Generalize Heisenberg Groups and Self-Duality

Abstract

This paper compares two generalizations of Heisenberg groups and studies their connection to one of the major open problems in the field of locally compact abelian groups, namely the description of the selfdual locally compact abelian groups ([12], [13]). The first generalization is presented by the so called generalized Heisenberg groups H(ω), defined in analogy with the classical Heisenberg group, and the second one is inspired by the construction proposed by Mumford in [19] and named after him as Mumford groups. These two families can be defined also in the framework of topological groups. We investigate the relationship between locally compact Mumford groups, locally compact generalized Heisenberg groups with center isomorphic to T, and symplectic self-dualities.

Heisenberg Groups via Algebra

“…One characteristic feature of nilquadratic groups is the symplectic structure they induce. We recall some basic facts, essentially following [10]. We call a subgroup…”

“…Without taking recourse to a section, one can in fact define a "commutator form" [, ] : H/Z(H) × H/Z(H) → [H, H] for an arbitrary group H. But it turns out that [, ] is bilinear precisely when H is nilquadratic; see Fact 2.4 in [10]. Since we are here only interested in the nilquadratic setting, the commutator form ω E is in fact bilinear as we shall now check.…”

“…For checking that α is a homomorphism, note that Next we must check that α H is consistent with the abelian bisections. In other words, we need to ensure (10)…”

“…Motivate "our" definition of Heisenberg groups as a natural algebraic analog of Mumford's definition in [34]. But also clarify the contrast to "Mumford groups" in the sense of [10,Def. 5.1]: The stipulation of a bijective instead of injective map into the dual (yielding a strong rather than a weak symplectic duality!)…”

“…References to be included are [10], [30], [36], [35], [48]. Two subspaces are called transverse if they span the given vector space, and, following [6, p. 21], we define a Lagrangian bisection of a symplectic vector space V as a pair of transverse Lagrangian subspaces L, L ′ ; then automatically L∔L ′ = V .…”