Differential algebras and simple Jordan superalgebras

St. Petersburg Math. J.

2013

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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.

mentioning

confidence: 99%

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

St. Petersburg Math. J.

2013

Self Cite

“…As it was proved in [7], the superalgebra J(A 1 , Δ 1 ) is simple. Therefore, B(A 1 , Δ 1 , Υ 1 ) is a simple (−1, 1)-superalgebra.…”

Section: Corollary Equalitiesmentioning

confidence: 81%

“…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].…”

mentioning

confidence: 97%

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

2015

Sib Math J

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We establish a connection between the integral domains, the projective finitely generated modules over them, the derivations of an integral domain and the (−1, 1)-superalgebras of vector type. We study the properties of the universal associative envelopings of the simple Jordan superalgebras of vector type. Some example of infinite-dimensional simple Jordan superalgebras can be obtained from an associative supercommutative superalgebra with a Jordan bracket using the Kantor doubling process (see [1][2][3]). If the Jordan bracket is given on an associative-commutative algebra then the even part of the so-obtained Jordan superalgebra is associative, and the odd part is a one-generated module over the even part. Analogously, the simple (−1, 1)-superalgebras are obtained from the associative-commutative algebras which are differentiably simple with respect to a derivation. The simple (−1, 1)-superalgebras of characteristic not 2 and 3 were described in [4]. It turned out that the even part A of this superalgebra is an associative-commutative algebra, and the odd part M is a finitely generated associative and commutative A-module. The product in M is given by some fixed finite sets of derivations and elements of A. Under some restrictions on A the odd part M is a one-generated A-module, and the initial (−1, 1)-superalgebra is a twisted superalgebra of vector type with respect to a derivation. The (−1, 1)-superalgebras obtained in [4] we also call the (−1, 1)-superalgebras of vector type. It should be noted that the adjoined Jordan superalgebra for a simple (−1, 1)-superalgebra is a unital simple special superalgebra with associative even part.In [5,6], the unital simple special Jordan superalgebras were described with an associative even part A and an odd part M , which is an associative A-module. In this case, if the superalgebra is not a superalgebra of nondegenerate bilinear superform then its even part A is a differentiably simple algebra, and the odd part M is a finitely generated projective A-module of rank 1. 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]. Note that in the constructed examples of simple Jordan superalgebras the odd part is a two-generated module over the even part. In [10], some prime Jordan superalgebras of vector type over the reals are given such that their odd part is generated as a module by more than two elements. In terms of the Lie algebra of derivations of an integral domain, some necessary and sufficient conditions were found in [11], which allow us to construct the Jordan superalgebras of vector type whose even part is the initial integral domain.

“…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.…”

Section: Proof Since a Is An Integral Domain The Annihilator Of Thementioning

confidence: 98%

“…This example may be found in [9]. 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].…”

mentioning

confidence: 96%

Jordan superalgebras of vector type and projective modules

2012

Sib Math J

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We study connection between the Jordan superalgebras of vector type and the finitely generated projective modules of rank 1 over an integral domain.Keywords: Jordan superalgebra of vector type, differentiably simple algebra, integral domain, field of quotients, polynomial algebra, projective module, Picard groupThe well-known Kantor construction [1, 2] allows us to construct a Jordan superalgebra that is based on an associative commutative algebra with derivation. The so-constructed Jordan superalgebra belongs to the class of superalgebras of vector type. The superalgebras of vector type play an important role in exhibition of some counterexamples. In [3], some examples are given of prime (−1, 1)-algebras and Jordan algebras with absolute zero-divisors, the so-called Pchelintsev's monsters. Other constructions of Pchelintsev's monsters were given in [4] by using the Jordan superalgebras of vector type. Some examples of prime algebras with absolute zero-divisors in the varieties of alternative, Jordan, and (−1, 1)-algebras were obtained in [5] with the help of the Jordan superalgebras of vector type. The Jordan superalgebras of vector type with one derivation were studied in [2,6]. Thus, the simplicity criterium for a Jordan superalgebra was found in [2]. The speciality of this Jordan superalgebra was proved in [6]. In [7] the universal associative enveloping algebra was constructed for a simple Jordan superalgebra of vector type with one derivation.In [8,9], the unital simple special Jordan superalgebras with associative even part A were described such that the odd part M is an associative A-module. If such superalgebra is not a superalgebra of nondegenerate bilinear superform then its even part A is a differentiably simple algebra with respect to some set of derivations, and the odd part M is a finitely generated projective A-module of rank 1. The multiplication on M is given by some fixed finite sets of derivations and elements of A. Furthermore, each of such Jordan superalgebras is a subsuperalgebra of vector type J(Γ, D) with derivation D. If the number of generators of the A-module M is equal to 1 then the initial Jordan superalgebra is a superalgebra of vector type J(Γ, D). Under some restriction on A, the odd part M is a one-generated A-module. For example, if A is a local algebra then M is a free (and, consequently, one-generated) A-module by the well-known Kaplansky theorem. If the ground field is of characteristic p > 2 then A is a local algebra by [10]; therefore, M is a one-generated A-module. If A is a unique factorization ring then it is known that M is a free (and, consequently, one-generated) A-module (for example, see [11]).The first example of a simple superalgebra of vector type over the reals with several derivations which is not isomorphic to J(Γ, D) was constructed by I. P. Shestakov. This example may be found in [9]. 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 exampl...