Abstract:
In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕αεG Aα graded by some group G, over a field of characteristic zero, has a nonzero component A1 (where 1 stands for the identity element of G), and A1 is a semisimple associative algebra. Let B = ⊕αεG Bα be a finite-dimensional semisimple Lie algebra over a prime field Fp, and let B be graded by a commutative group G. If B = Fp ⊗ℤ AL, where AL is the commutator algebra of a ℤ-algebra A = ⊕αεG Aα; if ℚ⊗ℤA is an algebra of associative type, then the 1-component of the algebra K⊗ℤB, where K stands for the algebraic closure of the field Fp, is the sum of some algebras of the form gl(ni,K). © 2010 Pleiades Publishing, Ltd.
