Simple Jordan superalgebras with associative nil-semisimple even part

Under study are the simple infinite-dimensional abelian Jordan superalgebras not isomorphic to the superalgebra of a bilinear form. We prove that the even part of such superalgebra is a differentially simple associative commutative algebra, and the odd part is a finitely generated projective module of rank 1. We describe unital simple Jordan superalgebras with associative nil-semisimple even part possessing two even elements which induce a nonzero derivation.     

Simple Special Jordan Superalgebras with Associative Even Part

We describe the simple special unital Jordan superalgebras with associative even part A whose odd part M is an associative module over A. We prove that each of these superalgebras, not isomorphic to a superalgebra of nondegenerate bilinear superform, is isomorphically embedded into a twisted Jordan superalgebra of vector type. We exhibit a new example of a simple special Jordan superalgebra. We also describe the superalgebras such that M [A,M] 0.     

On structure and TKK algebras for Jordan superalgebras

We compare a number of different definitions of structure algebras and TKK constructions for Jordan (super)algebras appearing in the literature. We demonstrate that, for unital superalgebras, all the definitions of the structure algebra and the TKK constructions reduce to one of two cases. Moreover, one can be obtained as the Lie superalgebra of superderivations of the other. We also show that, for non-unital superalgebras, more definitions become nonequivalent. As an application, we obtain the corresponding Lie superalgebras for all simple finite dimensional Jordan superalgebras over an algebraically closed field of characteristic zero.     

Centroids of Quadratic Jordan Superalgebras

The centroid of a Jordan superalgebra consists of the natural “superscalar multiplications” on the superalgebra. A philosophical question is whether the natural concept of “scalar” in the category of superalgebras should be that of superscalars or ordinary scalars. Basic examples of Jordan superalgebras are the simple Jordan superalgebras with semisimple even part, which were classified over an algebraically closed field of characteristic ≠ 2 by Racine and Zelmanov. Here, we determine the centroids of the analog of these superalgebras over general rings of scalars and show that they have no odd centroid, suggesting that ordinary scalars are the proper concept.     

Simple 5-Dimensional Right Alternative Superalgebras with Trivial Even Part

We study the simple right alternative superalgebras whose even part is trivial; i.e., the even part has zero product. A simple right alternative superalgebra with the trivial even part is singular. The first example of a singular superalgebra was given in [1]. The least dimension of a singular superalgebra is 5. We prove that the singular 5-dimensional superalgebras are isomorphic if and only if suitable quadratic forms are equivalent. In particular, there exists a unique singular 5-dimensional superalgebra up to isomorphism over an algebraically closed field.     

Noncommutative Jordan superalgebras of degree <Emphasis Type="Italic">n</Emphasis> > 2

We prove a coordinatization theorem for noncommutative Jordan superalgebras of degree n > 2, describing such algebras. It is shown that the symmetrized Jordan superalgebra for a simple finite-dimensional noncommutative Jordan superalgebra of characteristic 0 and degree n > 1 is simple. Modulo a “nodal” case, we classify central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0.     

On Right Alternative Superalgebras

Simple right alternative superalgebras which have a simple algebra as even part and, as odd part, an irreducible bimodule over the even part are investigated. Under these conditions, superalgebras with one dimensional even part are classified, as well as superalgebras having M ₂(F) as even part and a unital irreducible bimodule over M ₂(F) of dimension less than or equal to 6 as odd part. It is shown that there is only a unique non alternative simple right alternative superalgebra of the first type and, for the second type, there is a infinite family depending on a single parameter.     

Jordan Superderivations

Abstract

We extend Herstein's theorem on Jordan derivations of prime rings to superalgebras. Our main result states that a prime associative superalgebra admits a proper Jordan superderivation only in the case when its even part is commutative.     

Algebras of Jordan brackets and generalized Poisson algebras

We construct bases for free unital generalized Poisson superalgebras and for free unital superalgebras of Jordan brackets. Also, we prove an analogue of Farkas’ theorem for PI unital generalized Poisson algebras and PI unital algebras of Jordan brackets. Relations between generic Poisson superalgebras and superalgebras of Jordan brackets are studied.     

Differential algebras and simple Jordan superalgebras

In [14], a new example is constructed of a unital simple special Jordan superalgebra J over the field of reals. It turns out that J is a subsuperalgebra of a Jordan superalgebra of vector type but it cannot be isomorphic to a superalgebra of such a type. Moreover, the superalgebra of fractions of J is isomorphic to a Jordan superalgebra of vector type. In the present article, we find a similar example of a Jordan superalgebra. It is constructed over a field of characteristic 0 in which the equation t ² + 1 = 0 has no solutions.     

李超代数上的不变双线性型

本文通过定义李超代数上的形心和零次形心来考察其性质.证明了二次李超代数(G,B)上的不变数积的集合和其形心中的可逆B-超对称元素的集合之间存在一一对应.而对实单李超代数分为两种不同的类型:或者是一个忽略了复结构的复李超代数或者是一个复单李超代数的实形式.     

Jordan Superalgebras Defined by Brackets

Jordan superalgebras defined by brackets on associative commutativesuperalgebras are studied. It is proved that any such superalgebrais imbedded into a superalgebra defined by Poisson brackets.In particular, all Jordan superalgebras of brackets are i-special.The speciality of these superalgebras is also examined, andit is proved, in particular, that the Cheng–Kac superalgebrais special.     

Jordan centroids

Central simple triples are important for the classification of prime Jordan triples of Clifford type in arbitrary characterstics. For triples and pairs (or even for unital Jordan algebras of characteristic 2), there is no workable notion of center, and the concept of “central simple” system must be understood as “centroid-simple”. The centroid of a Jordan system (algebra, triple, or pair) consists of the “natural” scalars for that system: the largest unital, commutative ring Γ such that the system can be considered as a quadratic Jordan system over Γ. In this paper we will characterize the centroids of the basic simple Jordan algebras, triples, and pairs. (Consideration of the tangled ample outer ideals in Jordan algebras of quadratic forms will be left to a separate paper.) A powerful tool is the Eigenvalue Lemma, that a centroidal transformation on a prime system over φ which has an eigenvalue α in φ must actually be scalar multiplication by α. An important consequence is that a prime system over φ with reduced elements P_xJ = φx (or which grows reduced elements under controlled conditions) must already be central, Γ = φ.     

Alternative Superalgebras with DCC on Two-Sided Ideals#

We study the structure of alternative superalgebras that satisfy the descending chain condition (DCC) for two-sided ideals. The main results state that the Baer radical in an alternative superalgebra of characteristic ≠ 2, 3 with DCC on two-sided ideals is solvable and every such a semiprime superalgebra (of arbitrary characteristic) is isomorphic to a subdirect sum of an associative superalgebra with this property and a finite direct sum of simple alternative non-associative superalgebras.     

Gelfand-Kirillov Dimension and Local Finiteness of Symmetrizations of Associative Superalgebras

In this work we extend to superalgebras a result of Skosyrskii [Algebra and Logic, 18 (1) (1979) 49–57, Lemma 2] relating associative and Jordan structures. As an application, we show that the Gelfand-Kirillov dimension of an associative superalgebra coincides with that of its symmetrization, and that local finiteness is equivalent in associative superalgebras and in their symmetrizations. In this situation we obtain that having zero Gelfand-Kirillov dimension is equivalent to being locally finite.     

Gelfand–Kirillov Dimension and Local Finiteness of Jordan Superpairs Covered by Grids and Their Associated Lie Superalgebras

Abstract

In this paper we show that a Lie superalgebra L graded by a 3-graded irreducible root system has Gelfand–Kirillov dimension equal to the Gelfand–Kirillov dimension of its coordinate superalgebra A, and that L is locally finite if and only A is so. Since these Lie superalgebras are coverings of Tits–Kantor–Koecher superalgebras of Jordan superpairs covered by a connected grid, we obtain our theorem by combining two other results. Firstly, we study the transfer of the Gelfand–Kirillov dimension and of local finiteness between these Lie superalgebras and their associated Jordan superpairs, and secondly, we prove the analogous result for Jordan superpairs: the Gelfand–Kirillov dimension of a Jordan superpair V covered by a connected grid coincides with the Gelfand– Kirillov dimension of its coordinate superalgebra A, and V is locally finite if and only if A is so.