Deformations and Cohomologies of Relative Rota–Baxter Operators on Lie Algebroids and Koszul–Vinberg Structures

Meijun Liu ; Jiefeng Liu ; Yunhe Sheng

Symmetry, integrability and geometry: methods and applications, Tome 18 (2022)

Résumé

Given a Lie algebroid with a representation, we construct a graded Lie algebra whose Maurer–Cartan elements characterize relative Rota–Baxter operators on Lie algebroids. We give the cohomology of relative Rota–Baxter operators and study infinitesimal deformations and extendability of order $n$ deformations to order $n+1$ deformations of relative Rota–Baxter operators in terms of this cohomology theory. We also construct a graded Lie algebra on the space of multi-derivations of a vector bundle whose Maurer–Cartan elements characterize left-symmetric algebroids. We show that there is a homomorphism from the controlling graded Lie algebra of relative Rota–Baxter operators on Lie algebroids to the controlling graded Lie algebra of left-symmetric algebroids. Consequently, there is a natural homomorphism from the cohomology groups of a relative Rota–Baxter operator to the deformation cohomology groups of the associated left-symmetric algebroid. As applications, we give the controlling graded Lie algebra and the cohomology theory of Koszul–Vinberg structures on left-symmetric algebroids.

Détail

Keywords: cohomology, deformation, Rota–Baxter operator, left-symmetric algebroid.
Mots-clés : Lie algebroid, Koszul–Vinberg structure

Bibliographie

[1] Abouqateb A., Boucetta M., Bourzik C., “Contravariant pseudo-Hessian manifolds and their associated Poisson structures”, Differential Geom. Appl., 70 (2020), 101630, 26 pp., arXiv: 2020.10163 | DOI | MR

[2] Bai C., “Left-symmetric bialgebras and an analogue of the classical Yang–Baxter equation”, Commun. Contemp. Math., 10 (2008), 221–260, arXiv: 0708.1551 | DOI | MR

[3] Baxter G., “An analytic problem whose solution follows from a simple algebraic identity”, Pacific J. Math., 10 (1960), 731–742 | DOI | MR

[4] Benayadi S., Boucetta M., “On para-Kähler Lie algebroids and contravariant pseudo-Hessian structures”, Math. Nachr., 292 (2019), 1418–1443, arXiv: 1610.09682 | DOI | MR

[5] Burde D., “Left-symmetric algebras, or pre-Lie algebras in geometry and physics”, Cent. Eur. J. Math., 4 (2006), 323–357 | DOI | MR

[6] Cartier P., “On the structure of free Baxter algebras”, Adv. Math., 9 (1972), 253–265 | DOI | MR

[7] Chapoton F., Livernet M., “Pre-Lie algebras and the rooted trees operad”, Int. Math. Res. Not., 2001 (2001), 395–408, arXiv: math.QA/0002069 | DOI | MR

[8] Connes A., Kreimer D., “Renormalization in quantum field theory and the Riemann–Hilbert problem. I The Hopf algebra structure of graphs and the main theorem”, Comm. Math. Phys., 210 (2000), 249–273, arXiv: hep-th/9912092 | DOI | MR

[9] Crainic M., Moerdijk I., “Deformations of Lie brackets: cohomological aspects”, J. Eur. Math. Soc., 10 (2008), 1037–1059, arXiv: math.DG/0403434 | DOI | MR

[10] Das A., “Deformations of associative Rota–Baxter operators”, J. Algebra, 560 (2020), 144–180, arXiv: 1909.08320 | DOI | MR

[11] Ebrahimi-Fard K., Manchon D., Patras F., “A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov's recursion”, J. Noncommut. Geom., 3 (2009), 181–222, arXiv: 0705.1265 | DOI | MR

[12] Frégier Y., “A new cohomology theory associated to deformations of Lie algebra morphisms”, Lett. Math. Phys., 70 (2004), 97–107, arXiv: math.QA/0304323 | DOI | MR

[13] Frégier Y., Zambon M., “Simultaneous deformations and Poisson geometry”, Compos. Math., 151 (2015), 1763–1790, arXiv: 1202.2896 | DOI | MR

[14] Frégier Y., Zambon M., “Simultaneous deformations of algebras and morphisms via derived brackets”, J. Pure Appl. Algebra, 219 (2015), 5344–5362, arXiv: 1301.4864 | DOI | MR

[15] Gerstenhaber M., “The cohomology structure of an associative ring”, Ann. of Math., 78 (1963), 267–288 | DOI | MR

[16] Gerstenhaber M., “On the deformation of rings and algebras”, Ann. of Math., 79 (1964), 59–103 | DOI | MR

[17] Guo L., “Properties of free Baxter algebras”, Adv. Math., 151 (2000), 346–374, arXiv: math.RA/0407157 | DOI | MR

[18] Guo L., An introduction to Rota–Baxter algebra, Surveys of Modern Mathematics, 4, International Press, Somerville, MA; Higher Education Press, Beijing, 2012 | MR

[19] Guo L., Lang H., Sheng Y., “Integration and geometrization of Rota–Baxter Lie algebras”, Adv. Math., 387 (2021), 107834, 34 pp., arXiv: 2009.03492 | DOI | MR

[20] Huebschmann J., “Poisson cohomology and quantization”, J. Reine Angew. Math., 408 (1990), 57–113, arXiv: 1303.3903 | DOI | MR

[21] Kontsevich M., “Deformation quantization of Poisson manifolds”, Lett. Math. Phys., 66 (2003), 157–216, arXiv: q-alg/9709040 | DOI | MR

[22] Kosmann-Schwarzbach Y., Magri F., “Poisson–Lie groups and complete integrability. I Drinfel'd bialgebras, dual extensions and their canonical representations”, Ann. Inst. H. Poincaré Phys. Théor., 49 (1988), 433–460 | MR

[23] Kupershmidt B.A., “What a classical $r$-matrix really is”, J. Nonlinear Math. Phys., 6 (1999), 448–488, arXiv: math.QA/9910188 | DOI | MR

[24] Laurent-Gengoux C., Pichereau A., Vanhaecke P., Poisson structures, Grundlehren der mathematischen Wissenschaften, 347, Springer, Heidelberg, 2013 | DOI | MR

[25] Lichnerowicz A., “Les variétés de Poisson et leurs algèbres de Lie associées”, J. Differential Geometry, 12 (1977), 253–300 | DOI | MR

[26] Liu J., Sheng Y., Bai C., “Left-symmetric bialgebroids and their corresponding Manin triples”, Differential Geom. Appl., 59 (2018), 91–111, arXiv: 1705.08299 | DOI | MR

[27] Liu J., Sheng Y., Bai C., “Pre-symplectic algebroids and their applications”, Lett. Math. Phys., 108 (2018), 779–804, arXiv: 1604.00146 | DOI | MR

[28] Liu J., Sheng Y., Bai C., Chen Z., “Left-symmetric algebroids”, Math. Nachr., 289 (2016), 1893–1908, arXiv: 1312.6526 | DOI | MR

[29] Liu Z.-J., Weinstein A., Xu P., “Manin triples for Lie bialgebroids”, J. Differential Geom., 45 (1997), 547–574, arXiv: dg-ga/9508013 | DOI | MR

[30] Mackenzie K.C.H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005 | DOI | MR

[31] Mackenzie K.C.H., Xu P., “Lie bialgebroids and Poisson groupoids”, Duke Math. J., 73 (1994), 415–452 | DOI | MR

[32] Mokri T., “Matched pairs of Lie algebroids”, Glasgow Math. J., 39 (1997), 167–181 | DOI | MR

[33] Nguiffo Boyom M., “Cohomology of Koszul–Vinberg algebroids and Poisson manifolds. I”, Lie Algebroids and Related Topics in Differential Geometry (Warsaw, 2000), Banach Center Publ., 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001, 99–110 | DOI | MR

[34] Nguiffo Boyom M., “KV-cohomology of Koszul–Vinberg algebroids and Poisson manifolds”, Internat. J. Math., 16 (2005), 1033–1061 | DOI | MR

[35] Nijenhuis A., “On a class of common properties of some different types of algebras. II”, Nieuw Arch. Wisk., 17 (1969), 87–108 | MR

[36] Nijenhuis A., Richardson Jr. R.W., “Cohomology and deformations in graded Lie algebras”, Bull. Amer. Math. Soc., 72 (1966), 1–29 | DOI | MR

[37] Nijenhuis A., Richardson Jr. R.W., “Deformations of homomorphisms of Lie groups and Lie algebras”, Bull. Amer. Math. Soc., 73 (1967), 175–179 | DOI | MR

[38] Nijenhuis A., Richardson Jr. R.W., “Commutative algebra cohomology and deformations of Lie and associative algebras”, J. Algebra, 9 (1968), 42–53 | DOI | MR

[39] Pei J., Bai C., Guo L., “Splitting of operads and Rota–Baxter operators on operads”, Appl. Categ. Structures, 25 (2017), 505–538, arXiv: 1306.3046 | DOI | MR

[40] Rota G.-C., “Baxter algebras and combinatorial identities. I”, Bull. Amer. Math. Soc., 75 (1969), 325–329 | DOI | MR

[41] Rota G.-C., “Baxter algebras and combinatorial identities. II”, Bull. Amer. Math. Soc., 75 (1969), 330–334 | DOI | MR

[42] Shima H., “Homogeneous Hessian manifolds”, Ann. Inst. Fourier (Grenoble), 30 (1980), 91–128 | DOI | MR

[43] Shima H., The geometry of Hessian structures, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007 | DOI | MR

[44] Tang R., Bai C., Guo L., Sheng Y., “Deformations and their controlling cohomologies of $\mathcal{O}$-operators”, Comm. Math. Phys., 368 (2019), 665–700, arXiv: 1803.09287 | DOI | MR

[45] Uchino K., “Twisting on associative algebras and Rota–Baxter type operators”, J. Noncommut. Geom., 4 (2010), 349–379, arXiv: 0710.4309 | DOI | MR

[46] Vitagliano L., “Representations of homotopy Lie–Rinehart algebras”, Math. Proc. Cambridge Philos. Soc., 158 (2015), 155–191, arXiv: 1304.4353 | DOI | MR

[47] Wang Q., Liu J., Sheng Y., “Koszul–Vinberg structures and compatible structures on left-symmetric algebroids”, Int. J. Geom. Methods Mod. Phys., 17 (2020), 2050199, 28 pp., arXiv: 2004.01774 | DOI | MR