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Free loop spaces, Koszul duality and A-infinity algebras
Stockholm University, Faculty of Science, Department of Mathematics.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers on the topics of free loop spaces, Koszul duality and A-algebras. 

In Paper I we consider a definition of differential operators for noncommutative algebras. This definition is inspired by the connections between differential operators of commutative algebras, L-algebras and BV-algebras. We show that the definition is reasonable by establishing results that are analoguous to results in the commutative case. As a by-product of this definition we also obtain definitions for noncommutative versions of Gerstenhaber and BV-algebras. 

In Paper II we calculate the free loop space homology of (n-1)-connected manifolds of dimension of at least 3n-2. The Chas-Sullivan loop product and the loop bracket are calculated. Over a field of characteristic zero the BV-operator is determined as well. Explicit expressions for the Betti numbers are also established, showing that they grow exponentially. 

In Paper III we restrict our coefficients to a field of characteristic 2. We study the Dyer-Lashof operations that exist on free loop space homology in this case. Explicit calculations are carried out for manifolds that are connected sums of products of spheres. 

In Paper IV we extend the Koszul duality methods used in Paper II by incorporating A-algebras and A-coalgebras. This extension of Koszul duality enables us to compute free loop space homology of manifolds that are not necessarily formal and coformal. As an example we carry out the computations for a non-formal simply connected 7-manifold. 

Abstract [sv]

Denna avhandling består av fyra artiklar inom ämnena fria öglerum, Koszuldualitet och A-algebror.

I Artikel I behandlar vi en definition av differentialoperatorer för ickekommutativa algebror. Denna definition är inspirerad av kopplingar mellan differentialoperatorer för kommutativa algebror, L-algebror och BV-algebror. Vi visar att definitionen är rimlig genom att etablera resultat som är analoga med resultat i det kommutativa fallet. Som en biprodukt får vi också definitioner för ickekommutativa varianter av Gerstenhaber och BV-algebror.

I Artikel II beräknar vi den fria öglerumshomologin av (n-1)-sammanhängande mångfalder av dimension minst 3n-2. Chas-Sullivans ögleprodukt och öglehake beräknas. Över en kropp av karakteristik noll beräknas även BV-operatorn. Explicita uttryck för Bettitalen fastställs också, vilka visar att de växer exponentiellt.

I Artikel III begränsar vi koefficienterna till en kropp av karakteristik 2. Vi studerar Dyer- Lashofoperationer som existerar på den fria öglerumshomologin i detta fall. Explicita beräkningar görs för mångfalder som är sammanhängande summor av produkter av sfärer.

I Artikel IV utvidgar vi Koszuldualitetmetoden som används i Artikel II genom att inkorporera A-algebror och A-koalgebror. Denna utvidgning av Koszuldualitet gör det möjligt att beräkna fri öglerumshomologi för mångfalder som inte nödvändigtvis är formella och koformella. Som ett exempel utför vi beräkningar för en ickeformell enkelt sammanhängande 7-mångfald. 

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2017. , 17 p.
Keyword [en]
Koszul duality, free loop spaces, a-infinity algebras, BV-algebras, string topology
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-145403ISBN: 978-91-7649-925-2 (print)ISBN: 978-91-7649-926-9 (electronic)OAI: oai:DiVA.org:su-145403DiVA: diva2:1128921
Public defence
2017-09-22, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.

Available from: 2017-08-30 Created: 2017-07-31 Last updated: 2017-08-21Bibliographically approved
List of papers
1. A Algebras Derived from Associative Algebras with a Non-Derivation Differential
Open this publication in new window or tab >>A Algebras Derived from Associative Algebras with a Non-Derivation Differential
2015 (English)In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 9, no 1, 214Article in journal (Refereed) Published
Abstract [en]

Given an associative graded algebra equipped with a degree +1 differential delta we define an A-structure that measures the failure of delta to be a derivation. This can be seen as a non-commutative analog of generalized BV-algebras. In that spirit we introduce a notion of associative order for the operator delta and prove that it satisfies properties similar to the commutative case. In particular when it has associative order 2 the new product is a strictly associative product of degree +1 and there is compatibility between the products, similar to ordinary BV-algebras. We consider several examples of structures obtained in this way. In particular we obtain an A-structure on the bar complex of an A-algebra that is strictly associative if the original algebra is strictly associative. We also introduce strictly associative degree +1 products for any degree +1 action on a graded algebra. Moreover, an A∞-structure is constructed on the Hochschild cocomplex of an associative algebra with a non-degenerate inner product by using Connes’ B-operator.

Keyword
a-infinity algebras, Hochschild complex, BV-algebras, coassociative coalgebra
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-145397 (URN)10.4172/1736-4337.1000214 (DOI)
Available from: 2017-07-31 Created: 2017-07-31 Last updated: 2017-08-03Bibliographically approved
2. Free loop space homology of highly connected manifolds
Open this publication in new window or tab >>Free loop space homology of highly connected manifolds
2017 (English)In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 29, no 1, 201-228 p.Article in journal (Refereed) Published
Abstract [en]

We calculate the homology of the free loop space of (n - 1)-connected closed manifolds of dimension at most 3 n - 2 (n >= 2), with the Chas-Sullivan loop product and loop bracket. Over a field of characteristic zero, we obtain an expression for the BV-operator. We also give explicit formulas for the Betti numbers, showing they grow exponentially. Our main tool is the connection between formality, coformality and Koszul algebras that was elucidated by the first author [6].

Keyword
Free loop spaces, Koszul algebras, String topology, BV-algebras, Hochschild cohomology
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-138911 (URN)10.1515/forum-2015-0074 (DOI)000391192600010 ()
Available from: 2017-01-30 Created: 2017-01-30 Last updated: 2017-07-31Bibliographically approved
3. Restricted Gerstenhaber algebra structure on the free loop homology of (Sn×Sn)#m
Open this publication in new window or tab >>Restricted Gerstenhaber algebra structure on the free loop homology of (Sn×Sn)#m
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We compute the 2-primary restricted Gerstenhaber algebra structure on the free loop homology of (Sn×Sn)#m. To this end we construct a small complex with an explicit retract from the Hochschild cohomology complex of the cohomology algebra. The methods involved are Koszul duality, PBW-bases and the perturbation lemma. 

Keyword
Dyer-Lashof operations, string topology, Gerstenhaber algebra
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-145399 (URN)
Available from: 2017-07-31 Created: 2017-07-31 Last updated: 2017-08-03Bibliographically approved
4. Koszul A-algebras and free loop space homology
Open this publication in new window or tab >>Koszul A-algebras and free loop space homology
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We introduce a notion of Koszul A-algebra that generalizes Priddy’s notion of a Koszul algebra and we use it to construct small A- algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples. 

Keyword
Koszul duality, string topology, free loop spaces, a-infinity algebras
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-145400 (URN)
Available from: 2017-07-31 Created: 2017-07-31 Last updated: 2017-08-03Bibliographically approved

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