COMBINATORIAL ASPECTS OF FREE ASSOCIATIVE ALGEBRAS AND COHOMOLOGIES OF LIE p-ALGEBRAS WITH ONE DEFINING RELATION

摘要

Let R and Q be elements of a free, associative, finitely generated algebra A over a field k. Assume that a leading homogeneous part Q_v does not have two-sided divisors and RQ = QR and R_v = Q_v~t In this paper, the solutions of the equation ∑_iΧ_iRy_i = z in A are found and with their help, the identity theorem and Freiheitssatz for a finitely generated associative algebra k(X;R = 0) with one defining relation are proved. As a consequence, similar theorems for Lie p-algebras with one defining relation are proved; these results are applied to the proof of the periodicity of cohomologies of Lie p-algebras with one defining relation.