Simplicity of Jordan Superalgebras and Relations with Lie Structures (original) (raw)
2016
It is proved that the prime degenerate (-1,1) algebra constructed in [13] (the (-1,1)-monster) generates the same variety of algebras as the Grassman (-1,1)-algebra. Moreover, the same variety is generated by the Grassmann envelope of any simple nonassociative (-1,1)-superalgebra. The variety occurs to be the smallest variety of (-1,1)-algebras that contains prime nonassociative algebras. Similar results are obtained for Jordan algebras. Thus, the Jordan monster (the prime degenerate algebra constructed in [13]) and the Grassmann envelope of the prime Jordan superalgebra of vector type have the same ideals of identities. It is also shown that the Jordan monster generates a minimal variety that contains prime degenerate Jordan algebras. All the algebras and superalgebras are considered over a field of characteristic 0.
Nuclear Physics B, 1978
Lie structures which include Lie algebras and superalgebras (Z, graded Lie algebras) as special cases are defined. Several examples based on the ZN @ ZN grading are presented and two theorems on Zz @ Z2 e . . . e Z2 graded algebras are given. Some possible physical applications are discussed.
Speciality and non-speciality of two Jordan superalgebras
Journal of Algebra, 1992
In a recent paper, Medvedev and Zelmanov used two inlinite-dimensional Jordan superalgebras to cast light on ordinary Jordan algebras. In this note we investigate the speciality of the two examples. We show that (1) the superalgebra of vector fields on the line is special, with an imbedding into 2 x 2 matrices with entries in the algebra of differential operators on polynomials (D(g)= dg/dx), but that (2) quite unexpectedly, the superalgebra of Poisson brackets is NOT special. We carry out our analysis in a general framework of Kantor superalgebras J= F + F. c [f g =fg, f.(g.c)=fg.c, (f.c).(g.c)=fxg] built by doubling a unital commutative associative algebra F and a bracket product f x g, related to Kantor's Poisson algebras. These are special for algebras of vector type where f x g = D(f') g-fD(G) comes from a derivation D of F, but not for algebras of Poisson type where F x 1 = 0 and 1 E .$(Fx F). Throughout this paper all algebras and superalgebras will be over a fixed ring of scalars @ CONTAINING l/2. A graded algebra J= J, 0 J, is a Jordan superalgebra if its Grassman envelope T(J) = J, @ f 0 + J, @ Z-, is an ordinary (linear) Jordan algebra; in terms of associators [Ix, y, z] = (xy)z-x(yz) the condition that T(J) be a linear Jordan algebra,
General superalgebras of vector type and (γ, δ)-superalgebras
2001
A general superalgebra of vector type is a superalgebra obtained by a certain double process from an associative and commutative algebra A with fixed derivation D and elements λ, µ, ν. We prove that any such a superalgebra is a superalgebra of (γ, δ) type. Conversely, any simple finite dimensional nonassociative (γ, δ) superalgebra with (γ, δ) = (1,1) or (-1,0) is isomorphic to a certain general superalgebra of vector type.
A sketch of Lie superalgebra theory
Communications in Mathematical Physics, 1977
This article deals with the structure and representations of Lie superalgebras (Z 2-graded Lie algebras). The central result is a classification of simple Lie superalgebras over IR and C.
The even, the odd, the superalgebras and their derivations
Advances in Pure and Applied Mathematics, 2018
We give an introduction to superalgebra, founded on the difference between even (commuting) and odd (anti-commuting) variables. We give a sketch of Graßmann’s work, and show how derivations of those structures induce various superalgebra structures, Lie superalgebras of Cartan type being obtained with even derivations, while odd derivations induce Jordan-type superalgebras.
2007
We develop the theory of N -homogeneous algebras in a super setting, with particular emphasis on the Koszul property. To any Hecke operator R on a vector superspace, we associate certain superalgebras S R,N and Λ R,N generalizing the ordinary symmetric and Grassmann algebra, respectively. We prove that these algebras are N -Koszul. For the special case where R is the ordinary supersymmetry, we derive an N -generalized super-version of MacMahon's classical "master theorem".