In the first part of my talk I will explain a solution to the extension of Lie's problem on classification of "local continuous transformation groups of a finite-dimensional manifold" to the case of supermanifolds. (More precisely, the problem is to classify simple linearly compact Lie superalgebras, i.e. toplogical Lie superalgebras whose underlying space is a topological product of finite-dimensional vector spaces). In the second part I will explain how this result is used in a classification of superconformal algebras. The list consists of affine superalgebras and certain super extensions of the Virasoro algebra. In the third part I will discuss representation theory of affine superalgebras and its relation to "almost" modular forms. Furthermore, I will explain how the quantum reduction of these representations leads to a unified representation theory of super extensions of the Virasoro algebra. In the forth part I will discuss representation theory of exceptional simple infinite-dimensional linearly compact Lie superalgebras and will speculate on its relation to the Standard Model.
We consider Lie superalgebras under constraints of Hamiltonian reduction, yielding finite $W$-superalgebras which provide candidates for quadratic spacetime superalgebras. These have an undeformed bosonic symmetry algebra (even generators) graded by a fermionic sector (supersymmetry generators) with anticommutator brackets which are quadratic in the even generators. We analyze the reduction of several Lie superalgebras of type $gl(M|N)$ or $osp(M|2N)$ at the classical (Poisson bracket) level, and also establish their quantum (Lie bracket) equivalents. Purely bosonic extensions are also conside…
One-dimensional nonrelativistic systems are studied when time-independent potential interactions are involved. Their supersymmetries are determined and their closed subsets generating kinematical invariance Lie superalgebras are pointed out. The study of even supersymmetries is particularly enlightened through the already known symmetries of the corresponding Schrödinger equation. Three tables collect the even, odd, and total supersymmetries as well as the invariance (super)algebras.
Infinitesimal supersymmetries over classical Lie groups that do not necessarily integrate to Lie supergroups are described. They yield a notion of supersymmetry that is less rigid than the assumption of a Lie supergroup action but still implies an underlying action of a Lie group. In contrast to Lie supergroups, the arising representation-theoretical Lie supergroups (RTLSG) occur as families associated to Harish-Chandra superpairs. However morphisms of RTLSGs directly correspond to morphisms of Harish-Chandra superpairs. Particular RTLSGs can be derived from the explicit constructions of Lie s…
We classify all continuous degenerate irreducible modules over the exceptional linearly compact Lie superalgebra E(1, 6), and all finite degenerate irreducible modules over the exceptional Lie conformal superalgebra CK6, for which E(1, 6) is the annihilation algebra.
The kinematical and dynamical symmetries of equations describing the time evolution of quantum systems like the supersymmetric harmonic oscillator in one space dimension and the interaction of a non-relativistic spin one-half particle in a constant magnetic field are reviewed from the point of view of the vector field prolongation method. Generators of supersymmetries are then introduced so that we get Lie superalgebras of symmetries and supersymmetries. This approach does not require the introduction of Grassmann valued differential equations but a specific matrix realization and the concept …
Irreducible nonzero level modules with finite-dimensional weight spaces are studied for non-twisted affine Lie superalgebras. A complete classification is obtained for superalgebras A(m,n)^ and C(n)^. In other cases the classification problem is reduced to the classification of cuspidal modules over finite-dimensional cuspidal Lie superalgebras described by Dimitrov, Mathieu and Penkov. It is also shown that any irreducible weakly integrable (in the sense of Kac and Wakimoto) module is highest weight.
We introduce a class of novel $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded color superalgebras of infinite dimension. It is done by realizing each member of the class in the universal enveloping algebra of a Lie superalgebra which is a module extension of the Virasoro algebra. Then the complete classification of central extensions of the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded color superalgebras is presented. It turns out that infinitely many members of the class have non-trivial extensions. We also demonstrate that the color superalgebras (with and without central extensions) have adjoint and…
The notion of a Lie conformal superalgebra encodes an axiomatic descrption of singular parts of the operator product expansions of chiral fields in conformal field theory. In the paper we give a detailed proof of the classification of all finite simple Lie conformal superalgebras. We also classify all their central extensions.
A Vogan diagram is a Dynkin diagram with a Cartan involution of twisted affine superlagebras based on maximally compact Cartan subalgebras. This article construct the Vogan diagrams of twisted affine superalgebras. It is a part of completion of classification of vogan diagrams to superalgebras cases.
We continue the study of irreducible representations of the exceptional Lie superalgebra E(3,6). This is one of the two simple infinite-dimensional Lie superalgebras of vector fields which have a Lie algebra sl(3)\times sl(2)\times gl(1) as the zero degree component of its consistent ZZ-grading. We provide the classification of the singular vectors in the degenerate Verma modules over E(3,6), completing thereby the classification and construction of all irreducible E(3,6)- modules that are L_0-locally finite.
The purpose of this paper is to investigate finite-dimensional superbialgebras and Hopf superalgebras. We study connected superbialgebras and provide a classification of non-trivial superbialgebras and Hopf superalgebras in dimension $n$ with $n\leq 4$.
In this work we investigate the complex Leibniz superalgebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and with nilindex equal to $n+m.$ We prove that such superalgebras with the condition $m_2\neq0$ have nilindex less than $n+m$. Therefore the complete classification of Leibniz algebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and with nilindex equal to $n+m$ is reduced to the classification of filiform Leibniz superalgebras of nilindex equal to $n+m,$ which was provided in \cite{AOKh} and \cite{GKh}.
Recently one of the authors obtained a classification of simple infinite-dimensional Lie superalgebras of vector fields which extends the well-known classification of E. Cartan in the Lie algebra case. The list consists of many series defined by simple equations, and of several exceptional superalgebras, among them E(3,6). In the article we study irreducible representations of the exceptional Lie superalgebra E(3,6). This superalgebra has s\ell(3)\times s\ell(2)\times g\ell(1) as the zero degree component of its consistent \ZZ-grading which leads us to believe that its representation theory ha…
We show that we can combine the (complex, self-dual) BMS vector fields with the recently defined BMS twistors to obtain a new supersymmetric extension of the BMS symmetries at null infinity. We compare our construction to other supersymmetric extensions of the BMS algebra proposed in the context of supergravity. Unlike the standard constructions the anticommutator in our superalgebra generates all the BMS vector fields including the Lorentz transformations. We also show that there exists a projection from our BMS Lie superalgebra to the global subalgebra of the Neveu-Schwarz supersymmetries on…
In this paper, we give a generalization of results in \cite{PU07} and \cite{DPU10} by applying the tools of graded Lie algebras to quadratic Lie superalgebras. In this way, we obtain a numerical invariant of quadratic Lie superalgebras and a classification of singular quadratic Lie superalgebras, i.e. those with a nonzero invariant. Finally, we study a class of quadratic Lie superalgebras obtained by the method of generalized double extensions.
We present a classification of the hyperbolic Kac-Moody (HKM) superalgebras. The HKM superalgebras of rank larger or equal than 3 are finite in number (213) and limited in rank (6). The Dynkin-Kac diagrams and the corresponding simple root systems are determined. We also discuss a class of singular sub(super)algebras obtained by a folding procedure.
Generalized quantum statistics such as para-statistics is usually characterized by certain triple relations. In the case of para-Fermi statistics these relations can be associated with the orthogonal Lie algebra B_n=so(2n+1); in the case of para-Bose statistics they are associated with the Lie superalgebra B(0|n)=osp(1|2n). In a previous paper, a mathematical definition of ``a generalized quantum statistics associated with a classical Lie algebra G'' was given, and a complete classification was obtained. Here, we consider the definition of ``a generalized quantum statistics associated with a b…
Using the classification theorem due to Kac we prove that any finite dimensional simple Lie superalgebra over an algebraically closed field of characteristic 0 is generated by one element.
A Vogan superdiagram is a set of involution and painting on a Dynkin diagram. It selects a real form, or equivalently an involution, from a complex simple Lie superalgebra. We introduce the double Vogan superdiagram, which is two sets of Vogan superdiagrams superimposed on an affine Dynkin diagram. They correspond to pairs of commuting involutions on complex simple Lie superalgebras, and therefore provide an independent classification of the simple locally symmetric or symmetric superpairs.
Generalized quantum statistics (GQS) associated to a Lie algebra or Lie superalgebra extends the notion of para-Bose or para-Fermi statistics. Such GQS have been classified for all classical simple Lie algebras and basic classical Lie superalgebras. In the current paper we finalize this classification for all exceptional Lie algebras and superalgebras. Since the definition of GQS is closely related to a certain Z-grading of the Lie (super)algebra G, our classification reproduces some known Z-gradings of exceptional Lie algebras. For exceptional Lie superalgebras such a classification of Z-grad…
Using adjoint representation of Lie superalgebras, we obtain the matrix form of super-Jacobi and mixed super-Jacobi identities of Lie superbialgebras. By direct calculations of these identities, and use of automorphism supergroups of two and three dimensional Lie superalgebras, we obtain and classify all two and three dimensional Lie superbialgebras.
