Dual Hodge decompositions and derived Poisson brackets

Abstract

Abstract. We study general properties of Hodge-type decompositions of cyclic and Hochschild homology of universal enveloping algebras of (DG) Lie algebras. Our construction generalizes the operadic construction of cyclic homology of Lie algebras due to Getzler and Kapranov [19]. We give a topological interpretation of such Lie Hodge decompositions in terms of S 1 -equivariant homology of the free loop space of a simply connected topological space. We prove that the canonical derived Poisson structure on a universal enveloping algebra arising from a cyclic pairing on the Koszul dual coalgebra preserves the Hodge filtration on cyclic homology. As an application, we show that the Chas-Sullivan Lie algebra of any simply connected closed manifold carries a natural Hodge filtration. We conjecture that the Chas-Sullivan Lie algebra is actually graded, i.e. the string topology bracket preserves the Hodge decomposition.

Representation homology of simply connected spaces

“…For $$j\,\in \,\mathbb {N}$$ and $$\omega \,\in \,\overline{\mathrm{HC}}^{(m)}_\ast (\mathcal {A})$$, the coefficient of $$z^{j-1}$$ (respectively, $$z^{j-1}dz$$) in $$S(\omega )$$ (respectively, $$E(\omega )$$) determines a homogeneous linear functional $$\eta _j$$ (respectively, $$\nu _j$$) on $$\overline{\mathrm{HC}}^{(m)}_\ast (\mathcal {A})$$ of homological degree $$p+m(p-1)+d(j-1)$$ (respectively, $$m(p-1)+dj$$). The desired lemma now follows from [3, Proposition 7.8; 7, Theorem 4.2], which together imply …”

“…The next result proven in [7] provides a topological interpretation of the Lie–Hodge homology. Theorem The Goodwillie‐Jones isomorphism (4.7) restricts to isomorphisms $$\begin{equation*} \mathrm{HC}^{(d)}_{\ast }(\mathfrak {a}_X) \cong {\overline{\mathrm{H}}}^{S^1, \,(d-1)}_{\ast }({\mathcal L}X, \mathbb {Q})\ ,\quad \forall \, d\geqslant 1.$$…”

“…Hence, $$\overline{\mathrm{HC}}^{(0)}_\ast (\mathcal {A})\,\cong \,z\mathbb {C}[z]$$ and $$\overline{\mathrm{HC}}^{(i)}_\ast (\mathcal {A})=0$$ for $$i>0$$. The desired lemma now follows from [3, Proposition 7.8; 7, Theorem 4.2], which together imply $$\begin{equation*} \overline{\mathrm{H}}_\ast ^{S^1,(m)}({\mathcal L}X)\,\cong \,{\left(\overline{\mathrm{HC}}^{(m)}_\ast (\mathcal {A})\right)}^{\ast }[-1]. \end{equation*}$$$$\Box$$…”

“…Clearly, $$|\xi _{j}|=(d(r+1)-2)m+dj-1$$. Since $$\begin{equation*} \overline{\mathrm{H}}^{S^1,(m)}_\ast ({\mathcal L}X)\,\cong \,{\left(\overline{\mathrm{HC}}^{(m)}_\ast (\mathcal {A})\right)}^{\ast }[-1] \end{equation*}$$by [3, Proposition 7.8; 7, Theorem 4.2], the desired lemma follows.$$\Box$$…”

Derived Character Maps of Groups Representations

“…It follows that lim Sketch of proof of Corollary 6.1. Moreover (see [15,Section 4.2] for example), L X may be chosen so that its universal enveloping algebra U L X is equipped with a derived Poisson structure inducing the Chas-Sullivan bracket on its (reduced) cyclic homology (which is isomorphic to H S 1 * (LX; k)). More precisely, L X may be chosen to be Koszul dual to the (graded linear dual of) the Lambrechts-Stanley model of X (see [43]), which is equipped with a cyclic pairing.…”