Simple Jordan superalgebras with the even parts of Clifford type

TL;DR

The paper tackles the classification of simple infinite-dimensional Jordan superalgebras by reducing to cases determined by the structure of the even part. It first shows that a simple unital Jordan superalgebra with finite-dimensional even part is either finite-dimensional or a superalgebra of a nondegenerate supersymmetric bilinear form on a vector space with finite even and infinite odd dimensions. It then treats the Clifford-type (Clifford) case, proving that either the algebra remains finite-dimensional or again is a superalgebra of a superform. Finally, it rules out the direct-sum-even-part scenario as a source of new infinite-dimensional simple algebras, proving finiteness in that case. Collectively, these results move toward a full classification by demonstrating that the only simple infinite-dimensional Jordan superalgebras of the studied Clifford-type configurations are those arising from superforms, thereby narrowing the landscape of possible examples and clarifying the underlying structure.

Abstract

The purpose of this paper is a partial progress towards classification of simple infinite dimensional Jordan superalgebras. First, we prove that the only simple infinite dimensional Jordan superalgebras with finite dimensional even parts are the superalgebras of superforms. Then we consider the superalgebras whose even parts are infinite dimensional algebras of ``Clifford type'', that is, direct sums of algebras of bilinear forms. The results of \cite{RZ} show that the number of summonds in these sums is 1 or 2. We prove that the second case is impossible and that the simple infinite dimensional Jordan superalgebras of the first type are the superalgebras of superforms.

Theorems & Definitions (15)

• Conjecture 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5