Let $mathcal M$ be a factor von Neumann algebra. It is shown that every nonlinear $*$-Lie higher derivation$D={phi_{n}}_{ninmathbb{N}}$ on $mathcal M$ is additive. In particular, if $mathcal M$ is infinite type $I$factor, a concrete characterization of $D$ is given.

The study of Maps (X,G), the group of polynomial maps of a complex algebraic variety X into a complex algebraic group G, and its representations is only well developed in the case that X is a complex torus C. In this case Maps (X,G) is a loop group and the corresponding Lie-algebra Maps (X, ◦ G) is the loop algebra C[t, t]⊗ ◦ G. Here the representation comes to life only after one replaces Maps...

The Capelli identity [1] is one of the best exploited results of the classical invariant theory. It provides a set of distinguished generators C1, . . . ,CN for the centre of the enveloping algebra U(glN ) of the general linear Lie algebra. For any non-negative integer M consider the natural action of the Lie algebra glN in the vector space CN⊗CM . Extend it to the action of the algebra U(glN )...

In recent papers J. Bergen and D. S. Passman have applied so-called`Delta methods' to enveloping algebras of Lie superalgebras. This paper generalizes their results to the class of Lie colour algebras. The methods and results in this paper are very similar to those of Bergen and Passman, and many of their proofs generalize easily. However, at some points there are serious diiculties to overcome...

Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie algebra structure together with the Leibniz law. For finite-dimensional ones we show that if they are semisimple as associative algebras then they are standard, on the other hand, if they are semisimple as Lie algebras then their associative products are trivial. We also give the descriptions of ...