This work aims at introducing weaker versions of quasi-Hamiltonian and quasi-Poisson geometries which are suitable for any symmetric bilinear form. As the author puts it: “this paper is addressed to the expert” so that the text is both technical and interesting.
To give some context, fix a compact Lie group \(G\) and let \(g\) denote its Lie algebra. Recall that quasi-Hamiltonian geometry was introduced in [A. Alekseev et al., J. Differ. Geom. 48, 445–495 (1998; Zbl 0948.53045)] to develop a theory of spaces admitting a \(G\)-valued moment map analogous to the classical theory of symplectic spaces admitting a \(g^\ast\)-valued moment map. Thus, a quasi-Hamiltonian \(G\)-space \(M\) is a smooth \(G\)-manifold endowed with a \(G\)-invariant \(2\)-form \(\sigma\in \Omega^2(M)\) and an equivariant map \(\Phi:M\to G\) satisfying axioms that rely on the choice of a positive-definite \(\mathrm{Ad}\)-invariant symmetric form \((-,-):g\otimes_{\mathbb{R}} g\to \mathbb{R}\). While \(\sigma\) is (in general) neither closed nor non-degenerate, the theory allows for any conjugacy class \(\mathcal{C}\subset G\) to endow the reduced space \(\Phi^{-1}(\mathcal{C})/G\) (if smooth) with a structure of symplectic manifold. Shortly after, the paper [A. Alekseev et al., Can. J. Math. 54, 3–29 (2002; Zbl 1006.53072)] introduced the related notion of Hamiltonian quasi-Poisson \(G\)-manifolds. The smooth \(G\)-manifold \(M\) admits such a structure if it is endowed with a \(G\)-invariant bivector field \(P\in \bigwedge^2 TM\) and an equivariant map \(\Phi:M\to G\) again subject to axioms involving the form \((-,-)\) on \(g\). The bivector \(P\) is, in general, not Poisson because the associated bracket only satisfies the Jacobi identity up to some \(3\)-vector field obtained from the action of \(G\) on \(M\). Nevertheless this bracket is a genuine Poisson bracket when restricted to \(G\)-invariant functions on \(M\), and the reduced space \(\Phi^{-1}(\mathcal{C})/G\) inherits a Poisson structure. Interestingly, there is a notion of non-degeneracy for the quasi-Poisson bivector \(P\) which allows to build a correspondence between the structures of quasi-Hamiltonian \(G\)-spaces \((M,\sigma,\Phi)\) and of non-degenerate Hamiltonian quasi-Poisson manifolds \((M,P,\Phi)\). An important example for which this correspondence holds is the double \(G\times G\) for the action of \(G\times G\) through \((g_1,g_2)\cdot (a,b)=(g_1ag_2^{-1} , g_2bg_1^{-1})\) and the map \(\Phi \colon (a,b) \mapsto (ab,a^{-1}b^{-1})\).
With this in mind, a first objective of the present paper is to extend the quasi-Hamiltonian and quasi-Poisson theories to the case of an arbitrary \(\mathrm{Ad}\)-invariant symmetric form on the Lie algebra \(g\) of \(G\), where \(G\) is either a Lie group (in the real or complex analytic setting) or an algebraic group (over an algebraically closed field of characteristic 0). A second objective consists in reinterpreting the Poisson structure of wild character varieties. In those cases, \(G\) is a complex reductive group, and the varieties are constructed using a possibly not non-degenerate quasi-Poisson bivector and a possibly degenerate symmetric form on \(g\), thus generalizing the approach of [P. Boalch, Ann. Math. (2) 179, 301–365 (2014; Zbl 1283.53075)].
The main result of the paper can be outlined as follows. In Section 4, weakly \(G\)-quasi-Hamiltonian manifolds \((M,\sigma,\Phi)\) are introduced as generalizations of quasi-Hamiltonian spaces for any \(\mathrm{Ad}\)-invariant symmetric form on \(g\) and any (real, complex analytic, or algebraic) setting. Similarly, \(G\)-quasi-Poisson manifolds \((M,P)\) with a \(G\)-momentum mapping \(\Phi:M\to G\) are defined in Section 5 to generalize Hamiltonian quasi-Poisson manifolds. In both situations, important notions are extended, e.g., the structure of the double \(G\times G\) (cf. §4.3 and §6.3-6.5), fusion (cf. §4.4 and §6.2), and reduction (cf. §4.6 and §5.5). It is also explained how interesting examples that lead to (wild) character varieties can be built using the extended moduli space point of view, see §4.7 and Section 8. In order to relate the weakly quasi-Hamiltonian and quasi-Poisson approaches, one should consider a momentum duality presented in Section 7. In particular, it is proved that when the chosen symmetric form on \(g\) is non-degenerate, there is an equivalence between the two types of structures (Theorem 7.9). This last result is proved using methods from Dirac geometry inspired by A. Alekseev et al. [Astérisque 327, 131–199 (2009; Zbl 1251.53052)].