Split Dialgebras, Split Quasi-Jordan Algebras and Regular Elements

Abstract

Communicated by S. R. López-PermouthThe notions of annihilator ideal and split structure are studied in detail for both, dialgebras and quasi-Jordan algebras. It yields methods for additional units in the two structures. As a consequence the notion of regular element receives special attention. . Downloaded from www.worldscientific.com by PENNSYLVANIA STATE UNIVERSITY on 03/15/15. For personal use only.If the bracket is skew-symmetric, then L is a Lie algebra. Therefore Lie algebras are particular cases of Leibniz algebras. Example 2. Let L be a Lie algebra and let M be a L-module with action M ×L → M , (m, x) → mx. Let f : M → L be a L-equivariant linear map, this is f (mx) = [f (m), x], for all m ∈ M and x ∈ L, then one can put a Leibniz structure on M as follows [m, n] := mf (n), for all m, n ∈ M. Additionally, the map f defines a homomorphism between Leibniz algebras, since f ([m, n] ) = f (mf (n)) = [f (m), f(n)].Example 3. Let (A, d) be a differential associative algebra. So, by hypothesis, d(ab) = da b + a db and d 2 = 0. We define the bracket on A by the formula [a, b] := a db − db a, then the vector space A equipped with this bracket is a Leibniz algebra. J. Algebra Appl. 2009.08:191-218. Downloaded from www.worldscientific.com by PENNSYLVANIA STATE UNIVERSITY on 03/15/15. For personal use only.Split Dialgebras,

Restrictive, split and unital quasi-Jordan algebras

“…This relationship shows that every quasi-Jordan algebra is isomorphic to a subalgebra of a split quasi-Jordan algebra (see [14] for more details).…”

“…In [14], the notions of the annihilator ideal and split structure are studied in detail for both dialgebras and quasi-Jordan algebras. The authors also provide methods for additional units in the two structures.…”

“…Additionally, the right units in quasi-Jordan algebras are not unique. The following examples can be find in [13] and [14]. …”

“…For more details on the concepts and results that we discuss below we refer the reader to [14]. For a quasi-Jordan algebra we introduce…”

“…The reciprocal of this characterization is not true, that is, a split quasi-Jordan algebra with unital Jordan part, need not necessarily have a right unit (see [14] for more details).…”