Revista Integración, temas de matemáticas.
Vol. 39 No. 1 (2021): Revista integración, temas de matemáticas
Research and Innovation Articles

Some special types of determinants in graded skew P BW extensions.

Héctor Suárez
Universidad Pedagógica y Tecnológica de Colombia
Duban Cáceres
Universidad Pedagógica y Tecnológica de Colombia
Armando Reyes
Universidad Nacional de Colombia.

Published 2021-05-19

Keywords

  • Calabi-Yau algebra,
  • skew PBW extension,
  • double Ore extension,
  • homological determinant,
  • P-determinant,
  • Nakayama automorphism
  • ...More
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How to Cite

Suárez, H., Cáceres, D., & Reyes, A. (2021). Some special types of determinants in graded skew P BW extensions. Revista Integración, Temas De matemáticas, 39(1), 91–107. https://doi.org/10.18273/revint.v39n1-2021007

Abstract

In this paper, we prove that the Nakayama automorphism of a graded skew PBW extension over a finitely presented Koszul Auslanderregular algebra has trivial homological determinant. For A = σ(R)<x1, x2> a graded skew PBW extension over a connected algebra R, we compute its Pdeterminant and the inverse of σ. In the particular case of quasi-commutative skew PBW extensions over Koszul Artin-Schelter regular algebras, we show explicitly the connection between the Nakayama automorphism of the ring of coefficients and the extension. Finally, we give conditions to guarantee that A is Calabi-Yau. We provide illustrative examples of the theory concerning algebras of interest in noncommutative algebraic geometry and noncommutative differential geometry

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References

  1. Acosta J.P. and Lezama O., “Universal property of skew PBW extensions”, Algebra and Discrete Math., 20 (2015), No. 1, 1-12.
  2. Artamonov V.A., “Derivations of skew PBW extensions”, Commun. Math. Stat., 3 (2015), No. 4, 449-457. doi: 10.1007/s40304-015-0067-9
  3. Artin M. and Schelter W.F., “Graded algebras of global dimension 3”, Adv. Math., 66 (1987), 171-216. doi: 10.1016/0001-8708(87)90034-X
  4. Backelin J. and Fröberg R., “Koszul algebras, Veronese subrings and rings with linear resolutions”, Rev. Roumaine Math. Pures Appl., 30 (1985), No. 2, 85-97.
  5. Bell A. and Goodearl K., “Uniform rank over differential operator rings and PoincaréBirkhoff-Witt extensions”, Pacific J. Math., 131 (1988), No. 1, 13-37.
  6. Carvalho P., Lopes S. and Matczuk J., “Double Ore extensions versus iterated Ore extensions”, Comm. Algebra, 39 (2011), No. 8, 2838-2848. doi: 10.1080/00927872.2010.489532
  7. Fajardo W., Gallego C., Lezama O., Reyes A., Suárez H., and Venegas H., Skew PBW Extensions: Ring and Module-theoretic Properties, Matrix and Gro¨bner Methods, and Applications, Springer Nature, vol. 28, 2020.
  8. Gallego C. and Lezama O., “Gröbner bases for ideals of σ-PBW extensions”, Comm. Algebra, 39 (2011), No. 1, 50-75. doi: 10.1080/00927870903431209
  9. Gómez J.Y. and Suárez H., “Double Ore extensions versus graded skew PBW extensions”, Comm. Algebra, 48 (2020), No. 1, 185-197.
  10. Goodearl K.R. and Warfield R.B. Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 2nd ed., vol. 61, Cambridge, 2004. doi: 10.1017/CBO9780511841699
  11. Hashemi E., Khalilnezhad K. and Alhevaz A., “(Σ, ∆)-Compatible skew PBW extension ring”, Kyungpook Math. J., 57 (2017), No. 3, 401-417. doi: 10.5666/KMJ.2017.57.3.401
  12. Hashemi E., Khalilnezhad K. and Alhevaz A., “Extensions of rings over 2-primal rings”, Le Matematiche, 54 (2019), No. 1, 141-162. doi: 10.4418/2019.74.1.10
  13. Hashemi E., Khalilnezhad K. and Ghadiri Herati M., “Baer and quasi-Baer properties of skew PBW extensions”, J. Algebraic Systems, 7 (2019), No. 1, 1-24.
  14. Isaev A.P., Pyatov P.N. and Rittenberg V., “Diffusion algebras”, J. Phys. A., 34 (2001), No. 29, 5815-5834. doi: 10.1088/0305-4470/34/29/306
  15. Jordan D., “The graded algebra generated by two Eulerian derivatives”, Algebr. Represent. Theory, 4 (2001), No. 3, 249-275. doi: 10.1023/A:1011481028760 Jorgensen P. and Zhang J.J., “Gourmet’s Guide to Gorensteinness”, Adv. Math., 151 (2000), No. 2, 313-345. doi: 10.1006/aima.1999.1897
  16. Lezama O., “Computation of point modules of finitely semi-graded rings”, Comm. Algebra, 48 (2020), No. 2, 866-878. doi: 10.1080/00927872.2019.1666404
  17. Lezama O. and Gallego C., “d-Hermite rings and skew PBW extensions”, São Paulo J. Math. Sci., 10 (2016), No. 1, 60-72. doi: 10.1007/s40863-015-0010-8
  18. Lezama O. and Gómez J., “Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras”, Symmetry, 11 (2019), No. 7, 1-22. doi: 10.3390/sym11070881
  19. Lezama O. and Reyes A., “Some homological properties of skew PBW extensions”, Comm. Algebra, 42 (2014), No. 3, 1200-1230. doi: 10.1080/00927872.2012.735304
  20. Lezama O. and Venegas C., “Center of skew PBW extensions”, Internat. J. Algebra Comput., 30 (2020), No. 8, 1625-1650. doi: 10.1142/S0218196720500575
  21. Liu Y. and Ma W., “Nakayama automorphism of Ore extensions over polynomial algebras”, Glasg. Math. J., 62 (2020), No. 3, 518-530. doi: 10.1017/s0017089519000259
  22. Liu Y., Wang S. and Wu Q.-S., “Twisted Calabi-Yau property of Ore extensions”, J. Noncommut. Geom., 8 (2014), No. 2, 587-609. doi: 10.4171/JNCG/165
  23. Lu J., Mao X. and Zhang J.J., “Nakayama automorphism and applications”, Trans. Amer. Math. Soc., 369 (2017), No. 4, 2425-2460. doi: 10.1090/tran/6718
  24. Ore O., “Theory of non-commutative polynomials”, Ann. Math, 34 (1933), No. 3, 480-508. doi: 10.2307/1968173
  25. Polishchuk A. and Positselski L., Quadratic algebras, American Mathematical Society, vol. 37, Providence, RI, 2005. doi: 10.1090/ulect/037
  26. Reyes M., Rogalski D. and Zhang J.J., “Skew Calabi-Yau algebras and homological identities”, Adv. Math., 264 (2014), 308-354. doi: 10.1016/j.aim.2014.07.010
  27. Reyes A. and Suárez H., “Armendariz property for skew P BW extensions and their classical ring of quotients”, Rev. Integr. Temas Mat., 34 (2016), No. 2, 147-168. doi: 10.18273/revint.v34n2-2016004
  28. Reyes A. and Suárez H., “Enveloping Algebra and Skew Calabi-Yau algebras over Skew Poincaré-Birkhoff-Witt extensions”, Far East J. Math. Sci., 102 (2017), No. 2, 373-397. doi 10.17654/MS102020373
  29. Reyes A. and Suárez H., “Radicals and Köthe’s conjecture for skew PBW extensions”, Commun. Math. Stat., (2019). doi: 10.1007/s40304-019-00189-0
  30. Reyes A. and Suárez H., “Skew Poincaré-Birkhoff-Witt extensions over weak compatible rings”, J. Algebra Appl., 19 (2020), No. 12, 21. doi: 10.1142/S0219498820502254
  31. Reyes A. and Suárez H., “Skew Poincaré-Birkhoff-Witt extensions over weak zip rings”, Beitr. Algebra Geom., vol. 60 (2019), No. 2, 197-216. doi: 10.1007/s13366-018-0412-8
  32. Shen Y. and Lu D.-M., “Nakayama automorphisms of P BW deformations and Hopf actions”, Sci. China Math., 59 (2016), No. 4, 661-672. doi: 10.1007/s11425-015-5077-2
  33. Shen Y., Zhou G-S. and Lu D-M., “Nakayama automorphisms of twisted tensor products”, J. Algebra, 504 (2018), 445-478. doi: 10.1016/j.jalgebra.2018.02.025
  34. Suárez H., “Koszulity for graded skew PBW extensions”, Comm. Algebra, 45 (2017), No. 10, 4569-4580. doi: 10.1080/00927872.2016.1272694
  35. Suárez H., Lezama O. and Reyes A., “Calabi-Yau property for graded skew PBW extensions”, Rev. Colombiana Mat., 51 (2017), No. 2, 221-238. doi: 10.15446/recolma.v51n2.70902
  36. Suárez H. and Reyes A., “Koszulity for skew PBW extensions over fields”, JP J. Algebra Number Theory Appl., 39 (2017), No. 2, 181-203. doi: 10.17654/NT039020181
  37. Suárez H. and Reyes A., “A generalized Koszul property for skew PBW extensions”, Far East J. Math. Sci., 101 (2017), No. 2, 301-320. doi: 10.17654/MS101020301
  38. Suárez H. and Reyes A., “Nakayama automorphism of some skew PBW extensions”, Ingeniería y Ciencia, 15 (2019), No. 29, 157-177. doi: 10.17230/ingciencia.15.29.6
  39. Tumwesigye A.B., Richter J. and Silvestrov S., Centralizers in PBW extensions, Springer, vol. 317, Cham, 2020. doi: 10.1007/978-3-030-41850-2-20
  40. Wu Q. and Zhu C., “Skew group algebras of Calabi-Yau algebras”, J. Algebra, 340 (2011), 53-76. doi: 10.1016/j.jalgebra.2011.05.027
  41. Zambrano B.A., “Poisson brackets on some skew PBW extensions”, Algebra Discrete Math., 29 (2020), No. 2, 277-302.
  42. Zhang J.J. and Zhang J., “Double Ore extensions”, J. Pure Appl. Algebra, 212 (2008), No. 12, 2668-2690. doi: 10.1016/j.jpaa.2008.05.008
  43. Zhang J.J. and Zhang J., “Double extension regular algebras of type (14641)”, J. Algebra, 322 (2009), No. 2, 373-409. doi: 10.1016/j.jalgebra.2009.03.041
  44. Zhu C., Van Oystaeyen F. and Zhang Y., “Nakayama automorphisms of double Ore extensions of Koszul regular algebras”, Manuscripta Math., 152 (2017), No. 3-4, 555-584. doi: 10.1007/s00229-016-0865-8