CC-circuits and the expressive power of nilpotent algebras

We show that CC-circuits of bounded depth have the same expressive power as circuits over finite nilpotent algebras from congruence modular varieties. use this result to phrase and discuss a new algebraic version Barrington, Straubing Th\'erien's conjecture, which states need exponential size compute AND. Furthermore, we investigate complexity deciding identities solving equations in fixed algebra. Under assumption conjecture is true, obtain quasipolynomial algorithms for both problems. On other hand, if AND computable by uniform polynomial size, can construct algebra checking coNP-complete, NP-complete.