Explicit formulas for Eigenvalues of Capelli operators for the Lie superalgebra $\frak{osp}(1|2n)$

Cohomology of $\frak {osp}(1|2)$ acting on the space of bilinear differential operators on the superspace $\mathbb{R}^{1|1}$

We compute the first cohomology of the ortosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ on the (1,1)-dimensional real superspace with coefficients in the superspace $\frak{D}_{λ,ν;μ}$ of bilinear differential operators acting on weighted densities. This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on $\mathbb{R}\mathbb{P}^1$ acting on bilinear differential operators, International Journal of Geometric Methods in Modern Physics (2005), {\bf 2}; N 1, 23-40].

Representations of superconformal algebras and mock theta functions

It is well known that the normaized characters of integrable highest weight modules of given level over an affine Lie algebra $\hat{\frak{g}}$ span an $SL_2(\mathbf{Z})$-invariant space. This result extends to admissible $\hat{\frak{g}}$-modules, where $\frak{g}$ is a simple Lie algebra or $osp_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $\hat{\frak{g}}$-modules when $\frak{g} =sl_2$ (resp. $=osp_{1|2}$) one obtains minimal series modules over the Virasoro (resp. $N=1$ superconformal algebras), which form modular invariant families. Another instance of modular invari…

The Capelli problem for $\mathfrak{gl}(m|n)$ and the spectrum of invariant differential operators

The "Capelli problem" for the symmetric pairs $(\mathfrak{gl}\times \mathfrak{gl},\mathfrak{gl})$ $(\mathfrak{gl},\mathfrak{o})$, and $(\mathfrak{gl},\mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter. In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs $(\mathfrak{g},\mathfrak{g}):=(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$ and $(\mathfrak{gl}(m|n)\times\mathfrak{gl}(m|n),\mathfrak{gl}(m|n))$, acting on $W:=S^2(\mathbb C^{m|2n})$ and $\mathbb C^{m|n}\oti…

Cohomology of $\mathfrak {osp}(2|2)$ acting on the spaces of linear differential operators on the superspace $\mathbb{R}^{1|2}$

We compute the first differential cohomology of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2|2)$ with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1,\,2)-dimensional real superspace. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. This work is a simplest generalization of a result by Basdouri and Ben Ammar [Cohomology of $\frak {osp}(1|2)$ with coefficients in $\frak{D}_{λ,μ}$.

与李超代数 ${\ rak{osp}}(1|2n)$ 相关的 Yangian 的表示

Representations of the Yangians associated with Lie superalgebras ${\frak{osp}}(1|2n)$

本文借助 Drinfeld 多项式并基于 $n=1$ 的已有结果，完成了与李超代数 ${\ rak{osp}}_{1|2n}$ 相关 Yangian 的有限维不可约表示分类。

Symetries of modules of differential operators on $S^{1|1}$

Let $\frak F_ł$ be the space of tensor densities of degree $λ$ on the supercircle $S^{1|1}$. We consider the space $\mathfrak{D}_{λ,μ}^k$ of k-th order linear differential operators from $\frak F_ł$ to $\frak F_μ$ as $\mathfrak{osp}(1|2)$-module and we determine the space $\mathfrak{I}_{λ,μ}^k$ of linear maps on $\mathfrak{D}_{λ,μ}^k$ commuting with the $\mathfrak{osp}(1|2)$-action.