BC_N-graded Lie algebras arising from fermionic representations

扩张理论与费米子强融合2-范畴（附录由Thibault Didier Décoppet和Theo Johnson-Freyd撰写）

Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)

本文研究融合2-范畴的群分次扩张，并利用同伦论方法对费米子强融合2-范畴进行分类，同时详细分析了若干具体例子。

Fermionic formulas for (k, 3)-admissible configurations

We obtain the fermionic formulas for the characters of (k, r)-admissible configurations in the case of r=2 and r=3. This combinatorial object appears as a label of a basis of certain subspace $W(Λ)$ of level-$k$ integrable highest weight module of $\hat{sl}_{r}$. The dual space of $W(Λ)$ is embedded into the space of symmetric polynomials. We introduce a filtration on this space and determine the components of the associated graded space explicitly by using vertex operators. This implies a fermionic formula for the character of $W(Λ)$.

Rooted trees, Feynman graphs, and Hecke correspondences

We construct natural representations of the Connes-Kreimer Lie algebras on rooted trees/Feynman graphs arising from Hecke correspondences in the categories $\LRF, \LFG$ constructed by K. Kremnizer and the author. We thus obtain the insertion/elimination representations constructed by Connes-Kreimer as well as an isomorphic pair we term top-insertion/top-elimination. We also construct graded finite-dimensional sub/quotient representations of these arising from "truncated" correspondences.

由余张量余拟Hopf代数产生的拟量子群

Quasi-quantum groups arising from cotensor coquasi-Hopf algebras

本文证明余张量余代数具有分次余拟Hopf结构，并建立其与余拟三角、余ribbon性质的保持及嵌入态射，同时证明余拟对映体的双射性并构造分次拟余拟Hopf配对。

Functional equations in algebra

We study flat deformations of quotients of a polynomial algebra in a class of graded commutative associative algebras. Functional equations and their solutions in terms of theta functions play important role in these studies. An analog of this theory in a fermionic case is also briefly discussed.

Intertwining operators and fusion rules for vertex operator algebras arising from symplectic fermions

We determine fusion rules (dimensions of the space of intertwining operators) among simple modules for the vertex operator algebra obtained as an even part of the symplectic fermionic vertex operator superalgebra. By using these fusion rules we show that the fusion algebra of this vertex operator algebra is isomorphic to the group algebra of the Klein four group over Z.

迈向${\mathcal W}_3$代数分类的研究

Towards a classification of graded unitary ${\mathcal W}_3$ algebras

本文研究四维单位性对从四维通过SCFT/VOA对应得到的${\mathcal W}_3$顶点代数的约束，证明除了$(3,q+4)$极小模型外，其他中央荷值不兼容四维单位性。

Growth of Graded Twisted Calabi-Yau Algebras

We initiate a study of the growth and matrix-valued Hilbert series of non-negatively graded twisted Calabi-Yau algebras that are homomorphic images of path algebras of weighted quivers, generalizing techniques previously used to investigate Artin-Schelter regular algebras and graded Calabi-Yau algebras. Several results are proved without imposing any assumptions on the degrees of generators or relations of the algebras. We give particular attention to twisted Calabi-Yau algebras of dimension d at most 3, giving precise descriptions of their matrix-valued Hilbert series and partial results desc…

玻色子 - 费米子亚纯开弦顶点代数及其典范扭曲模

Boson-fermion meromorphic open-string vertex algebras and their canonically twisted modules

研究从有限维向量空间构造\(\frac{\mathbb{Z}}{2}\)分次的亚纯开弦顶点代数及其典范扭曲模，通过玻色子和费米子场的非交换推广生成，利用广义维克定理进行构造，是从自旋流形构造狄拉克型算子的代数部分。

关于分级酉顶点代数的半无穷上同调

On the semi-infinite cohomology of graded-unitary vertex algebras

本文研究了具有手性量子动量图的分级酉顶点代数的相对半无穷上同调，证明其具有类似紧致Kähler流形微分形式的结构且本身分级酉，为源自超对称量子场论的顶点算子代数确立了分级酉性，并建立了类似Lefschetz作用的USp(2)作用及形式性结果。

Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras

Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms of certain associated differential (bi)graded Lie and G(erstenhaber)-algebras; in particular the G-algebra arising from an almost complex structure is a d(ifferential) G-algebra iff the almost complex structure is integrable. Such G-algebras, endowed with a generator turning them …

Infinity Algebras, Massey Products, and Deformations

In this paper we give some examples of generalized Massey products, arising from deformations of A-infinity and L-infinity algebras. The generalized Massey products are given by certain graded commutative algebra structures, depending on the deformation problem considered. We show that the formal deformations of infinity algebras give rise to the same algebraic structure as the deformations of Z2-graded Lie algebras. However, the problem of deforming a Lie algebra into an L-infinity algebra, or an associative algebra into an A-infinity algebra gives rise to a new algebra. We also consider defo…

On the elliptic Kashiwara-Vergne Lie algebra

We recall the definitions of two independently defined elliptic versions of the Kashiwara-Vergne Lie algebra $\frak{krv}$, namely the Lie algebra $\frak{krv}^{(1,1)}$ constructed by A.Alekseev, N.Kawazumi, Y.Kuno and F.Naef arising from the study of graded formality isomorphisms associated to topological fundamental groups of surfaces, and the Lie algebra $\frak{krv}_{ell}$ defined using mould theoretic techniques arising from multiple zeta theory by E.Raphael and L.Schneps, and show that they coincide.

Noncommutative complete intersections and matrix integrals

We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive integers n such that the scheme Rep_n(A), of n-dimensional representations of A, is a complete intersection. We discuss examples of RCI algebras, including those arising from quivers. There is another interesting class of associative algebras called noncommutative complete intersections (NCCI). We prove that any graded RCI algebra is NCCI. We also obtain explicit formulas for the Hilbert series of …

On rigidity of Nichols algebras

We study deformations of graded braided bialgebras using cohomological methods. In particular, we show that many examples of Nichols algebras, including the finite-dimensional ones arising in the Andruskiewitsch-Schneider program of classification of pointed Hopf algebras, are rigid. This result can be regarded as nonexistence of "braided Lie algebras" with nontrivial bracket.