Multiplicative invariant lattices in Rn obtained by twisting of group algebras and some explicit characterizations

Let G be a finite group and RG be its group algebra defined over R. If we define in G a 2-cochain F, then we can consider the algebra RFG which is obtained from RG deforming the product, x.Fy=F(x,y)xy, ∀x,y∈G. Examples of RFn(Z2) algebras are Clifford algebras and Cayley algebras like octonions. In this paper we consider generalizations of lattices with complex multiplication in the context of these twisted group algebras. We explain how these induce the natural algebraic structure to endow any arbitrary finite-dimensional lattice whose real components stem from any finite algebraic field extension over Q with a multiplicative closed structure. Furthermore, we develop some fully explicit characterizations in terms of generalized trace and norm functions.