Gröbner-Shirshov bases for Sklyanin algebras

Vista sencilla de ítem
dc.contributor.advisorReyes Villamil, Milton Armando
dc.contributor.authorHerrera Cano, Karol Stefany
dc.date.accessioned2024-05-14T16:33:38Z
dc.date.available2024-05-14T16:33:38Z
dc.date.issued2024
dc.description.abstractIn this thesis we study the theory of Gröbner-Shirshov bases for three- dimensional and four-dimensional Sklyanin algebras. First, we present a brief construction of free algebras, and then describe the theory of Gröbner-Shirshov bases of these algebras. In addition, we present examples on the computation of the bases, and in particular, we consider some relations with PBW algebras. Next, we address the origin and review some of the properties of three-dimensional Sklyanin algebras, especially the PBW property. With this, we classify the three-dimensional Sklyanin algebras that are or not PBW algebras into at least eight families, and we compute their Gröbner-Shirshov bases, obtaining in some cases finite bases and in others, apparently infinite ones. In the same way, we study four-dimensional Sklyanin algebras, reviewing some of their algebraic properties, their classification into six families of degenerate algebras, and we compute their Gröbner-Shirshov bases obtaining only for one family, a finite basis. Finally, we use a code developed in MATLAB to review the hand-made computations of the Gröbner-Shirshov bases in the different families of the three-dimensional Sklyanin algebras, and at the same time test the correctness of the code. Once verified, we use it to perform the calculations for four-dimensional Sklyanin algebraseng
dc.description.abstractEn esta tesis estudiamos la teoría de bases de Gröbner-Shirshov para las álgebras de Sklyanin tridimensionales y cuatrodimensionales. En primer lugar, presentamos una breve construcción de las álgebras libres, para luego describir la teoría de bases de Gröbner- Shirshov de estas álgebras. Además, presentamos ejemplos sobre el cálculo de dichas bases, y en particular, conectamos estas bases con las álgebras PBW. Luego, abordamos el origen y revisamos algunas de las propiedades sobre las álgebras de Sklyanin tridimensionales, en especial, la propiedad PBW. Gracias a esta, clasificamos en al menos ocho familias las álgebras de Sklyanin tridimensionales que son o no son álgebras PBW, y calculamos sus bases de Gröbner-Shirshov, obteniendo en algunos casos bases finitas y en otros, al parecer, infinitas. De la misma manera, estudiamos las álgebras de Sklyanin cuatrodimensionales, revisando algunas de sus propiedades algebraicas y las clasificamos en seis familias de álgebras degeneradas, y calculamos sus bases de Gröbner-Shirshov obteniendo solo para una familia, una base finita. Finalmente, utilizamos un código hecho en MATLAB para revisar los cálculos hechos a mano de las bases de Gröbner-Shirshov en las diferentes familias de las álgebras de Sklyanin tridimensionales, y al mismo tiempo probar la veracidad del código. Una vez comprobado, lo usamos para realizar los cálculos para álgebras de Sklyanin cuatrodimensionales (Texto tomado de la fuente)spa
dc.description.degreelevelMaestríaspa
dc.format.extentvi, 189 páginasspa
dc.format.mimetypeapplication/pdfspa
dc.identifier.instnameUniversidad Nacional de Colombiaspa
dc.identifier.reponameRepositorio Institucional Universidad Nacional de Colombiaspa
dc.identifier.repourlhttps://repositorio.unal.edu.co/spa
dc.identifier.urihttps://repositorio.unal.edu.co/handle/unal/86074
dc.language.isoengspa
dc.publisherUniversidad Nacional de Colombiaspa
dc.publisher.branchUniversidad Nacional de Colombia - Sede Bogotáspa
dc.publisher.facultyFacultad de Cienciasspa
dc.publisher.placeBogotá, Colombiaspa
dc.publisher.programBogotá - Ciencias - Maestría en Ciencias - Matemáticasspa
dc.relation.referencesJ. Apel. Gröbnerbasen in Nichtkommutativen Algebren und ihre Anwendung. PhD thesis, Universität Leipzig, 1988.spa
dc.relation.referencesM. Artin and W. Schelter. Graded algebras of global dimension 3. Adv. in Math., 66(2):171–216, 1987spa
dc.relation.referencesM. Artin, J. Tate, and M. Van den Bergh. Some Algebras Associated to Automorphisms of Elliptic Curves. In The Grothendieck Festschrift, pages 33–85. Springer, 1990.spa
dc.relation.referencesM. Artin, J. Tate, and M. Van den Bergh. Modules over regular algebras of dimension 3. Invent. Math., 106(2):335–388, 1991spa
dc.relation.referencesV. V. Bavula. Description of bi-quadratic algebras on 3 generators with PBW basis. J. Algebra, 631:695–730, 2023.spa
dc.relation.referencesL. Bokut and Y. Chen. Gröbner-Shirshov bases and their calculation. Bull. Math. Sci., 4(3):325–395, 2014.spa
dc.relation.referencesL. Bokut and Y. Chen. Gröbner-Shirshov bases theory and Shirshov algorithm. Novosibirsk: RIZ NSU, 2014.spa
dc.relation.referencesY. Bai and Y. Chen. Gröbner-Shirshov bases theory for Leibniz superalgebras. Comm. Algebra, 50(8):3524–3542, 2022spa
dc.relation.referencesY. Bai, Y. Chen, and Z. Zhang. Gelfand-Kirillov dimension of bicommutative algebras. Linear Multilinear Algebra, 70(22):7623–7649, 2021spa
dc.relation.referencesG. M. Bergman. The Diamond lemma for ring theory. Adv. Math., 29(2):178–218, 1978spa
dc.relation.referencesL. Bokut, Y. Fong, and W.F. Ke. Gröbner-Shirshov bases and composition lemma for associative conformal algebras: an example. Contemp. Math., 264:63–90, 2000.spa
dc.relation.referencesA. Bell and K. Goodearl. Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions. Pacific J. Math., 131(1):13–37, 1988spa
dc.relation.referencesJ. Bueso, J. Gómez-Torrecillas, and A. Verschoren. Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups. Mathematical Modelling: Theory and Applications, Springer, 2003.spa
dc.relation.referencesL. Bokut and P. Malcolmson. Gröbner-Shirshov bases for quantum enveloping algebras. Israel J. Math., 96:97–113, 1996spa
dc.relation.referencesL. Bokut and L. Makar-Limanov. A basis of a free metabelian associative algebra. Sib. Math. J., 32(6):910–915, 1991spa
dc.relation.referencesL. Bokut. Unsolvability of the equality problem and subalgebras of finitely presented Lie algebras. Izv. Ross. Akad. Nauk Ser. Mat., 36(6):1173–1219, 1972.spa
dc.relation.referencesL. Bokut. Embeddings into simple associative algebras. Algebra Logika, 15(2):117–142, 1976spa
dc.relation.referencesG. Bellamy, D. Rogalski, T. Schedler, J. Stafford, and M. Wemyss. Noncom- mutative Algebraic Geometry, volume 64. Cambridge University Press, 2016.spa
dc.relation.referencesA. D. Bell and S. P. Smith. Some 3-dimensional skew polynomial rings. University of Wisconsin, Milwaukee, preprint, 1990.spa
dc.relation.referencesB. Buchberger. An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal (PhD thesis in German). Institute University of Innsbruck, 1965spa
dc.relation.referencesB. Buchberger. An algorithmical criteria for the solvability of algebraic systems of equations. Aequationes Math, 4:374–383, 1970spa
dc.relation.referencesI. V. Cherednik. On R-matrix quantization of formal loop groups. Group theoretical methods in physics, 2:161–180, 1986.spa
dc.relation.referencesA. Chirvasitu and S. P. Smith. Some algebras having relations like those for the 4-dimensional Sklyanin algebras. J. Korean Math. Soc., 60(4):745–777, 2023.spa
dc.relation.referencesT. Cassidy and M. Vancliff. Generalizations of Graded Clifford Algebras and of Complete Intersections. J. Lond. Math. Soc., 81(1):91–112, 2010.spa
dc.relation.referencesV. G. Drinfeld. Quantum Groups. Zapiski Nauchnykh Seminarov POMI, 155:18– 49, 1986.spa
dc.relation.referencesW. Fajardo, C. Gallego, O. Lezama, A. Reyes, H. Suárez, and H. Venegas. Skew PBW Extensions. Ring and Module-theoretic Properties, Matrix and Gröbner Methods, and Applications. Springer, Cham, 2020.spa
dc.relation.referencesB. L. Feigin and A. V. Odesskii. Sklyanin algebras associated with an elliptic curve. Preprint deposited with Institute of Theoretical Physics of the Academy of Sciences of the Ukrainian SSR, page 33, 1989.spa
dc.relation.referencesB. L. Feigin and A. V. Odesskii. Sklyanin elliptic algebras. Funct. Anal. Appl., 23(3):207–214, 1989spa
dc.relation.referencesA. V. Golovashkin and V. M. Maksimov. Skew Ore polynomials of higher orders generated by homogeneous quadratic relations. Russian Math. Surveys, 53(2):384–386, 1998.spa
dc.relation.referencesE. L. Green. Noncommutative Gröbner bases, and projective resolutions. In Computational Methods for Representations of Groups and Algebras: Euroconference in Essen (Germany), April 1–5, 1977, pages 29–60. Springer, 1999.spa
dc.relation.referencesP. P. Grivel. Une histoire du théorème de Poincare-Birkhoff-Witt. Expo. Math., 22(2):145–184, 2004.spa
dc.relation.referencesK. R. Goodearl and R. B. Warfield. An Introduction to Noncommutative Noetherian Rings. Cambridge University Press. London, 2004.spa
dc.relation.referencesJ. Huang and Y. Chen. Gröbner- Shirshov Bases Theory for Trialgebras. Mathematics, 9(11):1207, 2021spa
dc.relation.referencesA. P. Isaev, P. N. Pyatov, and V. Rittenberg. Diffusion algebras. J. Phys. A., 34(29):5815–5834, 2001.spa
dc.relation.referencesN. Iyudu and S. Shkarin. Three dimensional Sklyanin algebras and Gröbner bases. J. Algebra, 470:379–419, 2017.spa
dc.relation.referencesJ. C. Jantzen. Lectures on Quantum Groups, volume 6. American Mathematical Soc., 1996.spa
dc.relation.referencesE. G. Karpuz, F. Ateş, and A.S. Çevik. Gröbner-Shirshov bases of some Weyl groups. Rocky Mountain J. Math, 2015.spa
dc.relation.referencesA. Kandri-Rody and V. Weispfenning. Non-commutative Gröbner Bases in Algebras of Solvable Type. J. Symbolic Computation, 9(1):1–26, 1990.spa
dc.relation.referencesV. Levandovskyy. Non-commutative Computer Algebra for polynomial algebras: Gröbner bases, applications and implementation. PhD thesis, Technische Univer- sität Kaiserslautern, 2005.spa
dc.relation.referencesO. Lezama. Cuadernos de Álgebra: Anillos y Módulos, volume 2. Universidad Nacional de Colombia, Bogotá, D. C., 2023.spa
dc.relation.referencesH. Li. Noncommutative Gröbner Bases and Filtered-Graded Transfer. Lecture Notes in Mathematics, LNM, volume 1795, Springer, 2002.spa
dc.relation.referencesY. Li, Q. Mo, and L. Bokut. Gröbner-Shirshov bases for symmetric brace algebras. Comm. Algebra, 49(2):892–904, 2020spa
dc.relation.referencesT. Mora. An Introduction to Commutative and Noncommutative Gröbner bases. Theoret. Comput. Sci., 134(1):131–173, 1994.spa
dc.relation.referencesM. H. A. Newman. On theories with a combinatorial definition of “equiva- lence”. Ann. Math, pages 223–243, 1942.spa
dc.relation.referencesA. Niño and A. Reyes. On centralizers and pseudo-multidegree functions for non-commutative rings having PBW bases. J. Algebra Appl., https://doi. org/10.1142/S0219498825501099, 2023.spa
dc.relation.referencesA. Niño, M. C. Ramírez, and A. Reyes. A first approach to the Burchnall- Chaundy theory for quadratic algebras having PBW bases. https://arxiv. org/abs/2401.10023, 2023.spa
dc.relation.referencesO. Ore. Linear Equations in Non-commutative Fields. Ann. of Math. (2), 32(3):463–477, 1931.spa
dc.relation.referencesO. Ore. Theory of Non-commutative Polynomials. Ann. of Math. (2), 34(3):480– 508, 1933.spa
dc.relation.referencesA. Polishchuk and L. Positselski. Quadratic algebras, volume 37. University Lecture Series. American Mathematical Soc., 2005.spa
dc.relation.referencesI. T. Redman. The Non-Commutative Algebraic Geometry of some Skew Polynomial Algebras. PhD thesis, University of Wisconsin - Milwaukee, 1996.spa
dc.relation.referencesI. T. Redman. The homogenization of the three dimensional skew polynomial algebras of type I. Comm. Algebra, 27(11):5587–5602, 1999.spa
dc.relation.referencesD. Rogalski. Artin-Schelter Regular Algebras. https://arxiv.org/pdf/2307.03174v1.pdf, 2023.spa
dc.relation.referencesJ. J. Rotman. Advanced Modern Algebra. Second Edition, volume 114. Graduate Studies in Mathematics. Springer, 2010.spa
dc.relation.referencesA. Reyes and H. Suárez. Bases for Quantum Algebras and Skew Poincaré- Birkhoff-Witt extensions. Momento, 54(1):54–75, 2017.spa
dc.relation.referencesA. Reyes and C. Sarmiento. On the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras. Internat. J. Algebra Comput., 32(03):529–559, 2022.spa
dc.relation.referencesK. Saito. Extended affine root systems i (Coxeter transformations). Publ. Res. Inst. Math. Sci., 21(1):75–179, 1985.spa
dc.relation.referencesW. M. Seiler. Involution. The Formal Theory of Differential Equations and its Applications in Computer Algebra, volume 24. Algorithms and Computation in Mathematics (AACIM). Springer Berlin, Heidelberg, 2010.spa
dc.relation.referencesA. I. Shirshov. On free Lie rings, volume 87. Russian Academy of Sciences, Steklov Mathematical Institute of Russian, 1958.spa
dc.relation.referencesA. I. Shirshov. Some algorithmic problems for Lie algebras. Springer, 1962.spa
dc.relation.referencesE. K. Sklyanin. Some algebraic structures connected with the Yang-Baxter equation. Funct. Anal. Appl., 16(4):263–270, 1982.spa
dc.relation.referencesE. K. Sklyanin. Some algebraic structures connected with the Yang-Baxter equation. Representations of Quantum algebras. Funct. Anal. Appl., 17(4):273– 284, 1983.spa
dc.relation.referencesS. P. Smith. “Degenerate” 3-dimensional Sklyanin algebras are monomial algebras. J. Algebra, 358:74–86, 2012.spa
dc.relation.referencesS. P. Smith and J. T. Stafford. Regularity of the four dimensional Sklyanin algebra. Compos. Math., 83(3):259–289, 1992.spa
dc.relation.referencesK. Saito and T. Takebayashi. Extended affine root systems III (elliptic Weyl groups). Publ. Res. Inst. Math. Sci., 33(2):301–329, 1997.spa
dc.relation.referencesJ. T. Stafford. Regularity of algebras related to the Sklyanin algebra. Trans. Amer. Math. Soc., 341(2):895–916, 1994.spa
dc.relation.referencesD. R. Stephenson. Artin-Schelter Regular Algebras of Global Dimension Three. J. Algebra, 183(1):55–73, 1996.spa
dc.relation.referencesB. Shelton and M. Vancliff. Embedding a quantum rank three quadric in a quantum P3. Comm. Algebra, 27(6):2877–2904, 1999.spa
dc.relation.referencesD. R. Stephenson and M. Vancliff. Some finite quantum P3s that are infinite modules over their centers. J. Algebra, 297(1):208–215, 2006.spa
dc.relation.referencesM. Vancliff. Quadratic algebras associated with the union of a quadric and a line in P3. J. Algebra, 165(1):63–90, 1994.spa
dc.relation.referencesK. Van Rompay and M. Vancliff. Four-dimensional regular algebras with point scheme a nonsingular quadric in P3. Comm. Algebra, 28(5):2211–2242, 2000.spa
dc.relation.referencesK. Van Rompay, M. Vancliff, and L. Willaert. Some quantum P3 with finitely many points. Comm. Algebra, 26(4):1193–1208, 1998.spa
dc.relation.referencesM. Vancliff and K. Van Rompay. Embedding a Quantum Nonsingular Quadric in a Quantum P3. J. Algebra, 195(1):93–129, 1997.spa
dc.relation.referencesC. Walton. Degenerate Sklyanin algebras and generalized twisted homogeneous coordinate rings. J. Algebra, 322(7):2508–2527, 2009.spa
dc.relation.referencesX. Xiu. Non-commutative Gröbner bases and applications. PhD thesis, Universität Passau, 2012.spa
dc.relation.referencesG. Yunus, Z. Gao, and A. Obul. Gröbner-Shirshov basis of quantum groups. 22(03):495–516, 2015.spa
dc.relation.referencesG. Yunus and A. Obul. Gröbner-Shirshov basis of quantum group of type D4. Chinese Ann. Math., Ser. B, 32(4):581–592, 2011.spa
dc.relation.referencesZ. Zhang, Y Chen, and Bokut. L. Word problem for finitely presented metabelian Poisson algebras. arXiv: Rings and Algebras, 2019.spa
dc.relation.referencesX. Zhao. Jacobson’s Lemma via Gröbner-Shirshov Bases. Algebra Colloq., 24(02):309–314, 2017.spa
dc.relation.referencesJ. J. Zhang and J. Zhang. Double Ore extensions. J. Pure Appl. Algebra, 212(12):2668–2690, 2008.spa
dc.relation.referencesJ. J. Zhang and J. Zhang. Double extension regular algebras of type (14641). J. Algebra, 322(2):373–409, 2009.spa
dc.relation.referencesA. I. Shirshov. Some algorithmic problems for $\varepsilon$-algebras. Springer, 1962.spa
dc.relation.referencesY. Ren and A. Obul. Gröbner-Shirshov basis of quantum group of type ${G}_2$. Comm. Algebra, 39(5):1510–1518, 2011.spa
dc.relation.referencesC. Qiang and A. Obul. The Skew-commutator Relations and Gröbner-Shirshov Bases of Quantum Group of Type ${C}_3$. Acta Math. Appl. Sin. Engl. Ser., 36(4):825–835, 2020.spa
dc.relation.referencesL. Le Bruyn and S. P. Smith. Homogenized $\mathfrak{sl}_2$. Proc. Amer. Math. Soc., 118(3):725–730, 1993.spa
dc.relation.referencesM. Jimbo. A $q$-difference analogue of ${U}(\mathfrak{g})$ and the Yang-Baxter equation. Lett. Math. Phys., 10:63–69, 1985.spa
dc.rights.accessrightsinfo:eu-repo/semantics/openAccessspa
dc.rights.licenseReconocimiento 4.0 Internacionalspa
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/spa
dc.subject.proposalBase de Gröbner-Shirshovspa
dc.subject.proposalAlgoritmo de Shirshovspa
dc.subject.proposalLema del diamantespa
dc.subject.proposalÁlgebra PBWspa
dc.subject.proposalÁlgebra de Sklyaninspa
dc.subject.proposalGröbner-Shirshov basiseng
dc.subject.proposalShirshov's algorithmeng
dc.subject.proposalDiamond lemmaeng
dc.subject.proposalPBW algebraeng
dc.subject.proposalSklyanin algebraeng
dc.titleGröbner-Shirshov bases for Sklyanin algebraseng
dc.title.translatedBases de Gröbner-Shirshov para álgebras de Sklyaninspa
dc.typeTrabajo de grado - Maestríaspa
dc.type.coarhttp://purl.org/coar/resource_type/c_bdccspa
dc.type.coarversionhttp://purl.org/coar/version/c_ab4af688f83e57aaspa
dc.type.contentTextspa
dc.type.driverinfo:eu-repo/semantics/masterThesisspa
dc.type.redcolhttp://purl.org/redcol/resource_type/TMspa
dc.type.versioninfo:eu-repo/semantics/acceptedVersionspa
oaire.accessrightshttp://purl.org/coar/access_right/c_abf2spa

Archivos

Bloque original

Mostrando 1 - 1 de 1
Miniatura
Nombre:
1053587838.2024.pdf
Tamaño:
1.44 MB
Formato:
Adobe Portable Document Format
Descripción:
Tesis de Maestría en Ciencia - Matemáticas
Descargar

Bloque de licencias

Mostrando 1 - 1 de 1
No hay miniatura disponible
Nombre:
license.txt
Tamaño:
5.74 KB
Formato:
Item-specific license agreed upon to submission
Descripción:
Descargar

Colecciones

Maestría en Ciencias - Matemáticas