Xiaoqing Yue, Yucai Su
Let L be the Lie algebra of Block type over a field F of characteristic zero, defined with basis {L_{i,j} | i,j\in Z} and relations [L_{i,j},L_{k,l}]=((j+1)k-(l+1)i)L_{i+k,j+l}. Then L is Z^2-graded. In this paper, Z^2-graded L-modules of the intermediate series are classified.
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In this paper, we complete the classification of the {\bf Z}-graded modules of the intermediate series over the $q$-analog Virasoro-like algebra $L$. We first construct four classes of irreducible {\bf Z}-graded $L$-modules of the intermediate series. Then we prove that any {\bf Z}-graded $L$-modules of the intermediate series must be the direct sum of some trivial $L$-modules or one of the modules constructed by us.
In this paper, we consider the classification of irreducible ${\bf Z}$- and ${\bf Z}^2$-graded modules with finite dimensional homogeneous subspaces over the Virasoro-like algebra. We first prove that such a module is a uniformly bounded module or a generalized highest weight module. Then we determine all generalized highest weight irreducible modules. As a consequence, we also determine all the modules with nonzero center. Finally, we prove that there does not exist any nontrivial ${\bf Z}$-graded modules of intermediate series.
We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki's categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial Z-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.
We give an explicit graded cellular basis of the $\mathfrak{sl}_3$-web algebra $K_S$. In order to do this, we identify Kuperberg's basis for the $\mathfrak{sl}_3$-web space $W_S$ with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weight of flows on webs and the $q$-degree of foams. We use these observations to give a "foamy" version of Hu and Mathas graded cellular basis of the cyclotomic Hecke algebra which turns out to be a graded cellular basis of the $\mathfrak{sl}_3$-web algebra. We restrict ourselves to …
The main result of this paper shows that, over large enough fields of characteristic different from $2$, the alternating Hecke algebras are $\mathbb{Z}$-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke algebra of the symmetric group $\mathfrak{S}_n$. As a special case, this shows that the group algebra of the alternating group, over large enough fields of characteristic different from $2$, is a $\mathbb{Z}$-graded algebra. We give a homogeneous presentation for these algebras, compute their graded dimension and show that the blocks of the quiver Hecke algebras…
We review some recent advances in modular representation theory of symmetric groups and related Hecke algebras. We discuss connections with Khovanov-Lauda-Rouquier algebras and gradings on the blocks of the group algebras $FΣ_n$, which these connections reveal; graded categorification and connections with quantum groups and crystal bases; modular branching rules and the Mullineaux map; graded cellular structure and graded Specht modules; cuspidal systems for affine KLR algebras and imaginary Schur-Weyl duality, which connects representation theory of these algebras to the usual Schur algebras …
The present paper is another step toward the classification of $\Z\times\Z$-graded Lie algebras: we classify $\Z\times\Z$-graded Lie algebras $A=\oplus_{i,j\in\Z}A_{i,j}$ over a field $F$ of characteristic 0 with dim $A_{i,j}\le1$ for $i,j\in\Z$, satisfying: (1) $Å_0=\oplus_{i\in \Z}Å_{i,0}$ is isomorphic to the centerless Virasoro algebra, (2) $A_{0,-1},A_{0,0},A_{0,1}$ span the 3-dimensional simple Lie algebra $sl_2$, (3) $A$ is generated by the centerless Virasoro algebra and $sl_2$. These algebras include some rank 2 centerless Virasoro algebras, and some rank 2 Block algebras and their ex…
A notion of intermediate vertex subalgebras of lattice vertex operator algebras is introduced, as a generalization of the notion of principal subspaces. Bases and the graded dimensions of such subalgebras are given.As an application, it is shown that the characters of some modules of an intermediate vertex subalgebra between $E_7$ and $E_8$ lattice vertex operator algebras satisfy some modular differential equations.This result is an analogue of the result concerning the "hole" of the Deligne dimension formulas and the intermediate Lie algebra between the simple Lie algebras $E_7$ and $E_8$.
Quotients of graded quasi-Hopf algebras
研究强\(G -\)分次拟霍普夫代数的拟霍普夫理想分类，通过证明其与\((N,K)\)对应来分类，特定类拟霍普夫理想对应关系有改进，应用于余中心阿贝尔扩张等。
Category $\mathcal{O}$ for hybrid quantum groups and non-commutative Springer resolutions
本文研究了杂交量子群的Category $\mathcal{O}$，通过非交换Springer解析构建其主块模型，并证明主块与仿射Hecke范畴导出等价，同时揭示了简单模在泛化Kazhdan-Lusztig多项式中的分级乘数。
We analyze the BGG Category $\mathcal{O}$ over a large class of generalized Weyl algebras (henceforth termed GWAs). Given such a "triangular" GWA for which Category $\mathcal{O}$ decomposes into a direct sum of subcategories, we study in detail the homological properties of blocks with finitely many simples. As consequences, we show that the endomorphism algebra of a projective generator of such a block is quasi-hereditary, finite-dimensional, and graded Koszul. We also classify all tilting modules in the block, as well as all submodules of all projective and tilting modules. Finally, we prese…
The classical deformation theory of Lie algebras involves different kinds of Massey products of cohomology classes. Even the condition of extendibility of an infinitesimal deformation to a formal one-parameter deformation of a Lie algebra involves Massey powers of two dimensional cohomology classes which are not powers in the usual definition of Massey products in the cohomology of a differential graded Lie algebra. In the case of deformations with other local bases, one deals with other, more specific Massey products. In the present work a construction of generalized Massey products is given,…
Graded pseudo-traces for strongly interlocked modules for a vertex operator algebra and applications
本文研究了顶点算子代数中强互锁模块的分级伪迹定义及其在海森堡和维拉索洛代数中的应用。
In this paper, we introduce two kinds of Lie conformal algebras associated with the loop Schrödinger-Virasoro Lie algebra and the extended loop Schrödinger-Virasoro Lie algebra, respectively. The conformal derivations, the second cohomology groups of these two conformal algebras are completely determined. And nontrivial free conformal modules of rank one and Z-graded free intermediate series modules over these two conformal algebras are also classified in the present paper
The Lie conformal algebra of loop Virasoro algebra, denoted by $\mathscr{CW}$, is introduced in this paper. Explicitly, $\mathscr{CW}$ is a Lie conformal algebra with $\mathbb{C}[\partial]$-basis $\{L_i\,|\,i\in\mathbb{C}\}$ and $λ$-brackets $[L_i\, {}_λ\, L_j]=(-\partial-2λ) L_{i+j}$. Then conformal derivations of $\mathscr{CW}$ are determined. Finally, rank one conformal modules and $\mathbb{Z}$-graded free intermediate series modules over $\mathscr{CW}$ are classified.
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