8. Deformation Theory with Homotopy Algebra Structures on Tensor Products

  • Daniel Robert-Nicoud Universit\'e Paris 13, Lab. Analyse, G\'eometrie et Applications, 99 Avenue Jean Baptiste Cl\'ement, 93430 Villetaneuse, France
Keywords: Operads, homotopy Lie algebras, homotopy associative algebras, infinity-morphisms, Maurer--Cartan elements

Abstract

In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras and algebras. When considering tensor products, such algebraic structures extend the Lie algebra or associative algebra structures that can be obtained by means of the Manin products of operads. These new homotopy algebra structures are proven to be compatible with the concepts of homotopy theory: $\infty$-morphisms and the Homotopy Transfer Theorem. We give a conceptual interpretation of their Maurer{\textendash}Cartan elements. In the end, this allows us to construct the deformation complex for morphisms of algebras over an operad and to represent the deformation $\infty$-groupoid for differential graded Lie algebras.

References

\bibitem[BL15]{brown15}
C.~Brown and A.~Lazarev.
\newblock Unimodular homotopy algebras and {C}hern--{S}imons theory.
\newblock {\em Journal of Pure and Applied Algebra}, 219(11):5158--5194, 2015.
\newblock \href{https://arxiv.org/abs/1309.3219v3}{arXiv:1309.3219v3}.
DOI 10.1016/j.jpaa.2015.05.017;
zbl 06448470;
MR3351579

\bibitem[DR15]{dolgushev15}
V.~A. Dolgushev and C.~L. Rogers.
\newblock A version of the {G}oldman--{M}illson theorem for filtered
{$L_\infty$}-algebras.
\newblock {\em Journal of Algebra}, 430:260--302, 2015.
\newblock \href{https://arxiv.org/abs/1407.6735}{arXiv:1407.6735}.
DOI 10.1016/j.jalgebra.2015.01.032;
zbl 1327.17019;
MR3323983

\bibitem[GCTV12]{GCTV12}
I.~Galvez-Carrillo, A.~Tonks, and B.~Vallette.
\newblock Homotopy {B}atalin--{V}ilkovisky algebras.
\newblock {\em Journal of Noncommutative Geometry}, 6(3):539--602, 2012.
\newblock \href{https://arxiv.org/abs/0907.2246}{arXiv:0907.2246}.
DOI 10.4171/JNCG/99;
zbl 1258.18005;
MR2956319

\bibitem[Get09]{getzler09}
E.~Getzler.
\newblock Lie theory for nilpotent {$L_\infty$}-algebras.
\newblock {\em Annals of Mathematics}, 170(1):271--301, 2009.
\newblock \href{https://arxiv.org/abs/math/0404003}{arXiv:math/0404003}.
DOI 10.4007/annals.2009.170.271;
zbl 1246.17025;
MR2521116

\bibitem[GJ94]{getzler94}
E.~Getzler and J.~D.~S. Jones.
\newblock Operads, homotopy algebra and iterated integrals for double loop
spaces.
\newblock 1994.
\newblock \href{https://arxiv.org/abs/hep-th/9403055}{arXiv:hep-th/9403055}.

\bibitem[GK94]{ginzburg94}
V.~Ginzburg and M.~Kapranov.
\newblock Koszul duality for operads.
\newblock {\em Duke {M}athematical {J}ournal}, 76(1):203--272, 1994.
\newblock \href{https://arxiv.org/abs/0709.1228}{arXiv:{ }0709.1228}.
DOI 10.1215/S0012-7094-94-07608-4;
zbl 0855.18006;
MR1301191

\bibitem[Hin97a]{hinich97}
V.~Hinich.
\newblock Descent of {D}eligne groupoids.
\newblock {\em International Mathematics Research Notices}, 1997(5):223--239,
1997.
\newblock
\href{https://arxiv.org/abs/alg-geom/9606010}{arXiv:alg-geom/9606010}.
DOI 10.1155/S1073792897000160;
zbl 0948.22016;
MR1439623

\bibitem[Hin97b]{hinich97homological}
V.~Hinich.
\newblock Homological algebra of homotopy algebras.
\newblock {\em Communications in Algebra}, 25(10):3291--3323, 1997.
\newblock \href{https://arxiv.org/abs/q-alg/9702015}{arXiv:q-alg/9702015}.
DOI 10.1080/00927879708826055;
zbl 0894.18008;
MR1465117

\bibitem[Kad88]{kadeishvili88}
T.~Kadeishvili.
\newblock The ${A}(\infty)$-algebra structure and {H}ochschild and {H}arrison
cohomology.
\newblock {\em Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR},
91:19--27, 1988.
\newblock \href{https://arxiv.org/abs/math/0210331}{arXiv:math/0210331}.
zbl 0717.55011;
MR1029003

\bibitem[Kon03]{kontsevich03}
M.~Kontsevich.
\newblock Deformation quantization of {P}oisson manifolds.
\newblock {\em Letters in Mathematical Physics}, 66(3):157--216, 2003.
\newblock \href{https://arxiv.org/abs/q-alg/9709040}{arXiv:q-alg/9709040}.
DOI 10.1023/B:MATH.0000027508.00421.bf;
zbl 1058.53065;
MR2062626

\bibitem[Kon17]{kontsevich17}
M.~Kontsevich.
\newblock Derived {G}rothendieck--{T}eichm{\"u}ller group and graph complexes
(after {T}. {W}illwacher).
\newblock {\em Report of the {S}{\'e}minaire {N}icolas {B}ourbaki},
2016--2017(1126), 2017.

\bibitem[Lur14]{lurie14}
J.~Lurie.
\newblock Higher algebra.
\newblock Draft available from the author’s website, 2014.

\bibitem[LV12]{vallette12}
J.~L. Loday and B.~Vallette.
\newblock {\em Algebraic Operads}, volume 346 of {\em Grundlehren der
{M}athematischen {W}issenschaften}.
\newblock Springer Verlag, 2012.
DOI 10.1007/978-3-642-30362-3;
zbl 1260.18001;
MR2954392

\bibitem[Man87]{manin87}
Y.~I. Manin.
\newblock Some remarks on {K}oszul algebras and quantum groups.
\newblock {\em Universit\'e de Grenoble. Annales de l'Institut Fourier},
37(4):191--205, 1987.
DOI 10.5802/aif.1117;
zbl 0625.58040;
MR0927397

\bibitem[Man88]{manin89}
Y.~I. Manin.
\newblock Quantum groups and noncommutative geometry.
\newblock {\em Universit\'e de {M}ontr\'eal, {C}entre de {R}echerches
{M}ath\'ematiques, {M}ontreal, {QC}}, 1988.
MR1016381

\bibitem[Pri10]{pridham10}
J.~P. Pridham.
\newblock Unifying derived deformation theories.
\newblock {\em Advances in Mathematics}, 224(3):772--826, 2010.
\newblock \href{https://arxiv.org/abs/0705.0344}{arXiv:0705.0344}.
DOI 10.1016/j.aim.2009.12.009;
zbl 1195.14012;
MR2628795

\bibitem[Qui70]{quillen70}
D.~Quillen.
\newblock On the (co-)homology of commutative rings.
\newblock In {\em Proc. Symp. Pure Math}, volume~17, pages 65--87, 1970.
zbl 0234.18010;
MR0257068

\bibitem[RN17]{rn17cosimplicial}
D.~Robert-Nicoud.
\newblock Representing the {D}eligne--{H}inich--{G}etzler $\infty$-groupoid.
\newblock 2017.
\newblock \href{https://arxiv.org/abs/1702.02529}{arXiv:1702.02529}.

\bibitem[TW15]{turchin15}
V.~Turchin and T.~Willwacher.
\newblock Hochshild--{P}irashvili homology on suspensions and representations
of {$Out(F_n)$}.
\newblock {\em Annales scientifiques de l'{ENS} (to appear)}, 2015.
\newblock \href{https://arxiv.org/abs/1507.08483}{arXiv:1507.08483}.
MR3653316

\bibitem[Val08]{vallette08}
B.~Vallette.
\newblock Manin products, {K}oszul duality, {L}oday algebras and {D}eligne
conjecture.
\newblock {\em Journal f{\"u}r die reine und angewandte {M}athematik ({C}relles
{J}ournal)}, 620:105--164, 2008.
\newblock \href{https://arxiv.org/abs/0609002}{arXiv:0609002}.
DOI 10.1515/CRELLE.2008.051;
zbl 1159.18001;
MR2427978

\bibitem[Wie16]{wierstra16}
F.~Wierstra.
\newblock Algebraic {H}opf invariants and rational models for mapping spaces.
\newblock 2016.
\newblock \href{https://arxiv.org/abs/1612.07762}{arXiv:math/1612.07762}.
Published
2018-05-08
Section
Unassigned Articles