make it a generalization of the rank of a linear map. We demonstrate how to construct other "rank functors" for a quiver Q, which induce ring homomorphisms (called "rank functions") from the representation ring of Q to Z. These rank functions give discrete numerical invariants of quiver representations, useful for computing tensor product multiplicities of representations and determining some structure of the representation ring. We also show that in characteristic 0, rank f…

来自6D ${\cal N}=(2,0)$ SCFT的多个可规范同态的分类对称性与纤维函子

Categorical Symmetries and Fiber Functors from Multiple Gaugeable Homomorphisms from 6D ${\cal N}=(2,0)$ SCFTs

利用SymTFT/TO对应和6D SCFT的高阶范畴结构，证明内在非可逆对称性的相对可规范代数总量子维度更大，支持超选择扇区重数可超1。

Suzuki functor at the critical level

In this paper we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t=0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings i…

Naturality of FHT isomorphism

Freed, Hopkins and Teleman constructed an isomorphism between twisted equivariant K-theory of compact Lie group $G$ and the "Verlinde ring" of the loop group of $G$. We call this isomorphism FHT isomorphism. However, it does not hold naturality with respect to group homomorphisms. We construct two "quasi functors" $t.e.K$ (a modification of twisted equivariant K-theory) and $RL$ (a modification of representation group of loop groups) so that FHT isomorphism is natural transformation between two "quasi functors" for tori, that is, we construct two "induced homomorphisms" of the "quasi functors"…

Rational extensions of the representation ring global functor and a splitting of global equivariant $K$-theory

We identify the group of homomorphisms $\operatorname{Hom}_{\mathcal{GF}}(F,\mathbf{RU}_{\mathbb Q})$ in the category of ($\operatorname{fin}$)-global functors to the rationalization of the unitary representation ring functor and deduce that the higher $\operatorname{Ext}$-groups $\operatorname{Ext}^n_{\mathcal{GF}}(F,\mathbf{RU}_{\mathbb Q})$, $n\geq 2$ have to vanish. This leads to a rational splitting of the ($\operatorname{fin}$)-global equivariant $K$-theory spectrum into a sum of Eilenberg-MacLane spectra. Interpreted in terms of cohomology theories, it means that the equivariant Chern c…