Some Open Problems in the Context of Skew PBW Extensions and Semi-graded Rings

Abstract

In this paper, we discuss some open problems of non-commutative algebra and non-commutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings. More exactly, we will analyze the isomorphism arising in the investigation of the Gelfand–Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain. The Serre’s conjecture will be discussed for a particular class of skew PBW extensions. The questions about the Noetherianity and the Zariski cancellation property of Artin–Schelter regular algebras will be reformulated for semi-graded rings. Advances for the solution of some of the problems are included.

Keywords

Gelfand–Kirillov conjecture / Serre’s conjecture / Artin–Schelter regular algebras / Zariski cancellation problem / Skew PBW extensions / Semi-graded rings

References

[1] Abhyankar S, Eakin P, Heinzer W. On the uniqueness of the coefficient ring in a polynomial ring. J. Algebra. 1972, 23 310-342

[2] Acosta JP, Chaparro C, Lezama O, Ojeda I, Venegas C. Ore and Goldie theorems for skew $PBW$ extensions. Asian-Eur. J. Math.. 2013, 06 1350061

[3] Acosta JP, Lezama O, Reyes MA. Prime ideals of skew $PBW$ extensions. Rev. Un. Mat. Argentina. 2015, 56 39-55

[4] Alev J, Ooms A, Van den Bergh M. The Gelfand-Kirillov conjecture for Lie algebras of dimension at most eight. J. Algebra. 2000, 227 549-581

[5] Alev J, Dumas F. Sur le corps des fractions de certaines algèbres quantiques. J. Algebra. 1994, 170 229-265

[6] Artamonov V. Serre’s quantum problem. Russian Math. Surv.. 1998, 53 657-730

[7] Artamonov V. On projective modules over quantum polynomials. J. Math. Sci.. 1999, 93 135-148

[8] Artamonov VA, Lezama O, Fajardo W. Extended modules and Ore extensions. Commun. Math. Stat.. 2016, 4 189-202

[9] Artin M, Schelter W. Graded algebras of global dimension 3. Adv. Math.. 1987, 66 171-216

[10] Artin M, Zhang JJ. Noncommutative projective schemes. Adv. Mah.. 1994, 109 228-287

[11] Bass H. Projective modules over algebras. Ann. Math.. 1962, 73 532-542

[12] Bavula VV. The algebra of one-sided inverses of a polynomial algebra. J. Pure Appl. Algebra. 2010, 214 1874-1897

[13] Bell J, Zhang JJ. Zariski cancellation problem for noncommutative algebras. Selecta Math. (N.S.). 2017, 23 1709-1737

[14] Bois J-M. Gelfand-Kirillov conjecture in positive characteristics. J. Algebra. 2006, 305 820-844

[15] Cauchon G. Effacement des dérivations et spectres premiers des algèbres quantiques. J. Algebra. 2003, 260 476-518

[16] Ceken S, Palmieri J, Wang Y-H, Zhang JJ. The discriminant controls automorphism groups of noncommutative algebras. Adv. Math.. 2015, 269 551-584

[17] Ceken S, Palmieri J, Wang Y-H, Zhang JJ. The discriminant criterion and the automorphism groups of quantized algebras. Adv. Math.. 2016, 286 754-801

[18] Essen AVD. Polynomial Automorphisms and the Jacobian Conjecture, Progress in Mathematics 190,. 2000 Cham: Birkäuser

[19] Fajardo, W.: Extended modules over skew $PBW$ extensions. Universidad Nacional de Colombia, Bogotá (2018).. (Ph.D. Thesis)

[20] Fajardo W. A computational Maple library for skew PBW extensions. Fund. Inform.. 2019, 167 159-191

[21] Fajardo W, Lezama O. Elementary matrix-computational proof of Quillen-Suslin theorem for Ore extensions. Fund. Inform.. 2019, 164 41-59

[22] Fujita T. On Zariski problem. Proc. Japan Acad.. 1979, 55 106-110

[23] Futorny V, Molev A, Ovsienko S. The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite W-algebras. Adv. Math.. 2010, 223 773-796

[24] Gabriel P. Des catégories abéliennes. Bull. Soc. Math. France. 1962, 90 323-448

[25] Gaddis, J. D.: PBW deformations of Artin–Schelter regular algebras and their homogenizations, Ph.D. thesis, The University of Wisconsin-Milwaukee, (2013)

[26] Gaddis JD. PBW deformations of Artin-Schelter regular algebras. J. Algebra Appl.. 2016, 15 1650064-1-1650064-15

[27] Gallego C, Lezama O. Projective modules and Gröbner bases for skew $PBW$ extensions. Dissertationes Math.. 2017, 521 1-50

[28] Gelfand I, Kirillov A. Sur le corps liés aux algèbres enveloppantes des algèbres de Lie. Math. IHES. 1966, 31 509-523

[29] Grothendieck A. Sur quelques points d’algèbre homologique. Tôhoku Math. J. (2). 1957, 9 119-221

[30] Gupta N. On Zariski’s cancellation problem in positive characteristic. Adv. Math.. 2014, 264 296-307

[31] Gupta N. On the Cancellation Problem for the Affine Space ${{\mathbb{A}}}^3$ in characteristic $p$. Inventiones Math.. 2014, 195 279-288

[32] Joseph A. Proof of the Gelfand-Kirillov conjecture for solvable Lie algebras. Pro. Am. Math. Soc.. 1974, 45 1-10

[33] Joseph A. A generalization of the Gelfand–Kirillov conjecture. Am. J. Math.. 1977, 99 1151-1165

[34] Lam TY. Serre’s Problem on Projective Modules. Springer Monogr. Math. 2006 Berlin: Springer

[35] Lezama O. Some aplications of Gröbner bases in homological algebra. São Paulo J. Math. Sci.. 2009, 3 25-59

[36] Lezama O, Gallego C. Gröbner bases for ideals of $\sigma $-PBW extensions. Comm. Algebra. 2011, 39 50-75

[37] Lezama O, Gómez JA. Koszulity and point modules of finitely semi-graded rings and algebras. Symmetry. 2019, 11 1-22

[38] Lezama O, Latorre E. Non-commutative algebraic geometry of semi-graded rings. Int. J. Algebra Comput.. 2017, 27 361-389

[39] Lezama O, Reyes A. Some homological properties of skew $PBW$ extensions. Comm. Algebra. 2014, 42 1200-1230

[40] Lezama, O., Venegas, H.: Center of skew PBW extensions, to appear in Internat. J. Algebra Comput.; arXiv: 1804.05425 [math.RA]

[41] Lezama O, Wang Y-H, Zhang JJ. Zariski cancellation problem for non-domain noncommutative algebras. Math. Z.. 2019, 292 1269-1290

[42] Lu DM, Zhou G-S. Artin-Schelter regular algebras of dimension five with two generators. J. Pure Appl. Algebra. 2014, 218 937-961

[43] Lu DM, Shen G, Zhou G-S. Homogeneous PBW deformation for Artin-Schelter regular algebras. Bull. Aust. Math. Soc.. 2015, 91 53-68

[44] McConnell, J., Robson, J.: Non-commutative Noetherian Rings. Grad. Stud. Math, AMS (2001)

[45] Miyanishi M, Sugie T. Affine surfaces containing cylinderlike open sets. J. Math. Kyoto Univ.. 1980, 20 11-42

[46] Ooms A. The Gelfand-Kirillov conjecture for semi-direct products of Lie algebras. J. Algebra. 2006, 305 901-911

[47] Quillen D. Projective modules over polynomial rings. Invent. Math.. 1976, 36 167-171

[48] Reyes A. Armendariz modules over skew PBW extensions. Commun. Algebra. 2019, 47 1248-1270

[49] Reyes A, Rodríguez C. The McCoy condition on Skew Poincaré-Birkhoff-Witt Extensions. Commun. Math. Stat.. 2019

[50] Reyes A, Suárez H. PBW bases for some 3-dimensional skew polynomial algebras. Far East J. Math. Sci. (FJMS). 2017, 101 1207-1228

[51] Reyes A, Suárez H. $\sigma $-PBW extensions of skew Armendariz rings. Adv. Appl. Clifford Algebr.. 2017, 27 3197-3224

[52] Reyes A, Suárez H. A notion of compatibility for Armendariz and Baer properties over skew PBW extensions. Rev. Un. Mat. Argentina. 2018, 59 157-178

[53] Reyes A, Suárez H. Skew Poincaré-Birkhoff-Witt extensions over weak zip rings. Beitr. Algebra Geom.. 2019, 60 197-216

[54] Riaño, A.: Classical and quantum Gelfand-Kirillov conjecture. Universidad Nacional de Colombia, Bogotá, Tesis de Maestría (2019)

[55] Rogalski, D.: Noncommutative projective geometry. In Noncommutative algebraic geometry, vol. 64 of Mathematical Sciences Research Institute Publications is now published by Cambridge University Press, New York, pp. 13–70 (2016)

[56] Rosenberg AL. Noncommutative Algebraic Geometry and Representations of Quantized Algebras, Mathematics and Its Applications. 1995 Amsterdam: Kluwer Academic Publishers

[57] Russell P. On Affine-Ruled rational surfaces. Math. Ann.. 1981, 255 287-302

[58] Serre JP. Faisceaux algébriques cohérents. Ann. Math.. 1955, 61 191-278

[59] Silvesterov, J., Richter, J., Tumwesigye, A.B.: Centralizers of skew PBW extensions, arXiv:1910.11177v2 [math.RA]

[60] Smith, S.P.: Non-commutative Algebraic Geometry. University of Washington, Department of Mathematics (2000)

[61] Suslin AA. Projective modules over polynomial rings are free. Soviet Math. Dokl.. 1976, 17 1160-1164

[62] Tang, X., Venegas, H., Zhang, J.: Cancellation problem for $AS$-regular algebras of dimension three, arXiv:1904.07281v1

[63] Verevkin AB. Serré injective sheaves. Mat. Zametki. 1992, 52 35-41

[64] Xu Y, Huang H-L, Wang D. Realization of $PBW$-deformations of type ${\mathbb{A}}_n$ quantum groups via multiple Ore extensions. J. Pure Appl. Algebra. 2019, 233 1531-1547