The universal conservative superalgebra

Abstract

We introduce the class of conservative superalgebras, in particular, the superalgebra U(V ) of bilinear operations on a superspace V. Moreover, we show that each conservative superalgebra modulo its maximal Jacobian ideal is embedded into U(V ) for a certain superspace V.

Conservative algebras and superalgebras: a survey

“…In particular, every unital Jordan algebra of dimension n is a subalgebra of U(n). The analog of this theorem for superalgebras was proved in [21].…”

“…Assuming that the underlying spaces are Z 2 -graded and introducing signs in appropriate places, we can get a notion of a conservative superalgebra [21]. One may also use the general approach to define conservative superalgebras.…”

“…We have already seen that U(V ) has a left unity. It is also easy to check that its maximal Jacobi ideal is zero (see, for example, [21]). Therefore, in our case the space W is U(V )/J that we identify with V by the mapping A → A(a, a).…”

Conservative algebras of 2-dimensional algebras, III

“…Also a similar problem is solved for the algebra W 2 of all commutative algebras on the 2-dimensional vector space and for the algebra S 2 of all commutative algebras with zero multiplication trace on the 2-dimensional vector space. The papers [15], [17] are also devoted to the study of conservative algebras and superalgebras.…”

One-generated nilpotent terminal algebras

“…Therefore, the class of terminal algebras is a variety. Also, about terminal and conservative algebras see, for example [3,25,28,29,33,35,39].…”