# Symmetry and Reversibility Properties for Quantum Algebras and Skew Poincaré-Birkhoff-Witt Extensions

## Main Article Content

## Keywords

Symmetry; reversibility; quantum algebra; skew Poincaré- Birkhoff-Witt extension.

## Abstract

Our aim in this paper is to investigate symmetry and reversibility properties for quantum algebras and skew PBW extensions. Under certain conditions we prove that these properties transfer from a ring of coefficients to a quantum algebra or skew PBW extension over this ring. In this way we generalize several results established in the literature and consider algebras which have not been studied before. We illustrate our results with remarkable examples of theoretical physics.

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## References

[1] J. Lambek, “On the representation of modules by sheaves of factor modules,”Canad. Math. Bull., vol. 14, pp. 359–368, 1971. 30

[2] D. Anderson and V. Camillo, “Semigroups and rings whose zero productscommute,”Commun. Algebra, vol. 27, no. 6, pp. 2847–2852, 1999. 30

[3] N. K. Kim and Y. Lee, “Extensions of reversible rings,”J. Pure Appl. Algebra,vol. 185, pp. 207–223, 2003. 30, 32, 33, 42

[4] Z. Wang and L. Wang, “Polynomial rings over symmetric rings need not tobe symmetric,”Commun. Algebra, vol. 34, pp. 1043–1047, 2006. 30

[5] M. B. Rege and S. Chhawchharia, “Armendariz rings,”Proc. Japan. Acad.Ser. A Math. Sci., vol. 73, pp. 14–17, 1997. 30

[6] E. P. Armendariz, “A note on extensions of Baer and p.p.-rings,”J. Aus-tralian Math. Soc., vol. 18, pp. 470–473, 1974. 30

[7] O. Ore, “Theory of non-commutative polynomials,”Ann. of Math. (2),vol. 34, no. 3, pp. 480–508, 1933. 30

[8] J. Krempa, “Some examples of reduced rings,”Algebra Colloq., vol. 3, no. 4,pp. 289–300, 1996. 30

[9] C. Y. Hong, N. K. Kim, and T. K. Kwak, “Ore extensions of Baer and p.p.-rings,”J. Pure Appl. Algebra, vol. 151, no. 3, pp. 215–226, 2000. 31

[10] A. Reyes, “Skew PBW extensions of Baer, quasi-Baer, p.p. and p.q.-rings,”Rev. Integr. Temas Mat., vol. 33, no. 6, pp. 173–189, 2015. 31, 32, 35, 36

[11] A. Reyes and H. Suárez, “σ-PBW extensions of skew Armendariz rings,”Adv.Appl. Clifford Algebr., vol. 27, no. 4, pp. 3197–3224, 2017. 31, 32, 37, 39

[12] C. Y. Hong, N. K. Kim, and T. K. Kwak, “On Skew Armendariz Rings,”Commun. Algebra, vol. 31, no. 1, pp. 103–122, 2003. 31, 37

[13] C. Y. Hong, T. K. Kwak, and S. T. Rezvi, “Extensions of generalized Ar-mendariz rings,”Algebra Colloq., vol. 13, no. 2, pp. 253–266, 2006. 31, 32,33, 37, 42

[14] E. Hashemi and A. Moussavi, “On(σ,δ)-skew Armendariz rings,”J. KoreanMath. Soc., vol. 42, no. 2, pp. 353–363, 2005. 31

[15] ——, “Polynomial extensions of quasi-Baer rings,”Acta. Math. Hungar., vol.107, no. 3, pp. 207–224, 2005. 31

[16] M. Baser, C. Y. Hong, and T. K. Kwak, “On extended reversible rings,”Algebra Colloq., vol. 16, no. 1, pp. 37–48, 2009. 31, 32, 33, 42, 45

[17] T. K. Kwak, “Extensions of extended symmetric rings,”Bull. Korean Math.Soc., vol. 44, pp. 777–788, 2007. 31, 32, 33, 42, 45

[18] C. Gallego and O. Lezama, “Gröbner bases for ideals ofσ-PBW extensions,”Commun. Algebra, vol. 39, no. 1, pp. 50–75, 2011. 31, 33, 34, 35

[19] A. Reyes, “Ring and module theoretical properties of skew PBW extensions,”Ph.D. dissertation, Universidad Nacional de Colombia, Sede Bogotá, 2013.31, 45

[20] O. Lezama and A. Reyes, “Some homological properties of skew PBW ex-tensions,”Commun. Algebra, vol. 42, no. 3, pp. 1200–1230, 2014. 31, 34

[21] A. Reyes, “Jacobson’s conjecture and skew PBW extensions,”Rev. Integr.Temas Mat., vol. 32, no. 2, pp. 139–152, 2014. 31

[22] C. Gallego and O. Lezama, “Projective modules and Gröbner bases for skewPBW extensions,”Dissertationes Math., vol. 521, pp. 1–50, 2017. 31

[23] A. Reyes and H. Suárez, “A notion of compatibility for Armendariz and Baerproperties over skew PBW extensions,”Rev. Un. Mat. Argent., vol. 59, no. 1,pp. 157–178, 2018. 31, 32, 38, 39, 45

[24] H. L. Suárez and A. O. Reyes, “Some Relations between N-Koszul, Artin-Schelter regular and Calabi-Yau algebras with skew PBW extensions,”Cien-cia en Desarrollo, vol. 6, no. 2, pp. 205–213, 2015. 32

[25] ——, “Calabi-Yau property for graded skew PBW extensions,”Rev. Colom-biana Mat., vol. 51, no. 2, pp. 221–238, 2017. 32

[26] A. Suárez, H. Reyes, “Koszulity for skew PBW extensions over fields,”JP J.Algebra, Number Theory Appl., vol. 39, no. 2, pp. 181–203, 2017. 32

[27] ——, “A generalized Koszul property for skew PBW extensions,”Far EastJ. Math. Sci., vol. 101, no. 2, pp. 301–320, 2017. 32

[28] A. Reyes and H. Suárez, “Armendariz property for skew PBW extensionsand their classical ring of quotients,”Rev. Integr. Temas Mat., vol. 34, no. 2,pp. 147–168, 2016. 32, 37

[29] A. Nino and A. Reyes, “Some ring theoretical properties for skew Poincaré-Birkhoff-Witt extensions,”Bol. Mat. (N.S.), vol. 24, no. 2, pp. 131–148, 2017. 32, 36, 37.

[30] C. Huh, H. K. Kim, N. K. Kim, and Y. Lee, “Basic examples and extensionsof symmetric rings,”J. Pure Appl. Algebra, vol. 202, pp. 154–167, 2005. 32,33, 42

[31] L. B. Yakoub and M. Louzari, “Ore Extensions of Extended Symmetric andReversible Rings,”International Journal of Algebra, vol. 3, no. 9, pp. 423–433, 2009. 32, 33, 38, 39, 41, 42, 43, 44, 45

[32] O. Lezama, J. Acosta, and A. Reyes, “Prime ideals of skew PBW extensions,”Rev. Un. Mat. Argent., vol. 56, no. 2, pp. 39–55, 2015. 34

[33] A. Reyes, “σ-PBW extensions of skewΠ-Armendariz rings,”Far East J.Math. Sci., vol. 103, no. 2, pp. 401–428, 2018. 38

[34] A. Reyes and H. Suárez, “Bases for quantum algebras and skew Poincaré-Birkhoff-Witt extensions,”Momento, vol. 54, no. 1, pp. 54–75, 2017. 45,49

[35] A. Reyes and Y. Suárez, “On the ACCP in skew Poincaré-Birkhoff-Wittextensions,”Beitr. Algebra Geom., pp. 3–21, 2018. 45

[36] R. Berger, “The quantum Poincaré-Birkhoff-Witt theorem,”Comm. Math.Physics, vol. 143, no. 2, pp. 215–234, 1992. 46

[37] T. Hayashi, “Q-Analogues of Clifford and Weyl Algebras - Spinor and Os-cillator Representations of Quantum Enveloping Algebras,”Comm. Math.Phys., vol. 127, no. 6, pp. 129–144, 1990. 47

[38] A. Jannussi, A. Leodaris, and R. Mignani, “Non-Hermitian realization of aLie-deformed Heisenberg algebra,”Phys. Lett. A, vol. 197, no. 3, pp. 187–191,1995. 47

[39] D. A. Slavnov, “Possibility of reconciling quantum mechanics with generalrelativity theory,”Theoret. Math. Phys., vol. 171, no. 3, pp. 848–861, 2012.49

[40] A. Reyes and H. Suárez, “Some remarks about the cyclic homology of skewPBW extensions,”Ciencia en Desarrollo, vol. 7, no. 2, pp. 99–197, 2016. 49

[2] D. Anderson and V. Camillo, “Semigroups and rings whose zero productscommute,”Commun. Algebra, vol. 27, no. 6, pp. 2847–2852, 1999. 30

[3] N. K. Kim and Y. Lee, “Extensions of reversible rings,”J. Pure Appl. Algebra,vol. 185, pp. 207–223, 2003. 30, 32, 33, 42

[4] Z. Wang and L. Wang, “Polynomial rings over symmetric rings need not tobe symmetric,”Commun. Algebra, vol. 34, pp. 1043–1047, 2006. 30

[5] M. B. Rege and S. Chhawchharia, “Armendariz rings,”Proc. Japan. Acad.Ser. A Math. Sci., vol. 73, pp. 14–17, 1997. 30

[6] E. P. Armendariz, “A note on extensions of Baer and p.p.-rings,”J. Aus-tralian Math. Soc., vol. 18, pp. 470–473, 1974. 30

[7] O. Ore, “Theory of non-commutative polynomials,”Ann. of Math. (2),vol. 34, no. 3, pp. 480–508, 1933. 30

[8] J. Krempa, “Some examples of reduced rings,”Algebra Colloq., vol. 3, no. 4,pp. 289–300, 1996. 30

[9] C. Y. Hong, N. K. Kim, and T. K. Kwak, “Ore extensions of Baer and p.p.-rings,”J. Pure Appl. Algebra, vol. 151, no. 3, pp. 215–226, 2000. 31

[10] A. Reyes, “Skew PBW extensions of Baer, quasi-Baer, p.p. and p.q.-rings,”Rev. Integr. Temas Mat., vol. 33, no. 6, pp. 173–189, 2015. 31, 32, 35, 36

[11] A. Reyes and H. Suárez, “σ-PBW extensions of skew Armendariz rings,”Adv.Appl. Clifford Algebr., vol. 27, no. 4, pp. 3197–3224, 2017. 31, 32, 37, 39

[12] C. Y. Hong, N. K. Kim, and T. K. Kwak, “On Skew Armendariz Rings,”Commun. Algebra, vol. 31, no. 1, pp. 103–122, 2003. 31, 37

[13] C. Y. Hong, T. K. Kwak, and S. T. Rezvi, “Extensions of generalized Ar-mendariz rings,”Algebra Colloq., vol. 13, no. 2, pp. 253–266, 2006. 31, 32,33, 37, 42

[14] E. Hashemi and A. Moussavi, “On(σ,δ)-skew Armendariz rings,”J. KoreanMath. Soc., vol. 42, no. 2, pp. 353–363, 2005. 31

[15] ——, “Polynomial extensions of quasi-Baer rings,”Acta. Math. Hungar., vol.107, no. 3, pp. 207–224, 2005. 31

[16] M. Baser, C. Y. Hong, and T. K. Kwak, “On extended reversible rings,”Algebra Colloq., vol. 16, no. 1, pp. 37–48, 2009. 31, 32, 33, 42, 45

[17] T. K. Kwak, “Extensions of extended symmetric rings,”Bull. Korean Math.Soc., vol. 44, pp. 777–788, 2007. 31, 32, 33, 42, 45

[18] C. Gallego and O. Lezama, “Gröbner bases for ideals ofσ-PBW extensions,”Commun. Algebra, vol. 39, no. 1, pp. 50–75, 2011. 31, 33, 34, 35

[19] A. Reyes, “Ring and module theoretical properties of skew PBW extensions,”Ph.D. dissertation, Universidad Nacional de Colombia, Sede Bogotá, 2013.31, 45

[20] O. Lezama and A. Reyes, “Some homological properties of skew PBW ex-tensions,”Commun. Algebra, vol. 42, no. 3, pp. 1200–1230, 2014. 31, 34

[21] A. Reyes, “Jacobson’s conjecture and skew PBW extensions,”Rev. Integr.Temas Mat., vol. 32, no. 2, pp. 139–152, 2014. 31

[22] C. Gallego and O. Lezama, “Projective modules and Gröbner bases for skewPBW extensions,”Dissertationes Math., vol. 521, pp. 1–50, 2017. 31

[23] A. Reyes and H. Suárez, “A notion of compatibility for Armendariz and Baerproperties over skew PBW extensions,”Rev. Un. Mat. Argent., vol. 59, no. 1,pp. 157–178, 2018. 31, 32, 38, 39, 45

[24] H. L. Suárez and A. O. Reyes, “Some Relations between N-Koszul, Artin-Schelter regular and Calabi-Yau algebras with skew PBW extensions,”Cien-cia en Desarrollo, vol. 6, no. 2, pp. 205–213, 2015. 32

[25] ——, “Calabi-Yau property for graded skew PBW extensions,”Rev. Colom-biana Mat., vol. 51, no. 2, pp. 221–238, 2017. 32

[26] A. Suárez, H. Reyes, “Koszulity for skew PBW extensions over fields,”JP J.Algebra, Number Theory Appl., vol. 39, no. 2, pp. 181–203, 2017. 32

[27] ——, “A generalized Koszul property for skew PBW extensions,”Far EastJ. Math. Sci., vol. 101, no. 2, pp. 301–320, 2017. 32

[28] A. Reyes and H. Suárez, “Armendariz property for skew PBW extensionsand their classical ring of quotients,”Rev. Integr. Temas Mat., vol. 34, no. 2,pp. 147–168, 2016. 32, 37

[29] A. Nino and A. Reyes, “Some ring theoretical properties for skew Poincaré-Birkhoff-Witt extensions,”Bol. Mat. (N.S.), vol. 24, no. 2, pp. 131–148, 2017. 32, 36, 37.

[30] C. Huh, H. K. Kim, N. K. Kim, and Y. Lee, “Basic examples and extensionsof symmetric rings,”J. Pure Appl. Algebra, vol. 202, pp. 154–167, 2005. 32,33, 42

[31] L. B. Yakoub and M. Louzari, “Ore Extensions of Extended Symmetric andReversible Rings,”International Journal of Algebra, vol. 3, no. 9, pp. 423–433, 2009. 32, 33, 38, 39, 41, 42, 43, 44, 45

[32] O. Lezama, J. Acosta, and A. Reyes, “Prime ideals of skew PBW extensions,”Rev. Un. Mat. Argent., vol. 56, no. 2, pp. 39–55, 2015. 34

[33] A. Reyes, “σ-PBW extensions of skewΠ-Armendariz rings,”Far East J.Math. Sci., vol. 103, no. 2, pp. 401–428, 2018. 38

[34] A. Reyes and H. Suárez, “Bases for quantum algebras and skew Poincaré-Birkhoff-Witt extensions,”Momento, vol. 54, no. 1, pp. 54–75, 2017. 45,49

[35] A. Reyes and Y. Suárez, “On the ACCP in skew Poincaré-Birkhoff-Wittextensions,”Beitr. Algebra Geom., pp. 3–21, 2018. 45

[36] R. Berger, “The quantum Poincaré-Birkhoff-Witt theorem,”Comm. Math.Physics, vol. 143, no. 2, pp. 215–234, 1992. 46

[37] T. Hayashi, “Q-Analogues of Clifford and Weyl Algebras - Spinor and Os-cillator Representations of Quantum Enveloping Algebras,”Comm. Math.Phys., vol. 127, no. 6, pp. 129–144, 1990. 47

[38] A. Jannussi, A. Leodaris, and R. Mignani, “Non-Hermitian realization of aLie-deformed Heisenberg algebra,”Phys. Lett. A, vol. 197, no. 3, pp. 187–191,1995. 47

[39] D. A. Slavnov, “Possibility of reconciling quantum mechanics with generalrelativity theory,”Theoret. Math. Phys., vol. 171, no. 3, pp. 848–861, 2012.49

[40] A. Reyes and H. Suárez, “Some remarks about the cyclic homology of skewPBW extensions,”Ciencia en Desarrollo, vol. 7, no. 2, pp. 99–197, 2016. 49