Lie Superalgebra Research Articles

Snowflake modules and Enright functor for Kac–Moody superalgebras

We introduce a class of modules over Kac-Moody superalgebras; we call these modules snowflake. These modules are characterized by invariance property of their characters with respect to a certain subgroup of the Weyl group. Examples of snowflake modules appear as admissible modules in representation theory of affine vertex algebras and in classification of bounded weight modules. Using these modules we prove Arakawa's Theorem for the Lie superalgebra osp(1|2n).

Quantum Berezinian for a strange Lie superalgebra

We introduce a new family of central elements of the Yangian of the queer Lie superalgebra q1.

Perspectives on the pure spinor superfield formalism

In this note, we study, formalize, and generalize the pure spinor superfield formalism from a rather nontraditional perspective. To set the stage, we review the notion of a multiplet for a general super Lie algebra, working in the context of the BV and BRST formalisms. Building on this, we explain how the pure spinor superfield formalism can be viewed as constructing a supermultiplet out of the input datum of an equivariant graded module over the ring of functions on the nilpotence variety. We use the homotopy transfer theorem and other computational techniques from homological algebra to relate these multiplets to more standard component-field formulations. Physical properties of the resulting multiplets can then be understood in terms of algebrogeometric properties of the nilpotence variety. We illustrate our discussion with many examples in various dimensions.

Representations of twisted affine Lie superalgebras–A recognition theorem

Over the past three decades, representation theory of affine Lie (super)algebras has been a research spotlight in both areas Mathematics and Physics. The even parts of almost all affine Lie superalgebras L contain two affine Lie algebras; say L0(1) and L0(2). Irreducible finite weight modules (f.w.m's for short) are the first interesting class of modules over such an L. The structure of these modules strongly depends on whether of root vectors corresponding to real roots of L0(1) and L0(2) act locally nilpotently or not. We know that there is no nonzero irreducible f.w.m. of nonzero level over L for which root vectors corresponding to nonzero real roots of both L0(1) and L0(2) act locally nilpotently. This leads us to define quasi-integrable modules and study the case that root vectors corresponding to nonzero real roots of one of L0(1) and L0(2) act locally nilpotently. We give a recognition theorem for quasi-integrable modules over twisted affine Lie superalgebras.

Duflo–Serganova functor and superdimension formula for the periplectic Lie superalgebra

In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo-Serganova functor. Given a simple $\mathfrak{p}(n)$-module $L$ and a certain element $x\in \mathfrak{p}(n)$ of rank $1$, we give an explicit description of the composition factors of the $\mathfrak{p}(n-1)$-module $DS_x(L)$, which is defined as the homology of the complex $$\Pi M \xrightarrow{x} M \xrightarrow{x} \Pi M.$$ In particular, we show that this $\mathfrak{p}(n-1)$-module is multiplicity-free. We then use this result to give a simple explicit combinatorial formula for the superdimension of a simple integrable finite-dimensional $\mathfrak{p}(n)$-module, based on its highest weight. In particular, this reproves the Kac-Wakimoto conjecture for $\mathfrak{p}(n)$, which was proved earlier by the authors.

Some properties of isoclinism in n-Lie superalgebras

In this paper, we define isoclinism of [Formula: see text]-Lie superalgebras of parity [Formula: see text] and study some of its properties. Furthermore, we prove that notion of isoclinism and isomorphism of two finite dimensional [Formula: see text]-Lie superalgebras of same parity are equivalent.

First cohomology space of the orthosymplectic Lie superalgebra o s p ( n | 2 ) in the Lie superalgebra of superpseudodifferential operators

UDC 515.12 We investigate the first cohomology space associated with the embedding of the Lie orthosymplectic superalgebra o s p ( n | 2 ) on the ( 1 , n ) -dimensional superspace ℝ 1 | n in the Lie superalgebra 𝒮 Ψ D O ( n ) (for n ≥ 4 ) of superpseudodifferential operators with smooth coefficients. Following Ovsienko and Roger, we give explicit expressions of the basis cocycles.This work is the simplest generalization of a result by Basdouri [<em>First space cohomology of the orthosymplectic Lie superalgebra in the Lie superalgebra of superpseudodifferential operators</em>, Algebras and Representation Theory, <strong>16</strong>, 35-50 (2013)].

Quantum Toroidal Comodule Algebra of Type A<sub>n−1</sub> and Integrals of Motion

We introduce an algebra $\mathcal{K}_n$ which has a structure of a left comodule over the quantum toroidal algebra of type $A_{n-1}$. Algebra $\mathcal{K}_n$ is a higher rank generalization of $\mathcal{K}_1$, which provides a uniform description of deformed $W$ algebras associated with Lie (super)algebras of types BCD. We show that $\mathcal{K}_n$ possesses a family of commutative subalgebras.

The Duflo–Serganova Functor, Vingt Ans Après

We review old and new results concerning the \(DS\) functor and associated varieties for Lie superalgebras. These notions were introduced in the unpublished manuscript (Duflo and Serganova in On associated variety for Lie super-algebras, 2005) by Michel Duflo and the third author. This paper includes the results and proofs of the original manuscript, as well as a survey of more recent results.

Restricted Cohomology of Restricted Lie Superalgebras

Suppose the ground field \(\mathbb{F}\) is an algebraically closed field characteristic of p > 2. In this paper, we investigate the restricted cohomology theory of restricted Lie superalgebras. Algebraic interpretations of low dimensional restricted cohomology of restricted Lie superalgebra are given. We show that there is a family of restricted model filiform Lie superalgebra \(L_{p,p}^\lambda\) structures parameterized by elements \(\lambda \in {\mathbb{F}^p}\). We explicitly describe both the 1-dimensional ordinary and restricted cohomology superspaces of \(L_{p,p}^\lambda\) with coefficients in the 1-dimensional trivial module and show that these superspaces are equal. We also describe the 2-dimensional ordinary and restricted cohomology superspaces of \(L_{p,p}^\lambda\) with coefficients in the 1-dimensional trivial module and show that these superspaces are unequal.

On a family of vertex operator superalgebras

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight-32 homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra V is simple, then V(2) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and V(32) is naturally a V(2)-module equipped with a V(2)-valued symmetric bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that A is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that U is an A-module equipped with a symmetric A-valued bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra L(A,U) and a simple vertex operator superalgebra LL(A,U)(ℓ,0) for every nonzero number ℓ such that LL(A,U)(ℓ,0)(2)=A and LL(A,U)(ℓ,0)(32)=U.

On the semisimplicity of the category KLk for affine Lie superalgebras

We study the semisimplicity of the category KLk for affine Lie superalgebras and provide a super analog of certain results from [8]. Let KLkfin be the subcategory of KLk consisting of ordinary modules on which a Cartan subalgebra acts semisimply. We prove that KLkfin is semisimple when 1) k is a collapsing level, 2) Wk(g,θ) is rational, 3) Wk(g,θ) is semisimple in a certain category. The analysis of the semisimplicity of KLk is subtler than in the Lie algebra case, since in super case KLk can contain indecomposable modules. We are able to prove that in many cases when KLkfin is semisimple we indeed have KLkfin=KLk, which therefore excludes indecomposable and logarithmic modules in KLk. In these cases we are able to prove that there is a conformal embedding W↪Vk(g) with W semisimple (see Section 10). In particular, we prove the semisimplicity of KLk for g=sl(2|1) and k=−m+1m+2, m∈Z≥0. For g=sl(m|1), we prove that KLk is semisimple for k=−1, but for k a positive integer we show that it is not semisimple by constructing indecomposable highest weight modules in KLkfin.

Candidate for the crystal B(−∞) for the queer Lie superalgebra

It is shown that the direct limit of the semistandard decomposition tableau model for polynomial representations of the queer Lie superalgebra exists, which is believed to be the crystal for the upper half of the corresponding quantum group. An extension of this model to describe the direct limit combinatorially is given. Furthermore, it is shown that the polynomial representations may be recovered from the limit in most cases.

The second Betti number of doubly weighted homology groups of a certain pre Lie superalgebra

We already showed that Betti numbers are all zero when w is not equal to h for (w,h)-doubly weighted homology groups of some special pre Lie superalgebra and showed the first Betti number is 0 when w = h. In this note, we show that the second Betti number is 0 when w = h.

The generalized super Kaup–Newell equation and its super bi-Hamiltonian structure

In this work, from the generalized Kaup–Newell (KN) equation spectral problem, the generalized super KN equation and its bi-Hamiltonian structure are constructed based on a Lie super-algebras and the super-trace identity. Besides, a generalized super KN equation with self-consistent sources is also discussed.

On the Harish-Chandra Homomorphism for Quantum Superalgebras

In this paper, we introduce the Harish-Chandra homomorphism for the quantum superalgebra $$\mathrm {U}_q({\mathfrak {g}})$$ associated with a simple basic Lie superalgebra $${\mathfrak {g}}$$ and give an explicit description of its image. We use it to prove that the center of $$\mathrm {U}_q({\mathfrak {g}})$$ is isomorphic to a subring of the ring $$J({\mathfrak {g}})$$ of exponential super-invariants in the sense of Sergeev and Veselov, establishing a Harish-Chandra type theorem for $$\mathrm {U}_q({\mathfrak {g}})$$ . As a byproduct, we obtain a basis of the center of $$\mathrm {U}_q({\mathfrak {g}})$$ with the aid of quasi-R-matrix.

Nonlinear realizations of Lie superalgebras

For any decomposition of a Lie superalgebra into a direct sum of a subalgebra and a subspace without any further resctrictions on and we construct a nonlinear realization of on The result generalizes a theorem by Kantor from Lie algebras to Lie superalgebras. When is a differential graded Lie algebra, we show that it gives a construction of an associated -algebra.

The Heisenberg-Virasoro Lie conformal superalgebra

In this paper, we introduce a finite Lie conformal superalgebra called the Heisenberg-Virasoro Lie conformal superalgebra s by using a class of Heisenberg-Virasoro Lie conformal modules. The super Heisenberg-Virasoro algebra of Ramond type S is defined by the formal distribution Lie superalgebra of s. Then we construct a class of simple S-modules, which are induced from simple modules of some finite dimensional solvable Lie superalgebras. These modules are isomorphic to simple restricted S-modules, and include the highest weight modules, Whittaker modules and high order Whittaker modules. As a byproduct, we present a subalgebra of S, which is isomorphic to the super Heisenberg-Virasoro algebra of Neveu-Schwarz type.

BPS Algebras in 2D String Theory

We discuss a set of heterotic and type II string theory compactifications to 1+1 dimensions that are characterized by factorized internal worldsheet CFTs of the form V_1otimes bar{V}_2, where V_1, V_2 are self-dual (super) vertex operator algebras. In the cases with spacetime supersymmetry, we show that the BPS states form a module for a Borcherds–Kac–Moody (BKM) (super)algebra, and we prove that for each model the BKM (super)algebra is a symmetry of genus zero BPS string amplitudes. We compute the supersymmetric indices of these models using both Hamiltonian and path integral formalisms. The path integrals are manifestly automorphic forms closely related to the Borcherds–Weyl–Kac denominator. Along the way, we comment on various subtleties inherent to these low-dimensional string compactifications.

Tensor product representations

In this paper, we study representations of a direct sum of Lie superalgebras. We deal with the question of whether the irreducible representations of a finite direct sum of Lie superalgebras are obtained in case representations of each constituent are available. We give an affirmative answer to this question for finite weight representations of a class of Lie superalgebras containing basic classical simple Lie superalgebras as well as affine Lie (super)algebras.