The Annihilation theorem for the completely reducible Lie superalgebras

Complexity for Modules Over the Classical Lie Superalgebra gl(m|n)

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus \mathfrak{g}_{\bar{1}}$ be a classical Lie superalgebra and $\mathcal{F}$ be the category of finite dimensional $\mathfrak{g}$-supermodules which are completely reducible over the reductive Lie algebra $\mathfrak{g}_{\bar{0}}$. In an earlier paper the authors demonstrated that for any module $M$ in $\mathcal{F}$ the rate of growth of the minimal projective resolution (i.e., the complexity of $M$) is bounded by the dimension of $\mathfrak{g}_{\bar{1}}$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie su…

A Lie conformal superalgebra and duality of representations for E(4,4)

We construct a duality functor in the category of continuous representations of the Lie superalgebra E(4,4), the only exceptional simple linearly compact Lie superalgebra, for which it wasn't known. This is achieved by constructing a Lie conformal superalgebra of type (4,4), for which E(4,4) is the annihilation algebra. Along the way we obtain an explicit realization of E(4,4) by vector fields on a (4|4)-dimensional supermanifold.

The Capelli eigenvalue problem for Lie superalgebras

For a finite dimensional unital complex simple Jordan superalgebra $J$, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $\mathfrak g_\flat\cong \mathfrak g_\flat(-1)\oplus\mathfrak g_\flat(0)\oplus\mathfrak g_\flat(1)$, such that $\mathfrak g_\flat(-1)\cong J$. Set $V:=\mathfrak g_\flat(-1)^*$ and $\mathfrak g:=\mathfrak g_\flat(0)$. In most cases, the space $\mathcal P(V)$ of superpolynomials on $V$ is a completely reducible and multiplicity-free representation of $\mathfrak g$, with a decomposition $\mathcal P(V):=\bigoplus_{λ\inΩ}V_λ$, where $\left(V_λ\right)_{λ\inΩ}…

Fock representations of the Lie superalgebra q(n+1)

For the Lie superalgebra $q(n+1)$ a description is given in terms of creation and annihilation operators, in such a way that the defining relations of $q(n+1)$ are determined by quadratic and triple supercommutation relations of these operators. Fock space representations $V_p$ of $q(n+1)$ are defined by means of these creation and annihilation operators. These new representations are introduced as quotient modules of some induced module of $q(n+1)$. The representations $V_p$ are not graded, but they possess a number of properties that are of importance for physical applications. For $p$ a pos…

On a class of tensor product representations for the orthosymplectic superalgebra

We introduce the spinor representations for osp(m|2n). These generalize the spinors for so(m) and the symplectic spinors for sp(2n) and correspond to representations of the supergroup with supergroup pair (Spin(m) x Mp(2n),osp(m|2n)). We prove that these spinor spaces are uniquely characterized as the completely pointed osp(m|2n)-modules. Then the tensor product of this representation with irreducible finite dimensional osp(m|2n)-modules is studied. Therefore we derive a criterion for complete reducibility of tensor product representations. We calculate the decomposition into irreducible osp(m…

Computation of the homology of the complexes of finite Verma modules for $K'_4$

We compute the homology of the complexes of finite Verma modules over the annihilation superalgebra $\mathcal A(K'_{4})$, associated with the conformal superalgebra $K'_{4}$, obtained in \cite{K4}. We use the computation of the homology in order to provide an explicit realization of all the irreducible quotients of finite Verma modules over $\mathcal A(K'_{4})$.

Classification of finite irreducible conformal modules for $K'_4$

We classify the finite irreducible modules over the conformal superalgebra $K'_{4}$ by their correspondence with finite conformal modules over the associated annihilation superalgebra $\mathcal A(K'_{4})$. This is achieved by a complete classification of singular vectors in generalized Verma modules for $\mathcal A(K'_{4})$. We also show that morphisms between generalized Verma modules can be arranged in infinitely many bilateral complexes.

Homology of the complexes of finite Verma modules over $CK_6$

We compute the homology of the first and third quadrants of the complexes of finite Verma modules over the annihilation superalgebra $\mathcal{A}(CK_{6})\cong E(1,6)$, associated with the conformal superalgebra $CK_6$, obtained in \cite{ck6}. This computation allows us to explicitly realize the irreducible quotients of degenerate finite Verma modules over $\mathcal{A}(CK_{6})$.

Orthosymplectic Lie superalgebras, Koszul duality, and a complete intersection analogue of the Eagon-Northcott complex

We study the ideal of maximal minors in Littlewood varieties, a class of quadratic complete intersections in spaces of matrices. We give a geometric construction for a large class of modules, including all powers of this ideal, and show that they have a linear free resolution over the complete intersection and that their Koszul dual is an infinite-dimensional irreducible representation of the orthosymplectic Lie superalgebra. We calculate the algebra of cohomology operators acting on this free resolution. We prove analogous results for powers of the ideals of maximal minors in the variety of l…

Bernstein-Gelfand-Gelfand resolutions for basic classical Lie superalgebras

We study Kostant cohomology and Bernstein-Gelfand-Gelfand resolutions for finite dimensional representations of basic classical Lie superalgebras and reductive Lie superalgebras based on them. For each choice of parabolic subalgebra and irreducible representation of such a Lie superalgebra, there is a natural definition of the derivative and coderivative, which define the (co)homology groups. We prove that a necessary condition to have a resolution of an irreducible module in terms of Verma modules is complete reducibility of the cohomology groups. Essentially, if it exists, every such a resol…

Heisenberg-Virasoro型李超代数上$U(\mathfrak h)$-自由模的分类：基于多项式移位算子

Classification of \(U(\mathfrak h)\)-Free Modules over the Heisenberg-Virasoro type Lie superalgebra via Polynomial Shift Operators

本文利用多项式移位算子分类了Heisenberg-Virasoro型李超代数上超秩$1|1$的$U(\mathfrak h)$-自由模，确定了三角族$\Omega^+(\lambda,\beta,d)$及平凡模结构，并给出该代数的$2\times2$矩阵新实现。

N=1 Heisenberg-Virasoro 超代数的表示

Representations of the N=1 Heisenberg-Virasoro superalgebra

本文定义了 N=1 Heisenberg-Virasoro 超代数的一类非权模，给出了所有子模和不可约商模，并证明了这些模构成秩为2的 U(𝔥)-自由模的完全分类，同时研究了其四个子代数上的非权模。

Programming Realization of Symbolic Computations for Non-linear Commutator Superalgebras over the Heisenberg--Weyl Superalgebra: Data Structures and Processing Methods

We suggest a programming realization of an algorithm for verifying a given set of algebraic relations in the form of a supercommutator multiplication table for the Verma module, which is constructed according to a generalized Cartan procedure for a quadratic superalgebra and whose elements are realized as a formal power series with respect to non-commuting elements. To this end, we propose an algebraic procedure of Verma module construction and its realization in terms of non-commuting creation and annihilation operators of a given Heisenberg--Weyl superalgebra. In doing so, we set up a proble…

A duality between vertex superalgebras $L_{-3/2}(\mathfrak{osp}(1\vert 2))$ and $\mathcal V^{(2)}$ and generalizations to logarithmic vertex algebras

We introduce a subalgebra $\overline F$ of the Clifford vertex superalgebra ($bc$ system) which is completely reducible as a $L^{Vir} (-2,0)$-module, $C_2$-cofinite, but it is not conformal and it is not isomorphic to the symplectic fermion algebra $\mathcal{SF}(1)$. We show that $\mathcal{SF}(1)$ and $\overline{F}$ are in an interesting duality, since $\overline{F}$ can be equipped with the structure of a $\mathcal{SF}(1)$-module and vice versa. Using the decomposition of $\overline F$ and a free-field realization from arXiv:1711.11342, we decompose $L_k(\mathfrak{osp}(1\vert 2))$ at the crit…