Revista Integración, temas de matemáticas.

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.

PDF
  • Héctor Suárez,
  • Duban Cáceres,
  • Armando Reyes
Héctor Suárez
Universidad Pedagógica y Tecnológica de Colombia
Duban Cáceres
Universidad Pedagógica y Tecnológica de Colombia
Armando Reyes
Universidad Nacional de Colombia.
DOI: https://doi.org/10.18273/revint.v39n1-2021007

Published 2021-05-19

Keywords

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

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

Copyright (c) 2021 Revista Integración, temas de matemáticas

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

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

PDF

Downloads

Download data is not yet available.

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