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

Let G be a finite group and be its group algebra defined over . If we define in G a 2-cochain F, then we can consider the algebra which is obtained from deforming the product, x.Fy=F(x,y)xy, x,y G. Examples of 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 with a multiplicative closed structure. Furthermore, we develop some fully explicit characterizations in terms of generalized trace and norm functions.