$U_q(\widehat{sl_2})$中的Heisenberg型族

Quasi-particles in the principal picture of $\widehat{\mathfrak{sl}}_{2}$ and Rogers-Ramanujan-type identities

In their seminal work J. Lepowsky and R. L. Wilson gave a vertex-operator theoretic interpretation of Gordon-Andrews-Bressoud's generalization of Rogers-Ramanujan combinatorial identities, by constructing bases of vacuum spaces for the principal Heisenberg subalgebra of standard $\widehat{\mathfrak{sl}}_{2}$-modules, parametrized with partitions satisfying certain difference 2 conditions. In this paper we define quasi-particles in the principal picture of $\widehat{\mathfrak{sl}}_{2}$ and construct quasi-particle monomial bases of standard $\widehat{\mathfrak{sl}}_{2}$-modules for which princi…

Two Equivalent Realizations of Trigonometric Dynamical Affine Quantum Group $U_{q,x}(\widehat{sl_2})=U_{q,λ}(\widehat{sl_2})$, Drinfeld Currents and Hopf Algebroid Structures

Two new realizations, denoted $U_{q,x}(\widehat{gl_2})$ and $U(R_{q,x}(\widehat{gl_2}))$ of the trigonometric dynamical quantum affine algebra $U_{q,λ}(\widehat{gl_2})$ are proposed, based on Drinfeld-currents and $RLL$ relations respectively, along with a Heisenberg algebra $\left\{P,Q\right\}$, with $x=q^{2P}$. Here $P$ plays the role of the dynamical variable $λ$ and $Q=\frac{\partial}{\partial P}$. An explicit isomorphism from $U_{q,x}(\widehat{gl_2})$ to $U(R_{q,x}(\widehat{gl_2}))$ is established, which is a dynamical extension of the Ding-Frenkel isomorphism of $U_{q}(\widehat{gl_2})$ w…

Trigonometric Lie algebras, affine Kac-Moody Lie algebras, and equivariant quasi modules for vertex algebras

In this paper, we study a family of infinite-dimensional Lie algebras $\widehat{X}_{S}$, where $X$ stands for the type: $A,B,C,D$, and $S$ is an abelian group, which generalize the $A,B,C,D$ series of trigonometric Lie algebras. Among the main results, we identify $\widehat{X}_{S}$ with what are called the covariant algebras of the affine Lie algebra $\widehat{\mathcal{L}_{S}}$ with respect to some automorphism groups, where $\mathcal{L}_{S}$ is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted $\widehat{X}_{S}$-modules of level $\ell$ naturally co…

Weak splittings of quotients of Drinfeld and Heisenberg doubles

We investigate the fine structure of the simplectic foliations of Poisson homogeneous spaces. Two general results are proved for weak splittings of surjective Poisson submersions from Heisenberg and Drinfeld doubles. The implications of these results are that the torus orbits of symplectic leaves of the quotients can be explicitly realized as Poisson-Dirac submanifolds of the torus orbits of the doubles. The results have a wide range of applications to many families of real and complex Poisson structures on flag varieties. Their torus orbits of leaves recover important families of varieties su…

非退化的若干方面：中心型因子群、海森堡表示与阿达马矩阵

Facets of nondegeneracy: central type factor groups, Heisenberg representations and Hadamard matrices

该研究针对有限群的互补子群对，探讨双1-上同调与海森堡型射影表示、二阶上同调的关联，证明相关表示非退化的等价条件，并以辛幂零李代数为例验证结论。

通过模线性微分方程改进 Heisenberg 代数 2-adic 特征映射像的下界

An Improved Lower Bound on the Image of the 2-adic Character Map for the Heisenberg Algebra via Modular Linear Differential Equations

研究利用模线性微分方程构造 level one 模形式解，并借助 Serre 定理证明其 2-adic 收敛到 Γ_0(2) 的 Hauptmodul，从而改进 Heisenberg VOA 特征映射像的下界，证明其包含权零过收敛模形式空间。