Algebras Of Quotients And Crossed Modules For Lie Color Algebras

The notions of algebras of quotients and weak algebras of quotients for Lie color algebras are given. Some properties, such as primeness, semiprimeness and nondegeneracy are introduced. The primeness and semiprimeness are lifted from a Lie color algebra to its algebras of quotients. It is shown that if L is a subalgebra of a Lie color algebra Q, then Q is an algebra of quotients of L if and only if Q is ideally absorbed into L. For every semiprime Lie color algebra, a maximal algebra of quotients is constructed.Then the notions of crossed modules for Lie color algebras are introduced. Let L and P be Lie color algebras and let M be a graded module over P, the crossed modules which make M as their kernal and P as their cokernal are considered. It is shown that under a suitable equivalent relation, there is a bijection between the set of the equivalent classes CML(P, L;M) and the homogeneous components of degree zero of H~3(P, L; M).