arXiv subjects
Publications and source records attributed to Elena Poletaeva.
We classify irreducible representations of finite $W$-algebra of the queer Lie superalgebra $Q(n)$ associated with the principal nilpotent coadjoint orbits. We use this classification and our previous results to obtain a classification of simple finite-dimensional representations of the super Yangian $YQ(1)$.
In this paper we study the finite W-algebra for the queer Lie superalgebra Q(n) associated with the non-regular even nilpotent coadjoint orbits in the case when the corresponding nilpotent element has Jordan blocks each of size l. We prove that this finite W-algebra is isomorphic to a quotient of the super-Yangian of Q({n/l})
In this paper we study finite W-algebras for basic classical superalgebras and Q(n) associated to the regular even nilpotent coadjoint orbits. We prove that this algebra satisfies the Amitsur-Levitzki identity and therefore all its irreducible representations are finite-dimensional. In the case of Q(n) we give an explicit description of the W-algebra in terms of generators and relation and realize it as a quotient of the super-Yangian of Q(1).
We obtain a realization of the Lie superalgebra $D(2, 1 ; α)$ in differential operators on the supercircle $S^{1|2}$ and in $4\times 4$ matrices over a Weyl algebra. A contraction of $D(2, 1 ; α)$ is isomorphic to the universal central extension $\hat{\p\sł}(2|2)$ of $\p\sł(2|2)$. We realize it in $4\times 4$ matrices over the associative algebra of pseudodifferential operators on $S^1$. Correspondingly, there exists a three-parameter family of irreducible representations of $\hat{\p\sł}(2|2)$ in a $(2|2)$--dimensional complex superspace.
We describe the infinitesimal deformations of the standard embedding of the Lie superalgebra $D(2, 1 ; α)$ into the Poisson superalgebra of pseudodifferential symbols on $S^{1|2}$. We show that for the standard embedding of $D(2, 1 ; α)$ into the Poisson superalgebra of differential operators on $S^{1|2}$, the infinitesimal deformations correspond to formal deformations. For the embedding of $D(2, 1 ; α)$ into the derived contact superconformal algebra ${K}'(4)$, the infinitesimal deformations are formal deformations.
We give a general construction of realizations of the contact superconformal algebras $K(2)$ and $\hat{K}'(4)$, and the exceptional superconformal algebra $CK_6$ as subsuperalgebras of matrices over a Weyl algebra of size $2^N\times 2^N$, where $N = 1, 2$ and $3$. We show that there is no such a realization for $K(2N)$, if $N\geq 4$.
We obtain an embedding of a one-parameter family of exceptional simple Lie superalgebras $D(2, 1 ; α)$ into the Lie superalgebra of pseudodifferential symbols on the supercircle $S^{1|2}$. Correspondingly, there is an embedding of $D(2, 1 ; α)$ into a nontrivial central extension of the derived contact superconformal algebra $K'(4)$ realized in terms of $4\times 4$ matrices over a Weyl algebra.
The superalgebra $\hat{K}'(4)$ and the exceptional N = 6 superconformal algebra have ``small'' irreducible representations in the superspaces $V^μ = t^μ\C[t, t^{-1}]\otimesΛ(N)$, where N = 2 and 3, respectively. For ${μ\in \C\backslash \Z}$ they are associated to the embeddings of these superalgebras into the Lie superalgebras of pseudodifferential symbols on the supercircle S^{1|N}. In this work we describe $\hat{K}'(4)$ and the exceptional N = 6 superconformal algebra in terms of matrices over a Weyl algebra. Correspondingly, we obtain realizations of their representations in $V^μ$ for $μ= 0$.
The Spencer cohomology of certain graded Lie superalgebras are completely computed. This cohomology is interpreted as analogs of Riemann and Penrose tensors on supermanifolds. The results make it manifest that there is no simple generalization of Borel-Weil-Bott's theorem for Lie superalgebras.
We explicitly describe the defining relations for simple Lie algebra of vector fields with polynomial coefficients and its subalgebras of divergence free, hamiltonian and contact vector fields, and for the Poisson algebra (realized on polynomials). We consider generators of these Lie algebras corresponding to the systems of simple roots associated with the standard grading of these algebras. (These systems of simple roots are distinguished in the sense of Penkov and Serganova.
We realize the exceptional superconformal algebra $CK_6$, spanned by 32 fields, inside the Lie superalgebra of pseudodifferential symbols on the supercircle $S^{1|3}$. We obtain a one-parameter family of irreducible representations of $CK_6$ in a superspace spanned by 8 fields.
We observe a correspondence between the zero modes of superconformal algebras $S'(2, 1)$ and W(4) and the Lie superalgebras formed by classical operators appearing in the K{ä}hler and hyper-K{ä}hler geometry.
The analogs of Chevalley generators are offered for simple (and close to them) Z-graded complex Lie algebras and Lie superalgebras of polynomial growth without Cartan matrix. We show how to derive the defining relations between these generators and explicitly write them for a "most natural" ("distinguished" in terms of Penkov and Serganova) system of simple roots. The results are given mainly for Lie superalgebras whose component of degree zero is a Lie algebra (other cases being left to the reader). Observe presentations of exceptional Lie superalgebras and Lie superalgebras of hamiltonian vector fields. Now we can, at last, q-quantize the Lie Lie superalgebras of hamiltonian vector fields and Poisson superalgebras.
This is a brief overview of my work on the realization of contact superconformal algebras in terms of pseudodifferential symbols on a supercircle. Also the 2-cocycles on K'(4) are given for this realization.
We describe representations of certain superconformal algebras in the semi-infinite Weil complex related to the loop algebra of a complex finite-dimensional Lie algebra and in the semi-infinite cohomology. We show that in the case where the Lie algebra is endowed with a non-degenerate invariant symmetric bilinear form, the relative semi-infinite cohomology of the loop algebra has a structure, which is analogous to the classical structure of the de Rham cohomology in Kähler geometry.
In this work we construct an embedding of a nontrivial central extension of the contact superconformal algebra K'(4) into the Lie superalgebra of pseudodifferential symbols on the supercircle S^{1|2}. Associated with this embedding is a one-parameter family of spinor-like tiny irreducible representations of K'(4) realized just on 4 fields instead of the usual 16.