Self-similar associative algebras

Famous self-similar groups were constructed by Grigorchuk, Gupta and Sidki, these examples lead to interesting examples of associative algebras. The authors suggested examples of self-similar Lie algebras in terms of differential operators. Recently Sidki introduced an example of an associative algebra of self-similar matrices. We construct families of self-similar associative algebras C-Omega, D-Omega, generalizing the example of Sidki. We prove that our algebras are Z circle plus Z-graded and have polynomial growth. Our approach is the weight strategy developed by the authors for self-similar Lie algebras and their envelopes. In particular, we obtain similar triangular decompositions into direct sums of three subalgebras C = C+ circle plus C-0 D = D+ circle plus D-0 circle plus D-. We prove that some of our algebras are direct sums of two locally nilpotent subalgebras C = C+ circle plus C--,C- D = D+ circle plus D-0 D-. We show that in some cases the zero components C-0, D-0 are nontrivial and not nil algebras. We show that our construction includes the example of Sidki and the examples of self-similar Lie algebras and their associative hulls constructed by the authors before. (C) 2013 Elsevier Inc. All rights reserved.