“Real" and "complex" toric topology are similar in many ways, sometimes without a clear explanation. We study the analogy between wedges of infinite projective spaces and, more generally, between real and complex Davis-Januszkiewicz spaces. Here the "complex" space is simply connected, while the "real" space is the classifying space for the right-angled Coxeter group RC_K.

The lower central series of a group G allows to construct an associated Lie ring L(G), which is known for free groups and right-angled Artin groups. Motivated by the analogy, we construct a new operation in the commutator subalgebra L'(RC_K), which is analogous to the composition with the Hopf element in homotopy groups. This allows to represent L'(RC_K) as a quotient of an explicit Lie algebra related to loop homology of moment-angle complexes. We conjecture that the quotient map is an isomorphism. The talk is based on a joint work with Yakov Veryovkin.