Triality Groups Associated with Triple Systems and their Homotope Algebras

We introduce the notion of an (α, β, γ) triple system, which generalizes the familiar generalized Jordan triple system related to a construction of simple Lie algebras. We then discuss its realization by considering some bilinear algebras and vice versa. Next, as a new concept, we study triality relations (a triality group and a triality derivation) associated with these triple systems; the relations are a generalization of the automorphisms and derivations of the triple systems. Also, we provide examples of several involutive triple systems with a tripotent element.

On the universal envelope of a Jordan triple system of n × n matrices

We study the universal (associative) envelope of the Jordan triple system of all [Formula: see text] [Formula: see text] matrices with the triple product [Formula: see text] over a field of characteristic 0. We use the theory of non-commutative Gröbner–Shirshov bases to obtain the monomial basis and the center of the universal envelope. We also determine the decomposition of the universal envelope and show that there exist only five finite-dimensional inequivalent irreducible representations of the Jordan triple system of all [Formula: see text] matrices.

Quadratic Jordan Triple Systems

In this work, We show that every Jordan triple system can be viewed as a T∗extension of another one or an ideal of co-dimension one of a Jordan triple system whose represent the T∗extension of another Jordan triple system. Moreover, several result involving the structure of quadratic Jordan triple systems are given.

Representations of simple anti-Jordan triple systems of m × n matrices

We show that the universal associative envelope of the simple anti-Jordan triple system of all [Formula: see text] ([Formula: see text] is even, [Formula: see text]) matrices over an algebraically closed field of characteristic 0 is finite-dimensional. The monomial basis and the center of the universal envelope are determined. The explicit decomposition of the universal envelope into matrix algebras is given. The classification of finite-dimensional irreducible representations of an anti-Jordan triple system is obtained. The semi-simplicity of the universal envelope is shown. We also show that the universal associative envelope of the simple polarized anti-Jordan triple system of [Formula: see text] matrices is infinite-dimensional.

Hermitian generalized Jordan triple systems and certain applications to field theory

We define Hermitian generalized Jordan triple systems and prove a structure theorem. We also give some examples of the systems and study mathematical properties. We apply a Hermitian generalized Jordan triple system to a field theory and obtain a Chern–Simons gauge theory.

A STRUCTURE THEORY OF (−1,−1)-FREUDENTHAL KANTOR TRIPLE SYSTEMS

In this paper we discuss the simplicity criteria of (−1,−1)-Freudenthal Kantor triple systems and give examples of such triple systems, from which we can construct some Lie superalgebras. We also show that we can associate a Jordan triple system to any (ε,δ)-Freudenthal Kantor triple system. Further, we introduce the notion of δ-structurable algebras and connect them to (−1,δ)-Freudenthal Kantor triple systems and the corresponding Lie (super)algebra construction.

Parallel submanifolds of complex space forms II

This is a continuation of Part I, which appeared in this journal.In the previous paper I we have defined the following notions: orthogonal Jordan triple system (OJTS), orthogonal symmetric graded Lie algebra (OSGLA), orthogonal Jordan algebra (OJA), Hermitian symmetric graded Lie algebra (HSGLA). And we have shown that equivalent classes of OJTS naturally correspond to equivalent classes of OSGLA and through this correspondence we have naturally constructed HSGLA’s from the OJTS’s associated with OJA’s with unity.