Structure Of Lie Superalgebras Research Articles

Chirality, a new key for the definition of the connection and curvature of a Lie-Kac superalgebra

A natural generalization of a Lie algebra connection, or Yang-Mills field, to the case of a Lie-Kac superalgebra, for example SU(m/n), just in terms of ordinary complex functions and differentials, is proposed. Using the chirality χ which defines the supertrace of the superalgebra: STr(…) = Tr(χ…), we construct a covariant differential: D = χ(d + A) + Φ, where A is the standard even Lie-subalgebra connection 1-form and Φ a scalar field valued in the odd module. Despite the fact that Φ is a scalar, Φ anticommutes with (χA) because χ anticommutes with the odd generators hidden in Φ. Hence the curvature F = DD is a superalgebra-valued linear map which respects the Bianchi identity and correctly defines a chiral parallel transport compatible with a generic Lie superalgebra structure.

BiHom–Lie Superalgebra Structures and BiHom–Yang–Baxter Equations

In this paper, we first introduce the notion of BiHom–Lie superalgebras, which is a generalization of both BiHom–Lie algebras and Hom–Lie superalgebras. Also, we explore some general classes of BiHom–Lie admissible superalgebras and describe all these classes via G-BiHom-associative superalgebras, where G is a subgroup of the symmetric group $$S_{3}$$ . Finally, we obtain a method to construct the solutions of the BiHom–Yang–Baxter equation from BiHom–Lie superalgebras.

On the structure of split involutive Lie superalgebras

On the structure of split involutive Lie superalgebras

Hidden symmetries and Lie algebra structures from geometric and supergravity Killing spinors

We consider geometric and supergravity Killing spinors and the spinor bilinears constructed out of them. The spinor bilinears of geometric Killing spinors correspond to the antisymmetric generalizations of Killing vector fields which are called Killing–Yano forms. They constitute a Lie superalgebra structure in constant curvature spacetimes. We show that the Dirac currents of geometric Killing spinors satisfy a Lie algebra structure up to a condition on 2-form spinor bilinears. We propose that the spinor bilinears of supergravity Killing spinors give way to different generalizations of Killing vector fields to higher degree forms. It is also shown that those supergravity Killing forms constitute a Lie algebra structure in six- and ten-dimensional cases. For five- and eleven-dimensional cases, the Lie algebra structure depends on an extra condition on supergravity Killing forms.

Submaximal conformal symmetry superalgebras for Lorentzian manifolds of low dimension

We consider a class of smooth oriented Lorentzian manifolds in dimensions three and four which admit a nowhere vanishing conformal Killing vector and a closed two-form that is invariant under the Lie algebra of conformal Killing vectors. The invariant two-form is constrained in a particular way by the conformal geometry of the manifold. In three dimensions, the conformal Killing vector must be everywhere causal (or null if the invariant two-form vanishes identically). In four dimensions, the conformal Killing vector must be everywhere null and the invariant two-form vanishes identically if the geometry is everywhere of Petrov type N or O. To the conformal class of any such geometry, it is possible to assign a particular Lie superalgebra structure, called a conformal symmetry superalgebra. The even part of this superalgebra contains conformal Killing vectors and constant R-symmetries while the odd part contains (charged) twistor spinors. The largest possible dimension of a conformal symmetry superalgebra is realised only for geometries that are locally conformally flat. We determine precisely which non-trivial conformal classes of metrics admit a conformal symmetry superalgebra with the next largest possible dimension, and compute all the associated submaximal conformal symmetry superalgebras. In four dimensions, we also compute symmetry superalgebras for a class of Ricci-flat Lorentzian geometries not of Petrov type N or O which admit a null Killing vector.

Algebraic Approach to Slice Monogenic Functions

In recent years, the study of slice monogenic functions has attracted more and more attention in the literature. In this paper, an extension of the well-known Dirac operator is defined which allows to establish the Lie superalgebra structure behind the theory of slice monogenic functions. Subsequently, an inner product is defined corresponding to this slice Dirac operator and its polynomial null-solutions are determined. Finally, analogues of the Hermite polynomials and Hermite functions are constructed in this context and their properties are studied.

Singular quadratic Lie superalgebras

In this paper, we generalize some results on quadratic Lie algebras to quadratic Lie superalgebras, by applying graded Lie algebras tools. We establish a one-to-one correspondence between non-Abelian quadratic Lie superalgebra structures and nonzero even super-antisymmetric 3-forms satisfying a structure equation. An invariant number of quadratic Lie superalgebras is then defined, called the dup-number. Singular quadratic Lie superalgebras (i.e. those with nonzero dup-number) are studied. We show that their classification follows the classifications of O(m)-adjoint orbits of o(m) and Sp(2n)-adjoint orbits of sp(2n). An explicit formula for the quadratic dimension of singular quadratic Lie superalgebras is also provided. Finally, we discuss a class of 2-nilpotent quadratic Lie superalgebras associated to a particular super-antisymmetric 3-form.

Omni-Lie superalgebras and Lie 2-superalgebras

We introduce the notion of omni-Lie superalgebras as a super version of an omni-Lie algebra introduced by Weinstein. This algebraic structure gives a nontrivial example of Leibniz superalgebras and Lie 2-superalgebras. We prove that there is a one-to-one correspondence between Dirac structures of the omni-Lie superalgebra and Lie superalgebra structures on a subspace of a super vector space.

Hom–Lie superalgebra structures on infinite-dimensional simple Lie superalgebras of vector fields

This paper considers the multiplicative Hom-Lie superalgebra structures on infinite dimensional simple Lie superalgebras of vector fields with characteristic zero. The main result is that there is only the multiplicative Hom-Lie superalgebra structure on these Lie superalgebras.

BKM superalgebras from counting dyons in [formula omitted] supersymmetric type II compactifications

We study the degeneracy of quarter BPS dyons in N=4 type II compactifications of string theory. We find that the genus-two Siegel modular forms generating the degeneracies of the quarter BPS dyons in the type II theories can be expressed in terms of the genus-two Siegel modular forms generating the degeneracies of quarter BPS dyons in the CHL theories and the heterotic string. This helps us in understanding the algebra structure underlying the degeneracy of the quarter BPS states. The Conway group, Co0, plays a role similar to Mathieu group, M24, in the CHL models with eta quotients appearing in the place of eta products. We construct BKM Lie superalgebra structures for the ZN (for N=2,3,4) orbifolds of the type II string compactified on a six-torus.

On split Lie superalgebras

We study the structure of arbitrary split Lie superalgebras. We show that any of such superalgebras L is of the form L=U+∑jIj with U a subspace of the Abelian (graded) subalgebra H and any Ij, a well described (graded) ideal of L satisfying [Ij,Ik]=0 if j≠k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal (graded) ideals, each one being a simple split Lie superalgebra.

On the maximal superalgebras of supersymmetric backgrounds

In this paper we give a precise definition of the notion of a maximal superalgebra of certain types of supersymmetric supergravity backgrounds, including the Freund–Rubin backgrounds, and propose a geometric construction extending the well-known construction of its Killing superalgebra. We determine the structure of maximal Lie superalgebras and show that there is a finite number of isomorphism classes, all related via contractions from an orthosymplectic Lie superalgebra. We use the structure theory to show that maximally supersymmetric waves do not possess such a maximal superalgebra, but that the maximally supersymmetric Freund–Rubin backgrounds do. We perform the explicit geometric construction of the maximal superalgebra of and find that it is isomorphic to . We propose an algebraic construction of the maximal superalgebra of any background asymptotic to and we test this proposal by computing the maximal superalgebra of the M2-brane in its two maximally supersymmetric limits, finding agreement.

Triple products on 𝔤𝔩 n

Triple products in 𝔤𝔩 n whose related algebra is 𝔤𝔩 n itself or 𝔰𝔩 n are classified up to isomorphism. This classification is obtained using the intimate relation between triple products and Lie (super)algebra structures.

Supertrace and Superquadratic Lie Structure on the Weyl Algebra, and Applications to Formal Inverse Weyl Transform

Using the Moyal *-product and orthosymplectic supersymmetry, we construct a natural non trivial supertrace and an associated non degenerate invariant supersymmetric bilinear form for the Lie superalgebra structure of the Weyl algebra. We decompose adjoint and twisted adjoint actions. We define a renormalized supertrace and a formal inverse Weyl transform in a deformation quantization framework and develop some examples.

Structure of Lie Superalgebras of Block Type Related to Locally Finite Derivations

Xu introduced a class of infinite dimensional simple Lie superalgebras which are generalizations of the Block algebras. These Lie superalgebras are constructed based on some locally finite derivations and are in general nongraded. In this paper, the derivation superalgebras, the 2-cocycles and the isomorphism classes of these Lie superalgebras are determined.

Classical r-matrices for the osp(2/2) Lie superalgebra

The co-Lie structures compatible with the osp(2/2) Lie superalgebra structure are investigated and found to be all of the coboundary type. The corresponding classical r-matrices are classified into several disjoint families. The osp(1/2)⊕u(1) Lie superbialgebras are also classified.

Structure of some ℤ-graded lie superalgebras of vector fields

In this paper we classifyℤ-graded transitive Lie superalgebras with prescribed nonpositive parts listed in [K2]. The classification of infinite-dimensional simple linearly compact Lie superalgebras given in [K2] is based on this result. We also study the structure of the exceptionalℤ-graded transitive Lie superalgebras and give their geometric realization.

Super Hamiltonian operators and Lie superalgebras

A superversion of the formal calculus of variations as developed by I.M. Gel'fand et al. is presented. This superversion is essentially contained in the one given by B.A. Kupershmidt. However, the form presented here is more manageable and suitable for applications. Even as well as odd super Hamiltonian operators are considered. It is proved that with every linear super Hamiltonian operator one can associate a Lie superalgebra structure on the space of (reduced) 1-forms and vice versa. Also a theorem is proved about the connection between 2-cocycles on this Lie superalgebra and super Hamiltonian operators. Finally an application to the sKdV equation is given.

Structure of basic Lie superalgebras and of their affine extensions

We generalize to the case of superalgebras several properties of simple Lie algebras involving the use of Dynkin diagrams. If to a simple Lie algebra can be associated one Dynkin diagram, it is a finite set of nonequivalent ones which can be constructed for a basic superalgebra (or B.S.A.). The knowledge of these diagrams, which can be obtained for each B.S.A. in a systematic way, allows us to deduce the regular subsuperalgebras of a B.S.A. The symmetries of the Dynkin diagrams are related to outer automorphisms of B.S.A. and lead to some singular subsuperalgebras. Finally we consider the extended Dynkin diagrams in order to classify the affine B.S.A. and use their symmetries to construct the twisted basic superalgebras.

Algebra of operators for a system of composite and elementary particles

In this paper, a system made up of two different fermion species with two algebraical structures, a bigraded-Lie structure and a Lie superalgebra structure, are shown. The Lie superalgebra structure gives the normal commutation relations, while the bigraded Lie structure gives the abnormal commutation relations. The Lie superalgebra structure seems to be the most convenient for studying systems of composite and elementary particles.