E2 structures and derived Koszul duality in
string topology

Abstract Let X be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$ -equivariant homology $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ of the free loop space of X preserves the Hodge decomposition of $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7].

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String topology for Lie groups

Journal of Topology ◽

10.1112/jtopol/jtq012 ◽

2010 ◽

Vol 3 (2) ◽

pp. 424-442 ◽

Cited By ~ 7

Author(s):

Richard A. Hepworth

Keyword(s):

Lie Groups ◽

String Topology

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Loop coproducts in string topology and triviality of higher genus TQFT operations

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2009.07.011 ◽

2010 ◽

Vol 214 (5) ◽

pp. 605-615 ◽

Cited By ~ 4

Author(s):

Hirotaka Tamanoi

Keyword(s):

String Topology ◽

Higher Genus

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KOSZUL DUALITY FOR STRATIFIED ALGEBRAS II. STANDARDLY STRATIFIED ALGEBRAS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788710001497 ◽

2010 ◽

Vol 89 (1) ◽

pp. 23-49 ◽

Cited By ~ 9

Author(s):

VOLODYMYR MAZORCHUK

Keyword(s):

Complete Picture ◽

Projective Modules ◽

Koszul Duality ◽

Tilting Theory

AbstractWe give a complete picture of the interaction between the Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras, imposing the condition that standard filtrations of projective modules are finite, and develop a tilting theory for such algebras. Under the assumption on existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both the Ringel and Koszul duals belong to the same class, and that these two dualities on this class commute.

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