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 A α graded by some group G, over a field of characteristic zero, has a nonzero component A 1 (where 1 stands for the identity element of G), and A 1 is a semisimple associative algebra. Let B = ⊕αεG B α be a finite-dimensional semisimple Lie algebra over a prime field F p , and let B be graded by a commutative group G. If B = F p ⊗ ℤ A L , where A L is the commutator algebra of a ℤ-algebra A = ⊕αεG A α ; if ℚ ◯ ℤ A is an algebra of associative type, then the 1-component of the algebra K ◯ ℤ B, where K stands for the algebraic closure of the field F p , is the sum of some algebras of the form gl(n i ,K).
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Original Russian Text © N. A. Koreshkov, 2010, published in Matematicheskie Zametki, 2010, Vol. 88, No. 1, pp. 43–52.
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Koreshkov, N.A. Lie algebras and algebras of associative type. Math Notes 88, 39–47 (2010). https://doi.org/10.1134/S0001434610070047
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DOI: https://doi.org/10.1134/S0001434610070047