No abstract

The generalized Lie algebras, which have recently been introduced under the name of color (super) algebras, are investigated. The generalized Poincaré–Birkhoff–Witt and Ado theorems hold true. We discuss the so-called commutation factors which enter into the defining identities of these algebras. Moreover, we establish a close relationship between the generalized Lie algebras and ordinary Lie (super) algebras.

We illustrate through the examples of the osp(2,1) and spl(2,1) algebras the differences between the properties of the irreducible representations of simple graded Lie algebras and simple Lie algebras.

Fock-Space (annihilation/creation operator) methods are introduced to describe systems of identical classical objects. Specific examples to which this formalism is applied are branching processes (including age dependent ones), chemical reactions, deterministic (Hamiltonian) systems, and generalized kinetic equations. Finally, a generalization to stochastic quantum systems is proposed which is applied to a gas of spinning molecules.

The cohomology groups of Lie superalgebras and, more generally, of ε Lie algebras, are introduced and investigated. The main emphasis is on the case where the module of coefficients is non-trivial. Two general propositions are proved, which help to calculate the cohomology groups. Several examples are included to show the peculiarities of the super case. For L = sl(1|2), the cohomology groups H 1 (L, V ) and H 2 (L, V ), with V a finite-dimensional simple graded L-module, are determined, and the result is used to show that H 2 (L, U (L)) (with U (L) the enveloping algebra of L) is trivial. This implies that the superalgebra U (L) does not admit of any non-trivial formal deformations (in the sense of Gerstenhaber). Garland's theory of universal central extensions of Lie algebras is generalized to the case of ε Lie algebras.q-alg/9701037