Propiedades de simetría y reversibilidad para álgebras cuánticas y extensiones torcidas de Poincaré-Birkhoff-Witt
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Keywords
Simetría; reversibilidad; álgebra cuántica; extensión torcida de Poincaré-Birkhoff-Witt.
Resumen
Nuestro propósito en este artículo es investigar las propiedades de simetría y reversibilidad para álgebras cuánticas y extensiones PBW torcidas. Bajo ciertas condiciones mostramos que estas propiedades se transfieren de un anillo de coeficientes a un álgebra cuántica o extensión PBW torcida sobre este anillo. De esta manera generalizamos diversos resultados establecidos en la literatura, y los ampliamos a álgebras antes no estudiadas. Ilustramos nuestros resultados con ejemplos destacados de la física teórica.
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Referencias
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[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
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[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
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Publicado
jun 15, 2018
Article Details
Cómo citar
Reyes, A., & Jaramillo, J. (2018). Propiedades de simetría y reversibilidad para álgebras cuánticas y extensiones torcidas de Poincaré-Birkhoff-Witt. Ingeniería Y Ciencia, 14(27), 29–52. https://doi.org/10.17230/ingciencia.14.27.2
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Biografía del autor/a
Armando Reyes, Dr., Universidad Nacional
Doctor en Ciencias Matemáticas
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