Hochschild homology of structured algebras

Wahl, Nathalie; Westerland, Craig

doi:10.48550/arxiv.1110.0651

Cited by 5 publications

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“…Example 3.11 (Hochschild homology). Let A be an A ∞ -Frobenius algebra of degree d. Following Costello [12], Wahl and Westerland [44] construct an HCFT of degree d for which…”

Section: 2mentioning

confidence: 99%

“…The proposition shows that not every homological conformal field theory can be extended to an h-graph theory. For example, applying the results of Wahl and Westerland [44] to the group ring F[G] of a finite nonabelian group G produces a homological conformal field theory φ with φ…”

Section: Algebraic Consequencesmentioning

confidence: 99%

“…Using the above formulas to decompose fun(P 0 |{p}, G) ≈ G and fun(P 0 , G) ≈ G 2 into orbits under the respective actions, we obtain the following commutative diagram of homeomorphisms and homotopy equivalences. Here the bottom horizontal arrow is the one appearing in diagram (44) and the upper horizontal arrow is given by the projection of G 2 onto its first coordinate and the projection of Z/2 × G {p,q} onto Z/2 × G {p} ; O(g) denotes the orbit of the element g ∈ G in the Z/2 × G {p} -action on G, and likewise O(g, e) denotes the orbit of the element (g, e) ∈ G 2 in the Z/2 × G {p,q} -action on G 2 ; the disjoint unions in the top pentagon run over a set of representatives for the orbits of the Z/2 × G {p} -action on G; the upper vertical maps are induced by the inclusions of O(g) into G and O(g, e) into G 2 ; the top oblique arrow on the left is given by the inclusions of C e g into Z/2 × G {p} and the maps that send pt to g ∈ O(g); and finally the top oblique arrow on the right is given is given by the homomorphisms C e g − − → Z/2 × G {p,q} , (ε, g p ) −→ (ε, g p , g p g [ε] ) and the maps that send pt to the element (g, e) ∈ O(g, e). The claim now follows by combining diagrams (44) and (45).…”

Section: An Example Of Non-trivial Higher Operationsmentioning

confidence: 99%

“…The study of algebraic structures on Hochschild homology of algebras with appropriate structure has developed in parallel with string topology, and sometimes in direct connection with it. See for example the work of Cohen and Jones [11], Costello [12], Tradler and Zeinalian [40,41], Félix, Thomas and Vigué-Poirrier [18], Wahl and Westerland [44] and Wahl [43].…”

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On string topology of classifying spaces

Hepworth

Lahtinen

2015

Advances in Mathematics

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Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.Comment: 93 pages; v4: minor changes; to appear in Advances in Mathematic

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“…Example 3.11 (Hochschild homology). Let A be an A ∞ -Frobenius algebra of degree d. Following Costello [12], Wahl and Westerland [44] construct an HCFT of degree d for which…”

Section: 2mentioning

confidence: 99%

Section: Algebraic Consequencesmentioning

confidence: 99%

Section: An Example Of Non-trivial Higher Operationsmentioning

confidence: 99%

mentioning

confidence: 99%

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On string topology of classifying spaces

Hepworth

Lahtinen

2015

Advances in Mathematics

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“…A (unital) commutative differential graded algebra is a strong symmetric monoidal functor C om → Ch. In [WW11], the Hochschild complex C(Φ) for general functors Φ : C om → Ch is defined as…”

Section: Introductionmentioning

confidence: 99%

The complex of formal operations on the Hochschild chains of commutative algebras

Klamt

2014

Journal of the London Mathematical Society

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Universal operations in Hochschild homology

Wahl

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We provide a general method for finding all natural operations on the Hochschild complex of E-algebras, where E is any algebraic structure encoded in a prop with multiplication, as for example the prop of Frobenius, commutative or A∞-algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex of formal operations, for which we provide an explicit model that we can calculate in a number of cases. When E encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasi-isomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and Kontsevich-Soibelman via different methods identify with all formal operations. When E encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by TradlerZeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of non-trivial operations and use these to produce non-trivial higher string topology operations, which had so far been elusive.

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Hochschild homology of structured algebras

Cited by 5 publications

References 0 publications

On string topology of classifying spaces

On string topology of classifying spaces

The complex of formal operations on the Hochschild chains of commutative algebras

Universal operations in Hochschild homology

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