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Keywords:
Functor category; polynomial functor; free group; Lie operad; PROP; Poincaré-Birkhoff-Witt
Functor category; polynomial functor; free group; Lie operad; PROP; Poincaré-Birkhoff-Witt
Summary:
Working over a field $k$ of characteristic zero, the category $F_{\omega}(gr^(op);k)$ of analytic contravariant functors on the category gr of finitely-generated free groups is shown to be equivalent to the category $F_{(Lie)}$ of representations of the $k$-linear category Cat Lie associated to the Lie operad. Two proofs are given of this result. The first uses the original Ginzburg-Kapranov approach to Koszul duality of binary quadratic operads and the fact that the category of analytic contravariant functors is Koszul. The second proof proceeds by making the equivalence explicit using the $k$-linear category $Cat USS^(u)$ associated to the operad $USS^(u)$ encoding unital associative algebras, which provides the ‘twisting bimodule’ between modules over Cat Lie and modules over $kgr^(op)$. A key ingredient is the Poincaré-Birkhoff-Witt theorem. Using the explicit formulation, it is shown how this equivalence reflects the tensor product on the category of analytic contravariant functors, relating this to the convolution product for representations of Cat Lie.
References:
[1] Batanin, Michael, Markl, Martin: Operadic categories as a natural environment for Koszul duality. Compositionality, Vol. 5, Iss. 3, 46 MR 4599796
[2] Beilinson, A. A., Ginsburg, V. A., Schechtman, V. V.: Koszul duality. J. Geom. Phys., Vol. 5, Iss. 3, 317-350, https://doi.org/10.1016/0393-0440(88)90028-9 DOI 10.1016/0393-0440(88)90028-9
[3] Beilinson, Alexander, Ginzburg, Victor, Soergel, Wolfgang: Koszul duality patterns in representation theory. J. Amer. Math. Soc., Vol. 9, Iss. 2, 473-527, https://doi.org/10.1090/S0894-0347-96-00192-0, DOI:10.1090/S0894-0347-96-00192-0 DOI 10.1090/S0894-0347-96-00192-0
[4] Chmutov, S., Duzhin, S., Mostovoy, J.: Introduction to Vassiliev knot invariants. Cambridge University Press, Cambridge, DOI:10.1017/CBO9781139107846 DOI 10.1017/CBO9781139107846 | MR 2962302
[5] Djament, Aurélien, Pirashvili, Teimuraz, Vespa, Christine: Cohomologie des foncteurs polynomiaux sur les groupes libres. Doc. Math., Vol. 21, 205-222 MR 3505136
[6] Djament, Aurélien, Vespa, Christine: Sur l’homologie des groupes d’automorphismes des groupes libres à coefficients polynomiaux. Comment. Math. Helv., Vol. 90, Iss. 1, 33-58, http://dx.doi.org/10.4171/CMH/345, DOI:10.4171/CMH/345 DOI 10.4171/cmh/345 | MR 3317332
[7] Eilenberg, Samuel, Mac Lane, Saunders: On the groups H(Π,n). II. Methods of computation. Ann. of Math. (2), Vol. 60, 49-139, https://doi.org/10.2307/1969702, DOI:10.2307/1969702 DOI 10.2307/1969702
[8] Fresse, Benoit: Lie theory of formal groups over an operad. J. Algebra, Vol. 202, Iss. 2, 455-511, https://doi.org/10.1006/jabr.1997.7280, DOI:10.1006/jabr.1997.7280 DOI 10.1006/jabr.1997.7280
[9] Fresse, Benoit: Modules over operads and functors. Lecture notes in mathematics, Springer-Verlag, Berlin, https://doi.org/10.1007/978-3-540-89056-0, ISBN:978-3-540-89055-3, DOI:10.1007/978-3-540-89056-0 DOI 10.1007/978-3-540-89056-0 | MR 2494775
[10] Ginzburg, Victor, Kapranov, Mikhail: Koszul duality for operads. Duke Math. J., Vol. 76, Iss. 1, 203-272, https://doi.org/10.1215/S0012-7094-94-07608-4, DOI:10.1215/S0012-7094-94-07608-4 DOI 10.1215/S0012-7094-94-07608-4
[11] Hartl, Manfred, Pirashvili, Teimuraz, Vespa, Christine: Polynomial functors from algebras over a set-operad and nonlinear Mackey functors. Int. Math. Res. Not. IMRN, Iss. 6, 1461-1554, https://doi.org/10.1093/imrn/rnt242, DOI:10.1093/imrn/rnt242 DOI 10.1093/imrn/rnt242 | MR 3340364
[12] Kapranov, M., Manin, Yu.: Modules and Morita theorem for operads. Amer. J. Math., Vol. 123, Iss. 5, 811-838 DOI 10.1353/ajm.2001.0033 | MR 1854112 | Zbl 1001.18004
[13] Kawazumi, Nariya, Vespa, Christine: On the wheeled PROP of stable cohomology of Aut(F_n) with bivariant coefficients. Algebr. Geom. Topol., Vol. 23, Iss. 7, 3089-3128, https://doi.org/10.2140/agt.2023.23.3089, DOI:10.2140/agt.2023.23.3089 DOI 10.2140/agt.2023.23.3089 | MR 4647673
[14] Loday, Jean-Louis, Vallette, Bruno: Algebraic operads. Grundlehren der mathematischen wissenschaften [fundamental principles of mathematical sciences], Springer, Heidelberg, https://doi.org/10.1007/978-3-642-30362-3, ISBN:978-3-642-30361-6, DOI:10.1007/978-3-642-30362-3 DOI 10.1007/978-3-642-30362-3 | MR 2954392
[15] Mac Lane, Saunders: Categorical algebra. Bull. Amer. Math. Soc., Vol. 71, 40-106, https://doi.org/10.1090/S0002-9904-1965-11234-4, DOI:10.1090/S0002-9904-1965-11234-4 DOI 10.1090/S0002-9904-1965-11234-4
[16] Macdonald, I. G.: Symmetric functions and Hall polynomials. Oxford classic texts in the physical sciences, The Clarendon Press, Oxford University Press, New York, ISBN:978-0-19-873912-8 MR 3443860
[17] Mitchell, Barry: Rings with several objects. Advances in Math., Vol. 8, 1-161, https://doi.org/10.1016/0001-8708(72)90002-3, DOI:10.1016/0001-8708(72)90002-3 DOI 10.1016/0001-8708(72)90002-3
[18] Pirashvili, Teimuraz: Hodge decomposition for higher order Hochschild homology. Ann. Sci. École Norm. Sup. (4), Vol. 33, Iss. 2, 151-179, http://dx.doi.org/10.1016/S0012-9593(00)00107-5, DOI:10.1016/S0012-9593(00)00107-5 DOI 10.1016/S0012-9593(00)00107-5
[19] Pirashvili, Teimuraz: On the PROP corresponding to bialgebras. Cah. Topol. Géom. Différ. Catég., Vol. 43, Iss. 3, 221-239 MR 1928233
[20] Powell, Geoffrey: On the Passi and the Mal’cev functors. arXiv:2309.07605, DOI:10.48550/arXiv.2309.07605 DOI 10.48550/arXiv.2309.07605
[21] Powell, Geoffrey: Outer functors and a general operadic framework. J. Algebra, Vol. 644, 526-562, https://doi.org/10.1016/j.jalgebra.2024.01.015, DOI:10.1016/j.jalgebra.2024.01.015 DOI 10.1016/j.jalgebra.2024.01.015 | MR 4696223
[22] Powell, Geoffrey, Vespa, Christine: Higher Hochschild homology and exponential functors. arXiv:1802.07574 MR 4929346
[23] Turchin, Victor, Willwacher, Thomas: Hochschild-Pirashvili homology on suspensions and representations of Out(F_n). Ann. Sci. Éc. Norm. Supér. (4), Vol. 52, Iss. 3, 761-795, https://doi.org/10.24033/asens.2396, DOI:10.24033/asens.2396 DOI 10.24033/asens.2396 | MR 3982870
[24] Vespa, Christine: Extensions between functors from free groups. Bull. Lond. Math. Soc., Vol. 50, Iss. 3, 401-419, https://doi.org/10.1112/blms.12091, DOI:10.1112/blms.12091 DOI 10.1112/blms.12091 | MR 3829729
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