Abdelkader BEN HASSINE, Taoufik CHTIOUI, Sami MABROUK, Mohamed Ali MAALAOUI

On Hom-F-manifold algebras and quantization

The notion of a F -manifold algebras is an algebraic description of a F -manifold. In this paper, we introduce the notion of Hom-F -manifold algebras which is generalisation of F -manifold algebras and Hom-Poisson algebras. We develop the representation theory of Hom-F -manifold algebras and generalize the notion of Hom-pre-Poisson algebras by introducing the Hom-pre-F -manifold algebras which give rise to a Hom-F -manifold algebra through the subadjacent commutative Hom-associative algebra and the subadjacent Hom-Lie algebra. Using O -operators on a Hom-F -manifold algebras we construct a Hom-pre-F -manifold algebras on a module. Then, we study Hom-pre-Lie formal deformations of commutative Hom-associative algebra and we prove that Hom-F -manifold algebras are the corresponding semiclassical limits. Finally, we study Hom-Lie infinitesimal deformations and extension of Hom-pre-Lie n-deformation to Hom-preLie (n + 1)-deformation of a commutative Hom-associative algebra via cohomology theory.

PDF

___

[1] Aizawa N, Sato H. q-deformation of the Virasoro algebra with central extension. Physics Letters B 1991; 256 (2): 185-190.
[2] Aguiar M. Pre-poisson algebras. Letters in Mathematical Physics 2000; 54 (4): 263-277.
[3] Ammar F, Mabrouk S, Makhlouf A. Representations and cohomology of n-ary multiplicative Hom-Nambu-Lie algebras. Journal of Geometry and Physics 2011;61 (10):1898-913.
[4] Bai C, Guo L, Ni X. Generalizations of the classical Yang-Baxter equation and O-operators. Journal of mathematical physics 2011; 52 (6): 063515.
[5] Bai C, Guo L, Ni X. Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and Post-Lie algebras. Communications in Mathematical Physics 2010; 297(2): 553-96.
[6] Benayadi S, Makhlouf A. Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. Journal of Geometry and Physics 2014; 76: 38-60.
[7] Baxter G. An analytic problem whose solution follows from a simple algebraic identity. Pacific Journal of Mathematics 1960; 10 (3): 731-742.
[8] Cai L, Sheng Y, Purely Hom-Lie bialgebras. Science China Mathematics 2018; 61 (9): 1553-1566.
[9] Connes A, Kreimer D, Hopf algebras. renormalisation and noncommutative geometry. Quantum field theory: perspective and prospective. Springer, Dordrecht 1999: 59-109.
[10] Cartier P. On the structure of free Baxter algebras. Advances in Mathematics 1972 ; 9 (2): 253-65.
[11] David L, Strachan IAB. Dubrovin’s duality for F -manifolds with eventual identities. Advances in Mathematics 2011; 22 (5): 4031-4060.
[12] David L. Strachan IAB. Compatible metrics on a manifold and nonlocal bi-Hamiltonian structures. International Mathematics Research Notices 2004; 66 :3533-3557.
[13] Dotsenko V. Algebraic structures of F-manifolds via pre-Lie algebras. Annali di Matematica Pura ed Applicata 2019; 198 (2): 517-27.
[14] Ebrahimi-Fard K, Guo L, Kreimer D. Spitzer’s identity and the algebraic Birkhoff decomposition in pQFT. Journal of Physics A: Mathematical and General 2004; 37 (45): 11037-11052.
[15] Ebrahimi-Fard K, Guo L, Manchon D. Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion. Communications in mathematical physics 2006; 267 (3): 821-45.
[16] Guo L. What is a Rota-Baxter algebra. Notices of the AMS. 2009; 56 (11): 1436-1437.
[17] Guo L. Introduction to Rota-Baxter Algebra. International Press and Higher Education Press 2012.
[18] Guo L, Keigher W. Baxter algebras and shuffle products. Advances in Mathematics 2000; 150: 117-149.
[19] Guo L, Zhang B. Renormalization of multiple zeta values. Journal of Algebra 2008; 319 (9): 3770-809.
[20] Guo S, Zhang X, Wang S. Manin triples and quasitriangular structures of Hom-Poisson bialgebras. arXiv preprint arXiv 2018:1807.06412.
[21] Hartwig JT, Larsson D, Silvestrov SD. Deformations of Lie algebras using σ - derivations. Journal of Algebra 2006; 295 (2): 314-361.
[22] Hertling C. Frobenius Manifolds and Moduli Spaces for Singularities. Cambridge University Press: 2002.
[23] Hertling C, Manin YI. Weak Frobenius manifolds. International Mathematics Research Notices 1999; 6 : 277-286.
[24] Hu N. q -Witt algebras, q -Lie algebras, q -holomorph structure and representations. Algebra Colloquim 1999; 6 (1): 51-70.
[25] Kupershmidt BA. What a classical r -matrix really is. Journal of Nonlinear Mathematical Physics 1999; 6 (4): 448-488.
[26] Larsson D, Silvestrov SD. Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities. Journal of Algebra 2005; 288 (2): 321-44.
[27] Lee YP. Quantum K-theory, I: Foundations. Duke Mathematical Journal 2004; 12 1 (3): 389-424.
[28] Lorenzoni P, Pedroni M, Raimondo A. F -manifolds and integrable systems of hydrodynamic type. Archiv der Mathematik 2011; 47: 163-180.
[29] Liu S, Song L, Tang R. Representations and cohomologies of regular Hom-pre-Lie algebras. Journal of Algebra and its Applications, 19 (08): 2050149 (2020).
[30] Liu J, Sheng Y, Bai C. F-manifold algebras and deformation quantization via pre-Lie algebras. Journal of Algebra 2020; 559: 467-495.
[31] Makhlouf A, Silvestrov SD. Hom-algebras structures. Journal of Generalized Lie Theory and Applications 2008; 2 (2): 51–64.
[32] Makhlouf A, Silvestrov SD. Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras. Forum Mathematicum 2010; 22: 715-739.
[33] Makhlouf A, Silvestrov SD. Hom-algebras and Hom-coalgebras. Journal of Algebra and its Applications 2010;9(04): 553-589.
[34] Manchon D, Paycha S. Nested sums of symbols and renormalised multiple zeta values. International Mathematics Research Notices 2010; 24 : 4628-4697.
[35] Merkulov SA. Operads, deformation theory and F -manifolds. Vieweg Teubner Verlag 2004; 36 : 213-251.
[36] Ming D, Chen Z, Li J. F -manifold color algebras. arXiv 2020: 2101.00959.
[37] Rota GC. Baxter algebras and combinatorial identities I, II. Bulletin of the American Mathematical Society 1969; 75 (2): 325-334.
[38] Rota GC. Baxter operators, an introduction Gian-Carlo Rota on Combinatorics. Contemporary Mathematicians Springer 1995: 504-512.
[39] Sheng Y. Representations of hom-Lie algebras. Algebras and Representation Theory 2012; 15 (6): 1081-1098.
[40] Sheng Y, Chen D. Hom-Lie 2-algebras. Journal of Algebra 2013; 376 : 174-195.
[41] Yau D. Hom-algebras and homology. Journal of Lie Theory 2009; 19: 409-421.
[42] Yau D. Non-commutative hom-Poisson algebras. arXiv 2010 :1010.3408.
[43] Yau D. Hom-Malcev, Hom-alternative and Hom-Jordan algebras. International Journal of Algebra 2012; 11: 177- 217.