Sergei SILVESTROV, Abdoreza ARMAKAN, Mohammad Reza FARHADPOOR

Enveloping algebras of color hom-Lie algebras

In this paper, the universal enveloping algebra of color hom-Lie algebras is studied. A construction of the freeinvolutive hom-associative color algebra on a hom-module is described and applied to obtain the universal envelopingalgebra of an involutive hom-Lie color algebra. Finally, the construction is applied to obtain the well-known Poincaré–Birkhoff–Witt theorem for Lie algebras to the enveloping algebra of an involutive color hom-Lie algebra.

PDF

___

[1] Abdaoui K, Ammar F, Makhlouf A. Constructions and cohomology of hom-Lie color algebras. Comm Algebra 2015; 43: 4581-4612.
[2] Abramov V, Le Roy B, Kerner R. Hypersymmetry: a Z3 -graded generalization of supersymmetry. J Math Phys 1997; 38: 1650-1669.
[3] Aizawa N, Sato H. q -deformation of the Virasoro algebra with central extension. Phys Lett B 1991; 256: 185-190.
[4] Ammar F, Makhlouf A. Hom-Lie Superalgebras and Hom-Lie admissible superalgebras. J Algebra 2010; 324: 1513- 1528.
[5] Ammar F, Ejbehi Z, Makhlouf A. Cohomology and deformations of Hom-algebras. J Lie Theory 2011; 21: 813-836.
[6] Ammar F, Mabrouk S, Makhlouf A. Representations and cohomology of n-ary multiplicative Hom-Nambu-Lie algebras. J Geometry and Physics 2011; 61: 1898-1913 doi: 10.1016/j.geomphys.2011.04.022
[7] Ammar F, Makhlouf A, Saadaoui N. Chomology of hom-Lie superalgebras and q-deformed Witt superalgebra. Czechoslovak Math J 2013; 68: 721-761.
[8] Arnlind J, Kitouni A, Makhlouf A, Silvestrov S. Structure and Cohomology of 3-Lie algebras induced by Lie algebras. In ”Algebra, Geometry and Mathematical Physics”, Springer proceedings in Mathematics and Statistics 2014; pp. 123-144.
[9] Arnlind J, Makhlouf A, Silvestrov S. Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras. J Math Phys 2011; 52: 123502.
[10] Arnlind J, Makhlouf A, Silvestrov S, Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras, J Math Phys 2010; 51: 043515, 11 pp.
[11] Bahturin YA, Mikhalev AA, Petrogradsky VM, Zaicev MV. Infinite Dimensional Lie Superalgebras. Berlin, Germany: Walter de Gruyter, 1992.
[12] Benayadi S, Makhlouf A. Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. J Geom Phys 2014; 76: 38-60.
[13] Calderon A, Delgado JS. On the structure of split Lie color algebras. Linear Algebra Appl 2012; 436: 307-315.
[14] Cao Y, Chen L. On split regular hom-Lie color algebras. Comm Algebra 2012; 40: 575-592.
[15] Casas JM, Insua MA, Pacheco N. On universal central extensions of hom-Lie algebras. Hacet J Math Stat 2015, 44: 277-288.
[16] Chaichian M, Ellinas D, Popowicz Z. Quantum conformal algebra with central extension. Phys Lett B 1990; 248: 95-99.
[17] Chaichian M, Isaev AP, Lukierski J, Popowic Z, Prešnajder P. q -deformations of Virasoro algebra and conformal dimensions. Phys Lett B 1991; 262: 32-38.
[18] Chaichian M, Kulish P, Lukierski J, q -deformed Jacobi identity, q -oscillators and q -deformed infinite-dimensional algebras. Phys Lett B 1990; 237: 401-406.
[19] Chaichian M, Popowicz Z, Prešnajder P. q -Virasoro algebra and its relation to the q -deformed KdV system. Phys Lett B 1990; 249: 63-65.
[20] Chen CW, Petit T, Van Oystaeyen F. Note on cohomology of color Hopf and Lie algebras. J Algebra 2006; 299: 419-442.
[21] Curtright TL, Zachos CK. Deforming maps for quantum algebras. Phys Lett B 1990; 243: 237-244.
[22] Damaskinsky EV, Kulish PP. Deformed oscillators and their applications (in Russian) Zap Nauch Semin LOMI 1991; 189: 37-74.
[23] Daskaloyannis C. Generalized deformed Virasoro algebras. Modern Phys Lett A 1992; 7: 809-816.
[24] Guo L, ZhangB, Zheng S. Universal enveloping algebras and Poincare-Birkhoff-Witt theorem for involutive hom-Lie algebras, J Lie Theory 2018; 21: 739-759.
[25] Hartwig JT, Larsson D, Silvestrov S. Deformations of Lie algebras using -derivations. J of Algebra 2006; 259: 314-361.
[26] Hellström L, Makhlouf A, Silvestrov SD. Universal algebra applied to hom-associative algebras, and more. In: Makhlouf A, Paal E, Silvestrov S, Stolin, editors. Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics and Statistics, vol 85. Berlin, Germany, 2014, pp. 157-199.
[27] Hellström L, Silvestrov SD. Commuting Elements in q -Deformed Heisenberg Algebras. Singapore: World Scientific, 2000.
[28] Hu N. q -Witt algebras, q -Lie algebras, q -holomorph structure and representations. Algebra Colloq 1999; 6: 51-70.
[29] Kharchenko V. Quantum Lie Theory - A Multilinear Approach, Lecture Notes in Mathematics, 2150, Springer International Publishing, 2015.
[30] Kassel C. Cyclic homology of differential operators, the virasoro algebra and a q -analogue. Comm Math Phys 1992; 146: 343-356.
[31] Kerner R. Ternary algebraic structures and their applications in physics. In: Proceedings of BTLP 23rd International Colloquium on Group Theoretical Methods in Physics. 2000. arXiv:math-ph/0011023v1
[32] Kerner R. Z3 -graded algebras and non-commutative gauge theories. In: Oziewicz Z, Jancewicz B, Borowiec A, editors. Spinors, Twistors, Clifford Algebras and Quantum Deformations Vol. 53. Kluwer Academic Publishers, 1993, pp. 349-357.
[33] Kerner R. The cubic chessboard: geometry and physics. Classical Quantum Gravity 1997; 14: A203-A225.
[34] Kerner R, Vainerman L. On special classes of n-algebras. J Math Phys 1996; 37: 2553-2565.
[35] Lecomte P. On some sequence of graded Lie algebras associated to manifolds. Ann Global Analysis Geom 1994; 12: 183-192.
[36] Larsson D, Silvestrov S. Quasi-Hom-Lie algebras, Central Extensions and 2-cocycle-like identities. J Algebra 2005; 288: 321-344.
[37] Larsson D, Silvestrov S. Quasi-Lie algebras. In: Fuchs, editor. Noncommutative Geometry and Representation Theory in Mathematical Physics Contemp Math 391. Providence, RI, USA: Amer Math Soc, 2005, pp. 241-248.
[38] Larsson D, Sigurdsson G, Silvestrov S. Quasi-Lie deformations on the algebra F[t]/(tN) . J Gen Lie Theory Appl 2008; 2: 201-205.
[39] Larsson D, Silvestrov S. Graded quasi-Lie agebras. Czechoslovak J Phys 2005; 55: 1473-1478.
[40] Larsson D, Silvestrov S. Quasi-deformations of sl2(F) using twisted derivations. Comm in Algebra 2007; 35: 4303- 4318.
[41] Liu KQ. Quantum central extensions. CR Math Rep Acad Sci Canada 1991; 13: 135-140.
[42] Liu KQ. Characterizations of the quantum Witt algebra. Lett Math Phys 1992; 24: 257-265.
[43] Liu KQ. The quantum Witt algebra and quantization of some modules over Witt algebra, PhD, University of Alberta, Edmonton, Canada, 1992.
[44] Makhlouf A. Paradigm of nonassociative hom-algebras and hom-superalgebras. Proceedings of Jordan Structures in Algebra and Analysis Meeting 2010: 145-177.
[45] Makhlouf A, Silvestrov S. Hom-algebra structures. J Gen Lie Theory Appl 2008; 2: 51-64.
[46] Makhlouf A, Silvestrov S. Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras In: Silvestrov S, Paal E, Abramov V, Stolin A, editors. Generalized Lie theory in Mathematics, Physics and Beyond. Berlin, Germany: Springer-Verlag, 2009, pp. 189-206.
[47] Makhlouf A, Silvestrov S. Hom-Algebras and Hom-Coalgebras. J Algebra Appl 2010; 9: 553-589.
[48] Makhlouf A, Silvestrov S. Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras. Forum Math 2010; 22: 715-739.
[49] Mikhalev AA, Zolotykh AA. Combinatorial Aspects of Lie Superalgebras. Boca Raton, FL, USA: CRC Press, 1995.
[50] Piontkovski D, Silvestrov S. Cohomology of 3-dimensional color Lie algebras. J Algebra 2007; 316: 499-513.
[51] Richard L, Silvestrov S. Quasi-Lie structure of -derivations of C[t 1] . J Algebra 2008; 319: 1285-1304.
[52] Richard L, Silvestrov S. A note on quasi-Lie and Hom-Lie structures of -derivations of C[z 1 1 ; : : : ; z 1 n ] , In: Silvestrov SD, Paal E, Abramov V, Stolin A, editors. Generalized Lie Theory in Mathematics, Physics and Beyond. Berlin, Germany: Springer, 2009, pp. 257-262.
[53] Scheunert M. Generalized Lie algebras. J Math Phys 1979; 20: 712-720.
[54] Scheunert M. Graded tensor calculus. J Math Phys 1983; 24: 2658-2670.
[55] Scheunert M. Introduction to the cohomology of Lie superalgebras and some applications. Res Exp Math 2002; 25: 77-107.
[56] Scheunert M, Zhang RB, Cohomology of Lie superalgebras and their generalizations. J Math Phys 1998; 39: 5024- 5061.
[57] Sigurdsson G, Silvestrov S. Lie color and Hom-Lie algebras of Witt type and their central extensions In: Silvestrov SD, Paal E, Abramov V, Stolin A, editors. Generalized Lie Theory in Mathematics, Physics and Beyond. Berlin, Germany: Springer, 2009, pp. 247-255.
[58] Sigurdsson G, Silvestrov S. Graded quasi-Lie algebras of Witt type. Czech J Phys 2006; 56: 1287-1291.
[59] Silvestrov S. Paradigm of quasi-Lie and quasi-Hom-Lie algebras and quasi-deformations. In: Krähmer U, Caenepeel S, Van Oystayen F, editors. New Techniques in Hopf Algebras and Graded Ring Theory. Brussels, Belgium: K Vlaam Acad Belgie Wet Kunsten (KVAB), 2007, pp. 165-177.
[60] Sheng Y. Representations of hom-Lie algebras. Algebr Represent Theory 2012; 15: 1081-1098.
[61] Sheng Y. Chen D. Hom-Lie 2-algebras. J Algebra 2013; 376: 174-195.
[62] Sheng Y. Bai C. A new approach to hom-Lie bialgebras; J Algebra 2014; 399: 232-250.
[63] Sheng Y. Xiong Z. On hom-Lie algebras, Linear Multilinear Algebra 2015; 63: 2379-2395.
[64] Yau D. Enveloping algebra of Hom-Lie algebras. J Gen Lie Theory Appl 2008, 2: 95-108.
[65] Yau D. Hom-algebras and homology. J Lie Theory 2009; 19: 409-421.
[66] Yau D. Hom-bialgebras and comodule algebras. Int Electron J Algebra 2010; 8: 45-64.
[67] Yau D. Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras. J Phys A: Math Theor 2009; 42: 165202.
[68] Yuan L. Hom-Lie color algebra structures. Comm Algebra 2012; 40: 575-592.
[69] Zhou J, Chen L, Ma Y, Generalized derivations of hom-Lie superalgebras. Acta Math Sinica (Chin Ser) 2014; 58: 3737-3751.