Isotropy in group cohomology

Abstract

Abstract. The analogue of Lagrangians for symplectic forms over finite groups is studied, motivated by the fact that symplectic G-forms with a normal Lagrangian N ⊳G are in one-to-one correspondence, up to inflation, with bijective 1-cocycle data on the quotients G/N . This yields a method to construct groups of central type from such quotients, known as Involutive Yang-Baxter groups. Another motivation for the search of normal Lagrangians comes from a noncommutative generalization of Heisenberg liftings which require normality.Although it is true that symplectic forms over finite nilpotent groups always admit Lagrangians, we exhibit an example where none of these subgroups is normal. However, we prove that symplectic forms over nilpotent groups always admit normal Lagrangians if all their p-Sylow subgroups are of order less than p 8 .

The Morita Theory of Quantum Graph Isomorphisms

“…Remark 3. 13. In contrast to the classification of quantum isomorphic quantum graphs (see Remark 3.8), the condition in Theorem 3.12 does not only depend on the abstract monoidal category with fibre functor QAut(Γ).…”

“…A nondegenerate 2-cocycle ψ of a group of central type L gives rise to a symplectic form 5 4 Equivalently, the classical subcategories can be understood as the categories of finite-dimensional representations of the commutative algebra of functions on Aut(Γ). 5 See [13] for an introduction to symplectic forms on groups.…”

“…If (L, ψ) is a group of central type, we may therefore think of ρ ψ as a non-degenerate alternating form on L. In particular, we borrow the following definitions and terminology from the theory of symplectic forms on groups [13].…”

“…Let G be a finite group. Up to Morita equivalence, indecomposable,13 special dagger Frobenius monoids in Hilb G correspond to pairs (L, ψ) where L is a subgroup of G and ψ : L × L − → U(1) is a 2-cocycle up to the equivalence relation:…”