Self-similar associative algebras.
\textit{V. M. Petrogradsky} [J. Algebra 302, No. 2, 881-886 (2006; Zbl 1109.17008)] and \textit{I. P. Shestakov} and \textit{E. Zelmanov} [J. Eur. Math. Soc. (JEMS) 10, No. 2, 391-398 (2008; Zbl 1144.17013)] introduced a family of self-similar Lie algebras over fields of prime characteristic \(p>0\) whose properties resemble those of Grigorchuk and Gupta-Sidki groups. This Lie algebra is generated by the following derivations of the truncated polynomial ring \(K[t_i]/t_i^p=0\): \[ v_1=\partial_1+t_0^{p-1}(\partial_2+t_1^{p-1}(\partial_3+t_2^{p-1}(\partial_4+\cdots))), \] \[ v_2=\partial_2+t_1^{p-1}(\partial_3+t_2^{p-1}(\partial_4+t_3^{p-1}(\partial_5+\cdots))). \] In the cited papers it was proved that the constructed Lie algebra has a nil \(p\)-mapping, this property is an analog of the periodicity property of the Grigorchuk and Gupta-Sidki groups. Also in the cited papers the associative hull of these operators and the restricted enveloping algebra were studied. Recently \textit{S. N. Sidki} [J. Algebra 322, No. 12, 4408-4429 (2009; Zbl 1187.15021)] introduced an associative algebra of self-similar matrices. In the paper under review other examples of self-similar associative algebras are constructed and investigated. Two families of self-similar associative algebras \(C_\Omega\) and \(D_\Omega\) where \(\Omega\) is an array of parameters are defined. A weight function for these algebras is defined and triangular decompositions \(C_\Omega=C_+\oplus C_0\oplus C_-\) and \(D_\Omega=D_+\oplus D_0\oplus D_-\) are obtained. The degenerations of these decompositions are studied.
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