EUDML

Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms of certain associated differential (bi)graded Lie and G(erstenhaber)-algebras; in particular the G-algebra arising from an almost complex structure is a (strict) d(ifferential) G-algebra...

Poisson structures on certain moduli spaces for bundles on a surface

Johannes Huebschmann — 1995

Annales de l'institut Fourier

Let $Σ$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$ , and $ξ : P \to Σ$ a principal $G$ -bundle. In earlier work we have shown that the moduli space $N (ξ)$ of central Yang-Mills connections, with reference to appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a homeomorphism from $N (ξ)$ onto a certain representation space ${Rep}_{ξ} (Γ, G)$ , in fact a diffeomorphism, with reference to suitable smooth structures $C^{\infty} (N (ξ))$ and $C^{\infty} ({Rep}_{ξ} (Γ, G))$ , where $Γ$ denotes the universal central extension of...

Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras

Johannes Huebschmann — 1998

Annales de l'institut Fourier

For any Lie-Rinehart algebra $(A, L)$ , B(atalin)-V(ilkovisky) algebra structures $\partial$ on the exterior $A$ -algebra $Λ_{A} L$ correspond bijectively to right $(A, L)$ -module structures on $A$ ; likewise, generators for the Gerstenhaber algebra $Λ_{A} L$ correspond bijectively to right $(A, L)$ -connections on $A$ . When $L$ is projective as an $A$ -module, given a B-V algebra structure $\partial$ on $Λ_{A} L$ , the homology of the B-V algebra $(Λ_{A} L, \partial)$ coincides with the homology of $L$ with coefficients in $A$ with reference to the right $(A, L)$ -module structure determined by $\partial$ . When...

Singular Poisson-Kähler geometry of certain adjoint quotients

Johannes Huebschmann — 2007

Banach Center Publications

The Kähler quotient of a complex reductive Lie group relative to the conjugation action carries a complex algebraic stratified Kähler structure which reflects the geometry of the group. For the group SL(n,ℂ), we interpret the resulting singular Poisson-Kähler geometry of the quotient in terms of complex discriminant varieties and variants thereof.