Quasi-Lie structure of twisted derivations of Laurent polynomials

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…

Constructing 3-Lie algebras

3-Lie algebras are constructed by Lie algebras, derivations and linear functions, associative commutative algebras, whose involutions and derivations. Then the 3-Lie algebras are obtained from group algebras $F[G]$. An infinite dimensional simple 3-Lie algebra $(A, [,,]_{ω, δ_0})$ and a non-simple 3-Lie algebra $(A, [,,]_{ω_1, δ})$ are constructed by Laurent polynomials $A=F[t, t^{-1}]$ and its involutions $ω$ and $ω_1$ and derivations $δ$ and $δ_0$. At last of the paper, we summarize the methods of constructing $n$-Lie algebras for $n\geq 3$ and provide a problem.

Quantum algebraic tori

The author introduces the notion of a quantum form of an algebraic torus. In the case of diagonal algebraic torus we get the algebra of Laurent twisted polynomials. Quantum algebraic torus can be characterized in terms of exact sequences. The author study the center of QAT. In the case of roots of 1 the new examples of central simple algebras are constructed.

Quivers supporting twisted Calabi-Yau algebras

We consider graded twisted Calabi-Yau algebras of dimension 3 which are derivation-quotient algebras of the form $A = \kk Q/I$, where $Q$ is a quiver and $I$ is an ideal of relations coming from taking partial derivatives of a twisted superpotential on $Q$. We define the type $(M, P, d)$ of such an algebra $A$, where $M$ is the incidence matrix of the quiver, $P$ is the permutation matrix giving the action of the Nakayama automorphism of $A$ on the vertices of the quiver, and $d$ is the degree of the superpotential. We study the question of what possible types can occur under the additional as…

Twisted exterior derivatives for universal enveloping algebras I

Consider any representation $ϕ$ of a finite-dimensional Lie algebra $g$ by derivations of the completed symmetric algebra $\hat{S}(g^*)$ of its dual. Consider the tensor product of $\hat{S}(g^*)$ and the exterior algebra $Λ(g)$. We show that the representation $ϕ$ extends canonically to the representation $\tildeϕ$ of that tensor product algebra. We construct an exterior derivative on that algebra, giving rise to a twisted version of the exterior differential calculus with the enveloping algebra in the role of the coordinate algebra. In this twisted version, the commutators between the noncomm…

Twisted Hamiltonian Lie Algebras and Their Multiplicity-Free Representations

We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated to Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu's generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.

A quantum space and some associated quantum groups

In this paper, we first introduce a quantum $n$-space with a cocommutative Hopf algebra structure. Then it is shown that to this quantum $n$-space there corresponds a derivation algebra of $σ$-twisted derivations related to some algebra automorphisms $σ$ on the quantum $n$-space. Furthermore, we show that this derivation algebra is a noncommutative and non-cocommutative Hopf algebra, namely, a quantum group. Morever, for this the quantum $n$-space, we show how to construct a bicovariant differential calculus related to the derivation algebra.

准双曲李代数的变形映射

Deformation maps of quasi-twilled Lie algebras

本文通过引入准双曲李代数，统一研究李代数各类算子的上同调与变形理论，提出两种变形映射类型，涵盖修正r矩阵、交叉同态、导子、同态、相对罗塔-巴克斯算子等，并给出其控制代数与上同调，拓展了现有结果并获得新结论。

Supersymmetric WZ-Poisson sigma model and twisted generalized complex geometry

It has been shown recently that extended supersymmetry in twisted first-order sigma models is related to twisted generalized complex geometry in the target. In the general case there are additional algebraic and differential conditions relating the twisted generalized complex structure and the geometrical data defining the model. We study in the Hamiltonian formalism the case of vanishing metric, which is the supersymmetric version of the WZ-Poisson sigma model. We prove that the compatibility conditions reduce to an algebraic equation, which represents a considerable simplification with respe…

Quasi-hom-Lie Algebras, Central Extensions and 2-cocycle-like Identities

This paper begins by introducing the concept of a quasi-hom-Lie algebra which is a natural generalization of hom-Lie algebras introduced in a previous paper by the authors. Quasi-hom-Lie algebras include also as special cases (color) Lie algebras and superalgebras, and can be seen as deformations of these by homomorphisms, twisting the Jacobi identity and skew-symmetry. The natural realm for these quasi-hom-Lie algebras is as a generalization-deformation of the Witt algebra $\Witt$ of derivations on the Laurent polynomials $\C[t,t^{-1}]$. We also develop a theory of central extensions for qhl-…

On Harish-Chandra modules of the Lie algebra arising from the $2$-Dimensional Torus

Let $A=\mathbb{C}[t_1^{\pm1},t_2^{\pm1}]$ be the algebra of Laurent polynomials in two variables and $B$ be the set of skew derivations of $A$. Let $L$ be the universal central extension of the derived Lie subalgebra of the Lie algebra $A\rtimes B$. Set $\widetilde{L}=L\oplus\mathbb{C} d_1\oplus\mathbb{C} d_2$, where $d_1$, $d_2$ are two degree derivations. A Harish-Chandra module is defined as an irreducible weight module with finite dimensional weight spaces. In this paper, we prove that a Harish-Chandra module of the Lie algebra $\widetilde{L}$ is a uniformly bounded module or a generalized…

Twisted modules for vertex operator algebras

This contribution is mainly based on joint papers with Lepowsky and Milas, and some parts of these papers are reproduced here. These papers further extended works by Lepowsky and by Milas. Following our joint papers, I explain the general principles of twisted modules for vertex operator algebras in their powerful formulation using formal series, and derive general relations satisfied by twisted and untwisted vertex operators. Using these, I prove new "equivalence" and "construction" theorems, identifying a set of sufficient conditions in order to have a twisted module for a vertex operator al…

类型 $B_2$ 的 $A_1$ 是否为 $B_2$ 型？

Is $A_1$ of type $B_2$

本文研究 $U_q^+(B_2)$ 的 Gelfand-Kirillov 维数为 2 的单商的导子，证明特定族的所有导子均为内导子，其余为量子广义 Weyl 代数且一阶 Hochschild 上同调维数为 1。

Exel-Pardo algebras with a twist

Katsura associated a $C^*$-algebra $C^*_{A,B}$ to integral matrices $A\ge 0$ and $B$ of the same size, gave sufficient conditions on $(A,B)$ making it simple purely infinite (SPI), and proved that any separable $C^*$-algebra $KK$-isomorphic to a cone of an element $ξ\in KK(C(S^1)^n,C(S^1)^n)$ in Kasparov's $KK$ is $KK$-isomorphic to an SPI $C^*_{A,B}$. Here we introduce, for the data of a commutative ring $\ell$, matrices $A,B$ as above and $C$ of the same size with coefficients in the group $\mathcal{U}(\ell)$ of invertible elements, an $\ell$-algebra $\mathcal{O}_{A,B}^C$, the twisted Katsur…

A quantum algebra approach to multivariate Askey-Wilson polynomials

We study matrix elements of a change of base between two different bases of representations of the quantum algebra $U_q(su(1,1))$. The two bases, which are multivariate versions of Al-Salam--Chihara polynomials, are eigenfunctions of iterated coproducts of twisted primitive elements. The matrix elements are identified with Gasper and Rahman's multivariate Askey-Wilson polynomials, and from this interpretation we derive their orthogonality relations. Furthermore, the matrix elements are shown to be eigenfunctions of the twisted primitive elements after a change of representation, which gives a …

The first cohomology, derivations and the reductivity of a (meromorphic open-string) vertex algebra

We give a criterion for the complete reducibility of modules satisfying a composability condition for a meromorphic open-string vertex algebra $V$ using the first cohomology of the algebra. For a $V$-bimodule $M$, let $\hat{H}^{1}_{\infty}(V, M)$ be the first cohomology of $V$ with the coefficients in $M$. Let $\hat{Z}^{1}_{\infty}(V, M)$ be the subspace of $\hat{H}^{1}_{\infty}(V, M)$ canonically isomorphic to the space of derivations obtained from the zero mode of the right vertex operators of weight $1$ elements such that the difference between the skew-symmetric opposite action of the left…

On the geometry of $(σ,τ)$-algebras

We introduce $(σ,τ)$-algebras as a framework for twisted differential calculi over noncommutative, as well as commutative, algebras with motivations from the theory of $σ$-derivations and quantum groups. A $(σ,τ)$-algebra consists of an associative algebra together with a set of $(σ,τ)$-derivations, and corresponding notions of $(σ,τ)$-modules and connections are introduced. We prove that $(σ,τ)$-connections exist on projective modules, and introduce notions of both torsion and curvature, as well as compatibility with a hermitian form, leading to the definition of a Levi-Civita $(σ,τ)$-connect…