New examples of simple Jordan superalgebras over an arbitrary field of characteristic 0

Abstract

In a joint paper with the author, I. P. Shestakov constructed a new example of a unital simple special Jordan superalgebra over the real number field. It turned out that this superalgebra is a subsuperalgebra of a Jordan superalgebra of the vector type J(Γ, D), but it is not isomorphic to a superalgebra of this type. Moreover, the superalgebra of quotients of the constructed superalgebra is isomorphic to a Jordan superalgebra of vector type. Later, a similar example was constructed for Jordan superalgebras over a field of characteristic 0 in which the equation t 2 + 1 = 0 is unsolvable. In the present paper, an example is given for a Jordan superalgebra with the same properties over an arbitrary field of characteristic 0. A similar example was discovered also for a Cheng-Kac superalgebra. §1. Introduction Jordan algebras and superalgebras form an important class of algebras in the theory of rings. Simple Jordan superalgebras were studied in [1,2,3,4,5,6,7,8].In [9, 10], a description was given for the unital simple special Jordan superalgebras with an associative even part A, the odd part M of which being an associative A-module. The paper [11] in which the simple (−1, 1)-superalgebras of characteristic different from 2 and 3 were described influenced significantly the investigations carried out in [9]. In the Jordan case, if a superalgebra is not the superalgebra of a nondegenerate bilinear superform, then its even part A is a differentiably simple algebra relative to a certain set of derivations, and its odd part M is a finitely generated projective A-module of rank 1. Here, like in the case of (−1, 1)-superalgebras, multiplication in M is given with the help of fixed finite sets of derivations and elements of the algebra A. It turned out that each Jordan superalgebra is a subsuperalgebra of this sort in a superalgebra of vector type J (Γ, D). Under some restrictions on the algebra A, the odd part M is a one-generated A-module, and thus, the initial Jordan superalgebra is isomorphic to a superalgebra J(Γ, D). So, for example, if A is a local algebra, then, by the well-known Kaplansky theorem, the odd part M is a free, and thus, one-generated A-module. If the ground field is of characteristic p > 2, then, by [14], A is a local algebra; therefore, the odd part M is a one-generated A-module. If A is a ring of polynomials in a finite number of variables, then, by [15], the odd part M is a free and, thus, one-generated A-module.Naturally, the question arises as to whether the initial superalgebra is isomorphic to a superalgebra J(Γ, D). This is equivalent to the question as to whether the odd part M is a one-generated A-module. In [10,12,13], there are examples of a unital simple special 2010 Mathematics Subject Classification. Primary 16W10; Secondary 17A15. Key words and phrases. Jordan superalgebra, (−1, 1)-superalgebra, superalgebra of vector type, differentiably simple algebra, polynomial algebra, projective module.

Jordan superalgebras of vector type and projective modules

“…As was shown in [14], A is differentiably simple with respect to Δ, and J(A, Δ) is a simple Jordan superalgebra with even part A and odd part M . The element γ 12 is equal to 1 + y 4 .…”

“…Using (13), (14), and direct computation, we infer that φ is an embedding of J(A, Δ) into B (+)s . Therefore, J(A, Δ) is special.…”

“…Given a field F of characteristic zero in which the equation t 2 + 1 = 0 is unsolvable, some example of a similar superalgebra over F was given in [12][13]. Finally, the example of a simple superalgebra of vector type with several derivations over an arbitrary field of characteristic zero was constructed in [14]; this example of Jordan superalgebra is an answer to a question from [15]. …”

(−1, 1)-Superalgebras of vector type: Jordan superalgebras of vector type and their universal envelopings

“…By Proposition 5 it suffices to show that (7) and (8) imply (9) and (10). Assume (7) and (8). Then by Proposition 2…”

“…Here, as for the (−1, 1)-superalgebras, the product in M is given by some fixed finite sets of derivations and the elements of A. Shestakov constructed in [6] the new example of a unital simple Jordan superalgebra of vector type over the reals such that its odd part is not a one-generated module. Some examples of the new unital simple Jordan superalgebras of vector type over other fields were constructed in [7,8]. These examples answer the Cantarini-Kac question [9].…”