Lie Superalgebra Research Articles

A Superunitary Fock Model of the Exceptional Lie Supergroup mathbb {D}(2, 1; alpha )

We construct a Fock model of the minimal representation of the exceptional Lie supergroup {mathbb {D}}(2,1; alpha ). Explicit expressions for the action are given by integrating to group level a Fock model of the Lie superalgebra D(2,1; alpha ) constructed earlier by the authors. It is also shown that the representation is superunitary in the sense of de Goursac–Michel.

Resolution of Singularities of the Odd Nilpotent Cone of Orthosymplectic Lie Superalgebras

Resolution of Singularities of the Odd Nilpotent Cone of Orthosymplectic Lie Superalgebras

The Multiplier and Cohomology of Lie Superalgebras

Suppose that [Formula: see text] is a Lie superalgebra over a field [Formula: see text] of characteristic different from 2 and 3. In this paper, the so-called 5-sequence of cohomology for a central extension of [Formula: see text] is constructed and is proved to be exact. Moreover, the multiplier of [Formula: see text] is proved to be isomorphic to the second cohomology group with coefficients in the trivial module of [Formula: see text]. Finally, an upper bound of the superdimension of the second cohomology group is given in the situation when [Formula: see text] is nilpotent and finite-dimensional.

Left-Symmetric Superalgebra Structures on an Infinite-Dimensional Lie Superalgebra

In this note, the compatible left-symmetric superalgebra structures on an infinite-dimensional Lie superalgebra with some natural grading conditions are completely determined. The results of earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures.

Left-symmetric superalgebras on special linear Lie superalgebras

In this paper, we study the existence and classification problems of left-symmetric superalgebras on special linear Lie superalgebras sl(m|n) with m≠n. The main three results of this paper are: (i) a complete classification of the left-symmetric superalgebras on sl(2|1), (ii) sl(m|1) does not admit left-symmetric superalgebras for m≥3, and (iii) sl(m+1|m) admits a left-symmetric superalgebra for every m≥1. To prove these results we combine previous results on the existence and classification of left-symmetric algebras on the Lie algebras glm with a detailed analysis of small representations of the Lie superalgebras sl(m|1). We also conjecture that sl(m|n) admits left-symmetric superalgebras if and only if m=n+1.

A Drinfeld-Type Presentation of the Orthosymplectic Yangians

We use the Gauss decomposition of the generator matrix in the R-matrix presentation of the Yangian for the orthosymplectic Lie superalgebra ospN|2m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {osp}_{N|2m}$$\\end{document} to produce its Drinfeld-type presentation. The results rely on a super-version of the embedding theorem which allows one to identify a subalgebra in the R-matrix presentation which is isomorphic to the Yangian associated with ospN|2m-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {osp}_{N|2m-2}$$\\end{document}.

Hom-Gel'fand-Dorfman conformal superbialgebras

Gel'fand Dorfman superbialgebra, which is both a Lie superalgebra and a (left) Novikov superalgebra with some compatibility condition, appears in the study of Hamiltonian pairs in completely integrable systems and a class of special Lie conformal superalgebras called quadratic Lie conformal superalgebras. In the present paper, we generalize this algebraic structure to the Hom-conformal case. We introduce first, Hom-Novikov conformal superalgebras and exihibit several properties. Then we introduce Hom-Gel'fand Dorfman superbialgebra and provide some construction results.

Geometric properties of special orthogonal representations associated to exceptional Lie superalgebras

From an octonion algebra [Formula: see text] over a field [Formula: see text] of characteristic not two or three, we show that the fundamental representation [Formula: see text] of the derivation algebra [Formula: see text] and the spinor representation [Formula: see text] of [Formula: see text] are special orthogonal representations. They have particular geometric properties coming from their similarities with binary cubics and we show that the covariants of these representations and their Mathews identities are related to the Fano plane and the affine space [Formula: see text]. This also permits to give constructions of exceptional Lie superalgebras.

Simple binary Lie and non-Lie superalgebra has solvable even part

We prove that every simple finite dimensional binary Lie superalgebra over the complex numbers field C with non-zero odd part is either a Lie superalgebra or has a solvable even part.

On linked modules over the super-Yangian of the superalgebra Q(1)

Let Q(n) be the queer Lie superalgebra. We determine conditions under which two one-dimensional modules over the super-Yangian of Q(1) can be extended nontrivially, and thus belong to the same block of the subcategory of finite-dimensional YQ(1)-modules admitting generalized central character χ = 0. We use these results to determine conditions under which two one-dimensional modules over the finite W-algebra for Q(n) can be extended nontrivially. We describe blocks in the category of finite-dimensional modules over the finite W-algebra for Q(2). In certain cases, we determine conditions under which two simple finite-dimensional YQ(1)-modules admitting central character χ ≠ 0 can be extended nontrivially and propose a conjecture in the general case.

The Hurwitz-Hopf map and harmonic wave functions for integer and half-integer angular momentum

Harmonic wave functions for integer and half-integer angular momentum are given in terms of the Euler angles (θ, ϕ, ψ) that define a rotation in SO(3), and the Euclidean norm r in , keeping the usual meaning of the spherical coordinates (r, θ, ϕ). They form a Hilbert (super)-space decomposed in the form . Following a classical work by Schwinger, 2-dimensional harmonic oscillators are used to produce raising and lowering operators that change the total angular momentum eigenvalue of the wave functions in half units. The nature of the representation space is approached from the double covering group homomorphism SU(2) → SO(3) and the topology involved is taken care of by using the Hurwitz-Hopf map . It is shown how to reconsider H as a 2-to-1 group map, , translating it into an assignment (z 1, z 2) ↦ (r, θ, ϕ, ψ) whose domain consists of pairs (z 1, z 2) of complex variables, under the appropriate identification of with . It is shown how the Lie algebra of G 0 is coupled with two Heisenberg Lie algebras of 2-dimensional (Schwinger’s) harmonic oscillators generated by the operators and their adjoints. The whole set of operators gets algebraically closed either into a 13-dimensional Lie algebra or into a (4∣8)-dimensional Lie superalgebra. The wave functions in can be written in terms of polynomials in the complex coordinates (z 1, z 2) and their complex conjugates and the representations are explicitly constructed via the various highest weight (or lowest weight) vector representations of G 0. Finally, a new nonrelativistic quantum (Schrödinger-like) equation for the hydrogen atom that takes into account the electron spin is introduced and expressed in terms of (r, θ, ϕ, ψ) and the time t. The equation is succeptible to be solved exactly in terms of the harmonic wave functions hereby introduced.

Formula omitted]-graded modules for [formula omitted]-graded modular vertex superalgebras

In this paper, we study Z-graded vertex superalgebras over a general field of an odd prime characteristic. In particular, we study affine vertex superalgebras Vgˆ(ℓ,0) and a family of quotient vertex superalgebras Vgˆχ(ℓ,0) with g assumed to be a restricted Lie superalgebra. Among the main results, we obtain a necessary and sufficient condition for a gˆ-module of level ℓ to be a Vgˆχ(ℓ,0)-module. Furthermore, we study the special case with g=g1¯ and introduce a quotient vertex superalgebra Lgˆ(ℓ,0) of Vgˆ(ℓ,0). As the main results, we prove that for ℓ≠0, vertex superalgebra Lgˆ(ℓ,0) is simple, the adjoint module is the only irreducible N-graded module up to equivalence, and every Lgˆ(ℓ,0)-module is isomorphic to the direct sum of some copies of Lgˆ(ℓ,0).

Deformations of 𝔞𝔣𝔣(n|1)-module of weighted densities on ℝ1|n, n ≥ 2

Over the [Formula: see text]-dimensional real superspace, [Formula: see text], we study non-trivial deformations of the natural action of the affine Lie superalgebra [Formula: see text] on the direct sum of the superspaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this action and we prove that any formal deformation is equivalent to its infinitesimal part. This work is the simplest generalization of a result by Basdouri and Omri [Cohomology and deformation of [Formula: see text] acting on differential operators, Int. J. Geom. Methods Mod. Phys. 15 (2018) 1850072].

Classical Lie bialgebras for AdS/CFT integrability by contraction and reduction

Integrability of the one-dimensional Hubbard model and of the factorised scattering problem encountered on the worldsheet of AdS strings can be expressed in terms of a peculiar quantum algebra. In this article, we derive the classical limit of these algebraic integrable structures based on established results for the exceptional simple Lie superalgebra \mathfrak{d}(2,1;\epsilon)𝔡(2,1;ϵ) along with standard \mathfrak{sl}(2)𝔰𝔩(2) which form supersymmetric isometries on 3D AdS space. The two major steps in this construction consist in the contraction to a 3D Poincaré superalgebra and a certain reduction to a deformation of the \mathfrak{u}(2|2)𝔲(2|2) superalgebra. We apply these steps to the integrable structure and obtain the desired Lie bialgebras with suitable classical r-matrices of rational and trigonometric kind. We illustrate our findings in terms of representations for on-shell fields on AdS and flat space.

Vanishing Property of BRST Cohomology for Modified Highest Weight Modules

We construct certain modified highest weight modules which are called quasi highest weight modules in this paper. Using the quasi highest weight modules, we introduce a new category of modules over an affine Lie superalgebra which contains projective covers. We also prove that both these projective covers and the quasi highest weight modules satisfy the vanishing property of BRST cohomology.

Non-weight modules over N = 1 Lie superalgebras of Block type

Abstract In this paper, a class of non-weight modules over N = 1 {N=1} Lie superalgebras of Block type (without center) is studied. These modules over Ramond algebras of Block type are free of rank 1 when restricted to the universal enveloping algebra of the Cartan subalgebra, while such modules are free of rank 2 over the Neveu–Schwarz algebras of Block type. We present the classification of these modules. Furthermore, the simplicity of these modules is determined.

Superbiderivations of Simple Modular Lie Superalgebras of Witt Type and Special Type

Let [Formula: see text] be a simple modular Lie superealgebra of Witt type or special type over an algebraically closed field of characteristic [Formula: see text]. In this paper, all the superbiderivations of [Formula: see text] are studied. By means of weight space decompositions with respect to a suitable torus and the standard [Formula: see text]-grading structures of [Formula: see text], we show that the supersymmetric superbiderivations of [Formula: see text] are zero. Generalizing a result on the skewsymmetric biderivation of Lie algebras to the super case, we find that all the superbiderivations of [Formula: see text] are inner. As applications, the linear supercommuting maps and the supercommutative post-Lie superalgebra structures on [Formula: see text] are described.

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Over algebraically closed fields of positive characteristic, for simple Lie (super)algebras, and certain Lie (super)algebras close to simple ones, with symmetric root systems (such that for each root, there is minus it of the same multiplicity) and of ranks less than or equal to 8—most needed in an approach to the classification of simple vectorial Lie superalgebras (i.e., Lie superalgebras realized by means of vector fields on a supermanifold),—we list the outer derivations and nontrivial central extensions. When the conjectural answer is clear for the infinite series, it is given for any rank. We also list the outer derivations and nontrivial central extensions of one series of non-symmetric (except when considered in characteristic 2), namely periplectic, Lie superalgebras—the one that preserves the nondegenerate symmetric odd bilinear form, and of the Lie algebras obtained from them by desuperization. We also list the outer derivations and nontrivial central extensions of an analog of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several results indigenous to positive characteristic are of particular interest being unlike known theorems for characteristic 0, some results are, moreover, counterintuitive.

Deformations of Symmetric Simple Modular Lie (Super)Algebras

We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank <9, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycleis integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.

A sum-bracket theorem for simple Lie algebras

Let g be an algebra over K with a bilinear operation [⋅,⋅]:g×g→g not necessarily associative. For A⊆g, let Ak be the set of elements of g written combining k elements of A via + and [⋅,⋅].We show a “sum-bracket theorem” for simple Lie algebras over K of the form g=sln,son,sp2n,e6,e7,e8,f4,g2: if char(K) is not too small, we have growth of the form |Ak|≥|A|1+ε for all generating symmetric sets A away from subfields of K. Over Fp in particular, we have a diameter bound matching the best analogous bounds for groups of Lie type [2].As an independent intermediate result, we prove also an estimate of the form |A∩V|≤|Ak|dim⁡(V)/dim⁡(g) for linear affine subspaces V of g. This estimate is valid for all simple algebras, and k is especially small for a large class of them including associative, Lie, and Mal'cev algebras, and Lie superalgebras.