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Register a new userOn Kostant Root Systems for Lie Superalgebras
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We study the eigenspace decomposition of a basic classical Lie superalgebra under the adjoint action of a toral subalgebra, thus extending results of Kostant. In recognition of Kostants contribution we refer to the eigenspaces appearing in the decomposition as Kostant roots. We then prove that Kostant root systems inherit the main properties of classical root systems. Our approach is combinatorial in nature and utilizes certain graphs naturally associated with Kostant root systems. In particular, we reprove Kostants results without making use of the Killing form.
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Let $mathfrak{r}$ be a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form, let $mathfrak{p}$ be a finite dimensional complex super vector space with a non-degenerate super-symmetric bilinear form, and let $ u: mathfrak{r}rightarrowmathfrak{osp}(mathfrak{p})$ be a Lie superalgebra homomorphism. In this paper, we give a necessary and sufficient condition for $mathfrak{r}oplusmathfrak{p}$ to be a quadratic Lie superalgebra. The criterion obtained is an analogue of a constancy condition given by Kostant in the Lie algebra setting. As an application, we prove an analogue of the Parthasarathys formula for the square of the Dirac operator attached to a pair of quadratic Lie superalgebras.
Using Howe duality we compute explicitly Kostant-type homology groups for a wide class of representations of the infinite-dimensional Lie superalgebra $hat{frak{gl}}_{infty|infty}$ and its classical subalgebras at positive integral levels. We also obtain Kostant-type homology formulas for the Lie algebra $ widehat{frak{gl}}_infty$ at negative integral levels. We further construct resolutions in terms of generalized Verma modules for these representations.
In this paper the authors introduce a class of parabolic subalgebras for classical simple Lie superalgebras associated to the detecting subalgebras introduced by Boe, Kujawa and Nakano. These parabolic subalgebras are shown to have good cohomological properties governed by the Bott-Borel-Weil theorem involving the zero component of the Lie superalgebra in conjunction with the odd roots. These results are later used to verify an open conjecture given by Boe, Kujawa and Nakano pertaining to the equality of various support varieties.
For a finite dimensional unital complex simple Jordan superalgebra $J$, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $mathfrak g_flatcong mathfrak g_flat(-1)oplusmathfrak g_flat(0)oplusmathfrak g_flat(1)$, 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 $mathcal P(V):=bigoplus_{lambdainOmega}V_lambda$, where $left(V_lambdaright)_{lambdainOmega}$ is a family of irreducible $mathfrak g$-modules parametrized by a set of partitions $Omega$. In these cases, one can define a natural basis $left(D_lambdaright)_{lambdainOmega}$ of Capelli operators for the algebra $mathcal{PD}(V)^{mathfrak g}$. In this paper we complete the solution to the Capelli eigenvalue problem, which is to determine the scalar $c_mu(lambda)$ by which $D_mu$ acts on $V_lambda$. We associate a restricted root system $mathit{Sigma}$ to the symmetric pair $(mathfrak g,mathfrak k)$ that corresponds to $J$, which is either a deformed root system of type $mathsf{A}(m,n)$ or a root system of type $mathsf{Q}(n)$. We prove a necessary and sufficient condition on the structure of $mathit{Sigma}$ for $mathcal{P}(V)$ to be completely reducible and multiplicity-free. When $mathit{Sigma}$ satisfies the latter condition we obtain an explicit formula for the eigenvalue $c_mu(lambda)$, in terms of Sergeev-Veselovs shifted super Jack polynomials when $mathit{Sigma}$ is of type $mathsf{A}(m,n)$, and Okounkov-Ivanovs factorial Schur $Q$-polynomials when $mathit{Sigma}$ is of type $mathsf{Q}(n)$.
We investigate a new cohomology of Lie superalgebras, which may be compared to a de Rham cohomology of Lie supergroups involving both differential and integral forms. It is defined by a BRST complex of Lie superalgebra modules, which is formulated in terms of a Weyl superalgebra and incorporates inequivalent representations of the bosonic Weyl subalgebra. The new cohomology includes the standard Lie superalgebra cohomology as a special case. Examples of new cohomology groups are computed.
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