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 CΩ, DΩ, generalizing the example of Sidki. We prove that our algebras are Z⊕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+⊕C0⊕C−, D=D+⊕D0⊕D−. We prove that some of our algebras are direct sums of two locally nilpotent subalgebras C=C+⊕C−, D=D+⊕D−. We show that in some cases the zero components C0, D0 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.