文摘
In this chapter, we demonstrate that every character Hopf algebra has a PBW basis. A Hopf algebra H is referred to as a character Hopf algebra if the group G of all group-like elements is commutative and H is generated over k [G] by skew-primitive semi-invariants, whereas a well-ordered subset \(V \subseteq H\) is a set of PBW generators of H if there exists a function \(h: V \rightarrow \mathbf{Z^{+}} \cup \{\infty \},\) called the height function, such that the set of all products $$\displaystyle{gv_{1}^{n_{1} }v_{2}^{n_{2} }\,\cdots \,v_{k}^{n_{k} },}$$ where \(g \in G,\ \ v_{1} < v_{2} <\ldots < v_{k} \in V,\ \ n_{i} < h(v_{i}),1 \leq i \leq k\) is a basis of H.
Page %P Close Plain text Look Inside Chapter Metrics Provided by Bookmetrix Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions About this Book Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Supplementary Material (0) References (236) References1.Abe, E.: Hopf Algebras. Cambridge University Press, Cambridge (1980)MATH2.Andruskiewitsch, N., Graña, M.: Braided Hopf algebras over non abelian finite group. Bol. Acad. Nac. Cienc. Cordoba 63, 46–78 (1999)MATH3.Andruskiewitsch, N., Schneider, H.-J.: Pointed Hopf algebras. In: Montgomery, S., Schneider, H.-J. (eds.) New Directions in Hopf Algebras. MSRI Publications, vol. 43, pp. 1–69. Cambridge University Press, Cambridge (2002)4.Andruskiewitsch, N., Schneider, H.-J.: On the classification of finite-dimensional pointed Hopf algebras. Ann. Math. 171(2)(1), 375–417 (2010)5.Andruskiewitsch, N., Heckenberger, I., Schneider, H.-J.: The Nichols algebra of a semisimple Yetter-Drinfeld mo
dule. Am. J. Math. 132(6), 1493–1547 (2010)MathSciNetMATH6.Apel J., Klaus, U.: FELIX (1991). http://felix.hgb-leipzig.de 7.Ardizzoni, A.: A Milnor-Moore type theorem for primitively generated braided bialgebras. J. Algebra 327, 337–365 (2011)MathSciNetMATHCrossRef8.Ardizzoni, A.: On the combinatorial rank of a graded braided bialgebra. J. Pure Appl. Algebra 215(9), 2043–2054 (2011)MathSciNetMATHCrossRef9.Ardizzoni, A.: On primitively generated braided bialgebras. Algebr. Represent. Theor. 15(4), 639–673 (2012)MathSciNetMATHCrossRef10.Ardizzoni, A.: Universal enveloping algebras of the PBW type. Glasgow Math. J. 54, 9–26 (2012)MathSciNetMATHCrossRef11.Ardizzoni, A., Menini, C.: Associated graded algebras and coalgebras. Commun. Algebra 40(3), 862–896 (2012)MathSciNetMATHCrossRef12.Ardizzoni, A., Stumbo, F.: Quadratic lie algebras. Commun. Algebra 39(8), 2723–2751 (2011)MathSciNetMATHCrossRef13.Ardizzoni, A., Menini, C., Ştefan, D.: Braided bialgebras of Hecke type. J. Algebra 321, 847–865 (2009)MathSciNetMATHCrossRef14.Bahturin, Y.A., Mikhalev, A.A., Petrogradsky, V.M., Zaicev, M.V.: Infinite-Dimensional Lie Superalgebras. De Gruyter Expositions in Mathematics, vol. 7. Walter de Gruyter, Berlin (1992)15.Bahturin, Y., Fishman, D., Montgomery, S.: On the generalized Lie structure of associative algebras. Israel J. Math. 96, 27–48 (1996)MathSciNetMATHCrossRef16.Bahturin, Y., Mikhalev, A.A., Zaicev, M.: Infinite-Dimensional Lie Superalgebras. Handbook of Algebra, vol. 2, pp. 579–614. North-Holland, Amsterdam (2000)17.Bahturin, Y., Fishman, D., Montgomery, S.: Bicharacters, twistings and Scheunert’s theorem for Hopf algebras. J. Algebra 236, 246–276 (2001)MathSciNetMATHCrossRef18.Bahturin, Y., Kochetov, M., Montgomery, S.: Polycharacters of cocommutative Hopf algebras. Can. Math. Bull. 45, 11–24 (2002)MathSciNetMATHCrossRef19.Bai, X.,
Hu, N.: Two-parameter quantum groups of exceptional type E-series and convex PBW-type basis. Algebra Colloq. 15(4), 619–636 (2008)MathSciNetMATHCrossRef20.Barratt, M.G.: Twisted Lie algebras. In: Barratt, M.G., Mahowald, M.E. (eds.) Geometric Applications of Homotopy Theory II. Lecture Notes in Mathematics, vol. 658, pp. 9–15. Springer, Berlin (1978)CrossRef21.Bautista, C.: A Poincaré-Birkhoff-Witt theorem for generalized Lie color algebras. J. Math. Phys. 39(7), 3828–3843 (1998)MathSciNetMATHCrossRef22.Benkart, G., Witherspoon, S.: Representations of two-parameter quantum groups and Sc
hur-Weyl
duality. Hopf Algebras. Lecture Notes in Pure and Applied Mathematics, vol. 237, pp. 65–92. Dekker, New York (2004)23.Benkart, G., Witherspoon, S.: Two-parameter quantum groups and Drinfel’d doubles. Algebr. Represent. Theor. 7(3), 261–286 (2004)MathSciNetMATHCrossRef24.Benkart, G., Kang, S.-J., Melville, D.: Quantized enveloping algebras for Borcherds superalgebras. Trans. Am. Math. Soc. 350(8), 3297–3319 (1998)MathSciNetMATHCrossRef25.Berezin, F.A.: Method of the Secondary Quantification. Nauka, Moscow (1965)26.Berezin, F.A.: Mathematical foundations of the supersymmetric field theories. Nucl. Phys. 29(6), 1670–1687 (1979)MathSciNet27.Berezin, F.A.: Intro
duction to Superanalysis. Mathematical Physics and Applied Mathematics, vol. 9. D. Reidel, Dordrecht/Boston, MA (1987)28.Berger, R.: The quantum Poincaré-Birkhoff-Witt theorem. Commun. Math. Phys. 143(2), 215–234 (1992)MATHCrossRef29.Bergeron, N., Gao, Y., Hu, N.: Drinfel’d doubles and Lusztig’s symmetries of two-parameter quantum groups. J. Algebra 301, 378–405 (2006)MathSciNetMATHCrossRef30.Bergman, G.M.: The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978)MathSciNetCrossRefMATH31.Boerner, H.: Representation of Groups with Special Consideration for the Needs of Modern Physics. North-Holland, New York, NY (1970)MATH32.Bokut, L.A.: Imbedding into simple associative algebras. Algebra Log. 15, 117–142 (1976)MathSciNetMATHCrossRef33.Bokut, L.A.: Associative Rings I. Library of the Department of Algebra and Logic, vol. 18. Novosibirsk State University, Novosibirsk (1977)34.Bokut, L.A.: Gröbner–Shirshov basis for the braid group in the Artin-Garside generators. J. Symb. Comput. 43(6–7), 397–405 (2008)MathSciNetMATHCrossRef35.Bokut, L.A., Chen, Y.: Gröbner–Shirshov bases and their calculation. Bull. Math. Sci. 4(3), 325–395 (2014)MathSciNetMATHCrossRef36.Bokut, L.A., Kolesnikov, P.S.: Gröbner–Shirshov bases from their inception to the present time. Int. J. Math. Sci. NY 116(1), 2894–2916 (2003). Translation from Zap. Nauchn. Sem. POMI 272 26–67 (2000)37.Bokut’, L.A., Kukin, G.P.: Algorithmic and Combinatorial Algebra. Mathematics and Its Applications, vol. 255. Kluwer, Dordrecht/Boston/London (1994)38.Bokut, L.A., Vesnin, A.: Gröbner–Shirshov bases for some braid groups. J. Symb. Comput. 41(3–4), 357–371 (2006)MathSciNetMATHCrossRef39.Borowiec, A., Kharchenko, V.K.: Algebraic approach to calculuses with partial derivatives. Sib. Adv. Math. 5(2), 10–37 (1995)MathSciNetMATH40.Borowiec, A., Kharchenko, V.K.: First order optimum calculi. Bulletin de la Société des sciences et des lettres de Łódź 45, 75–88 (1995). Recher. Deform. XIX41.Borowiec, A., Kharchenko, V.K., Oziewicz, Z.: On free differentials on associative algebras. In: Gonzáles, A. (ed.) Non Associative Algebras and Its Applications. Mathematics and Applications, pp. 43–56. Kluwer, Dordrecht (1994)42.Braverman, A., Gaitsgory, D.: Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type. J. Algebra 181, 315–328 (1996)MathSciNetMATHCrossRef43.Buchberger, B.: An algorithm for finding a basis for the resi
due class ring of a zero-dimensional ideal. Ph.D. Thesis, University of Innsbruck (1965)44.Buchberger, B.: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleic
hungssystems. Aequtiones Math. 4, 374–383 (1970)MathSciNetMATHCrossRef45.Cameron, P.J.: Permutation Groups, London Mathematical Society. Student Texts, vol. 45. University Press, Cambridge (1999)46.Chaichian, M., Demichev, A.: Intro
duction to Quantum Groups. World Scientific, Singapore (1996)MATHCrossRef47.Chari, V., Xi, N.: Monomial basis of quantized enveloping algebras. In: Recent Development in Quantum Affine Algebras and Related Topics. Contemporary Mathematics, vol. 248, pp. 69–81. American Mathematical Society, Providence, RI (1999)48.Chen, Y.Q., Shao, H.S., S
hum, K.P.: On Rosso–Yamane theorem on PBW basis of U
q (A
N ). CUBO Math. J. 10, 171–194 (2008)MathSciNetMATH49.C
huang, C.-L.: Identities with skew derivations. J. Algebra 224, 292–335 (2000)MathSciNetMATHCrossRef50.Cohen, M., Fishman, D., Westreich, S.: Sc
hur’s double centralizer theorem for triangular Hopf algebras. Proc. Am. Math. Soc. 122(1), 19–29 (1994)MATH51.Cohn, P.M.: Sur le critère de Friedrichs pour les commutateur dans une algèbre associative libre. C. R. Acad. Sci. Paris 239(13), 743–745 (1954)MathSciNetMATH52.Cohn, P.M.: The class of rings embeddable in skew fields. Bull. Lond. Math. Soc. 6, 147–148 (1974)MATHCrossRefMathSciNet53.Cohn, P.M.: Free Rings and Their Relations, 2nd edn. Academic Press, London (1985)MATH54.Connes, A.: Non-commutative Geometry. Academic Press, New York (1994)MATH55.Constantini, M., Varagnolo, M.: Quantum double and multiparameter quantum groups. Commun. Algebra 22(15), 6305–6321 (1994)CrossRefMathSciNetMATH56.Cotta-Ramusino, P., Rinaldi, M.: Multiparameter quantum groups related to link diagrams. Commun. Math. Phys. 142, 589–604 (1991)MathSciNetMATHCrossRef57.Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Interscience/Wiley, New York/London (1962)MATH58.De Concini, C., Kac, V.G., Procesi, C.: Quantum coadjoint action. J. Am. Math. Soc. 5, 151–189 (1992)MATHCrossRefMathSciNet59.Deng, B., Du, J.: Bases of quantized enveloping algebras. Pac. J. Math. 220(1), 33–48 (2005)MathSciNetMATHCrossRef60.Deng, B., Du, J.: Monomial bases for quantum affine \(\mathfrak{s}l_{n}\). Adv. Math. 191(2), 276–304 (2005)MathSciNetMATHCrossRef61.Diaconis, P.: Group Representations in Probability and Statistics. Lecture Notes Monograph Series, vol. 11. Institute of Mathematical Statistics, Hayward, CA (1988)62.Diaz Sosa, M.L., Kharchenko, V.K.: Combinatorial rank of \(u_{q}(\mathfrak{s}\mathfrak{o}_{2n})\). J. Algebra 448, 48–73 (2016)MathSciNetCrossRefMATH63.Dito, G., Flato, M., Sternheimer, D., Takhtadjian, L.: Deformation quantization and Nambu mechanics. Commun. Math. Phys. 183, 1–22 (1997)MATHCrossRefMathSciNet64.Dixon, D., Mortimer, B.: Permutation Groups. Springer, Berlin (1996)MATHCrossRef65.Drinfeld, V.G.: Hopf algebras and the Yang–Baxter equation. Soviet Math. Dokl. 32, 254–258 (1985)MathSciNet66.Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematics, Berkeley, CA, vol. 1, pp. 798–820 (1986)MathSciNet67.Drinfeld, V.G.: On almost cocommutative Hopf algebras. Algebra i Analiz 1(2), 30–46 (1989). English translation: Leningr. Math. J. 1, 321–342 (1990)68.Etingof, P., Gelaki, S.: The classification of finite-dimensional triangular Hopf algebras over an algebraically closed field. Int. Math. Res. Not. 5, 223–234 (2000)MathSciNetCrossRefMATH69.Etingof, P., Gelaki, S.: On cotriangular Hopf algebras. Am. J. Math. 123, 699–713 (2001)MathSciNetMATHCrossRef70.Etingof, P., Gelaki, S.: The classification of triangular semisimple and cosemisimple Hopf algebras over an algebraically closed field of characteristic zero. Mosc. Math. J. 3(1), 37–43 (2003)MathSciNetMATH71.Etingof, P., Schiffman, O.: Lectures on Quantum Groups. International Press Incorporated, Boston (1998)MATH72.FELIX – an assistant for algebraists. In: Watt, S.M. (ed.) ISSACi91, pp. 382–389. ACM Press, New York (1991)73.Ferreira, V.O., Murakami, L.S.I., Paques, A.: A Hopf-Galois correspondence for free algebras. J. Algebra 276 407–416 (2004)MathSciNetMATHCrossRef74.Filippov, V.T.: n-Lie algebras. Sib. Math. J. 26(6), 126–140 (1985)75.Finkelshtein, D.: On relations between commutators. Commun. Pure. Appl. Math. 8, 245–250 (1955)CrossRefMathSciNet76.Fischman, D., Montgomery, S.: A Sc
hur double centralizer theorem for cotriangular Hopf algebras and generalized Lie algebras. J. Algebra 168, 594–614 (1994)MathSciNetMATHCrossRef77.Flores de Chela, D.: Quantum symmetric algebras as braided Hopf algebras. In: De la Peña, J.A., Vallejo, E., Atakishieyev, N. (eds.) Algebraic Structures and Their Representations. Contemporary Mathematics, vol. 376, pp. 261–271. American Mathematical Society, Providence, RI (2005)CrossRef78.Flores de Chela, D., Green, J.A.: Quantum symmetric algebras II. J. Algebra 269, 610–631 (2003)79.Fox, R.H.: Free differential calculus. I. Ann. Math. 57(3), 517–559 (1953); II. Ann. Math. 59, 196–210 (1954); III. Ann. Math. 64, 407–419 (1956); IV. Ann. Math. 68, 81–95 (1958); IV. Ann. Math. 71, 408–422 (1960)80.Friedrichs, K.O.: Mathematical aspects of the quantum theory of fields V. Commun. Pure Applied Math. 6, 1–72 (1953)MathSciNetMATHCrossRef81.Frønsdal, C.: On the Classification of q-algebras. Lett. Math. Phys. 53, 105–120 (2000)MathSciNetCrossRefMATH82.Frønsdal, C., Galindo, A.: The ideals of free differential algebras. J. Algebra 222, 708–746 (1999)MathSciNetCrossRefMATH83.Gabber, O., Kac, V.G.: On defining relations of certain infinite-dimensional Lie algebras. Bull. Am. Math. Soc. 5, 185–189 (1981)MathSciNetMATHCrossRef84.Gavarini, F.: A PBW basis for Lusztig’s form of untwisted affine quantum groups. Commun. Algebra 27(2), 903–918 (1999)MathSciNetMATHCrossRef85.Gelaki, S.: On the classification of finite-dimensional triangular Hopf Algebras. In: Montgomery, S., Schneider, H.-J. (eds.) New Directions in Hopf Algebras. Cambridge University Press, Cambridge, MSRI Publications, vol. 43, pp. 69–116 (2002)MathSciNet86.Gomez, X., Majid, S.: Braided Lie algebras and bicovariant differential calculi over co-quasitriangular Hopf algebras. J. Algebra 261(2), 334–388 (2003)MathSciNetMATHCrossRef87.Graña, M., Heckenberger, I.: On a factorization of graded Hopf algebras using Lyndon words. J. Algebra 314(1), 324–343 (2007)MathSciNetMATHCrossRef88.Gurevich, D.: Generalized translation operators on Lie Groups. Sov. J. Contemp. Math. Anal. 18, 57–70 (1983). Izvestiya Akademii Nauk Armyanskoi SSR. Matematica 18(4), 305–317 (1983)89.Gurevich, D.: Algebraic aspects of the quantum Yang-Baxter equation. Leningr. Math. J. 2(4), 801–828 (1991)MATHMathSciNet90.Heckenberger, I.: Weyl equivalence for rank 2 Nichols algebras of diagonal type. Ann. Univ. Ferrara Sez. VII (NS) 51(1), 281–289 (2005)91.Heckenberger, I.: The Weyl groupoid of a Nichols algebra of diagonal type. Invent. Math. 164, 175–188 (2006)MathSciNetMATHCrossRef92.Heckenberger, I.: Classification of arithmetic root systems. Adv. Math. 220, 59–124 (2009)MathSciNetMATHCrossRef93.Heckenberger, I., Kolb, S.: Right coideal subalgebras of the Borel part of a quantized enveloping algebra. Int. Math. Res. Not. 2, 419–451 (2011)MathSciNetMATH94.Heckenberger, I., Kolb, S.: Homogeneous right coideal subalgebras of quantized enveloping algebras. Bull. Lond. Math. Soc. 44(4), 837–848 (2012)MathSciNetMATHCrossRef95.Heckenberger, I., Schneider, H.-J.: Root systems and Weyl groupoids for Nichols algebras. Proc. Lond. Math. Soc. 101(3), 623–654 (2010)MathSciNetMATHCrossRef96.Heckenberger, I., Schneider, H.-J.: Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid. Isr. J. Math. 197(1), 139–187 (2013)MathSciNetMATHCrossRef97.Heckenberger, I., Yamane, H.: Drinfel’d doubles and Shapovalov determinants. Rev. Un. Mat. Argentina 51(2), 107–146 (2010)MathSciNetMATH98.Herstein, I.: Noncommutative Rings. Carus Mathematical Monographs, vol. 15, The Mathematical Association of America, USA (1968)99.Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. Math. (2) 79, 109–203, 205–326 (1964)100.Hofmann, K.H., Strambach, K.: Topological and analytic loops. In: Chein, O., Pflugfelder, H., Smith, J.D.H (eds.) Quasigroups and Loops: Theory and Applications, pp. 205–262. Heldermann, Berlin (1990)101.Hopf, H.: Über die Topologeie der Gruppenmannigfaltigkeiten und ihre Verallgemeinerungen. Ann. Math. 42, 22–52 (1941)MathSciNetCrossRefMATH102.Humphreys, J.E.: Intro
duction to Lie Algebras and Representation Theory. Springer, New York-Heidelberg-Berlin (1972)MATHCrossRef103.Ion, B.: Relative PBW type theorems for symmetrically braided Hopf algebras. Commun. Algebra 39(7), 2508–2518 (2011)MathSciNetMATHCrossRef104.Jacobson, N.: Lie Algebras. Interscience, New York (1962)MATH105.Jantzen, J.C.: Lectures on Quantum Groups. Gra
duate Studies in Mathematics, vol. 6. American Mathematical Society, Providence, RI (1996)106.Jimbo, M.: A q-difference analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–69 (1985)MathSciNetMATHCrossRef107.Joseph, A.: Quantum Groups and Their Primitive Ideals. Springer, Berlin, Heidelberg (1995)MATHCrossRef108.Joyal, A.: Foncteurs analytiques et espèces de structures. Combinatoire énumérative. Lecture Notes in Mathematics, vol. 1234, pp. 126–159. Springer, Berlin (1986)109.Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–87 (1993)MathSciNetMATHCrossRef110.Jumphreys, J.E.: Linear Algebraic Groups. Gra
duate Texts in Mathematics 21. Springer-Verlag, New York (1975)111.Kac, G.I.: Ring groups and
duality principle. Proc. Mosc. Math. Soc. 12, 259–301 (1963)MathSciNet112.Kac, V.G.: Simple irre
ducible graded Lie algebras of finite growth. Izvestija AN USSR (Ser. Mat.) 32, 1923–1967 (1968). English translation: Math. USSR-Izvestija, 2, 1271–1311 (1968)113.Kac, V.G.: Classification of Simple Lie Superalgebras. Funct. Analys i ego prilozh. 9(3), 91–92 (1975); Letter to the editor, 10(2), 93 (1976); English translation: Funct. Anal. Appl. 9, 263–265 (1975)114.Kac, V.G.: A sketch of Lie superalgebra theory. Commun. Math. Phys. 53, 31–64 (1977)MATHCrossRefMathSciNet115.Kac, V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)MATHCrossRef116.Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1995)117.Kac, V.G.: Classification of infinite-dimensional simple linearly compact Lie superalgebras. Adv. Math. 139, 1–55 (1998)MathSciNetMATHCrossRef118.Kang, S.-J.: Quantum deformations of generalized Kac-Moody algebras and their mo
dules. J. Algebra 175, 1041–1066 (1995)MathSciNetMATHCrossRef119.Kashiwara, M.: On crystal bases of the q-analog of universal enveloping algebras. Duke Math. J. 63(2), 465–516 (1991)MathSciNetMATHCrossRef120.Kassel, C.: Quantum Groups. Springer, New York (1995)MATHCrossRef121.Kharchenko, V.K.: Automorphisms and Derivations of Associative Rings. Mathematics and Its Applications (Soviet Series), vol. 69. Kluwer, Dordrecht/Boston/London (1991)122.Kharchenko, V.K.: Noncommutative Galois Theory. Nauchnaja Kniga, Novosibirsk (1996)MATH123.Kharchenko, V.K.: An algebra of skew primitive elements. Algebra i Logika 37(2), 181–224 (1998). English translation: Algebra and Logic 37(2), 101–126 (1998)124.Kharchenko, V.K.: A quantum analog of the Poincaré-Birkhoff-Witt theorem. Algebra i Logika 38(4), 476–507 (1999). English translation: Algebra and Logic 38(4), 259–276 (1999)125.Kharchenko, V.K.: An existence condition for multilinear quantum operations. J. Algebra 217, 188–228 (1999)MathSciNetMATHCrossRef126.Kharchenko, V.K.: Multilinear quantum Lie operations. Zap. Nauchn. Sem. S.-Petersburg. Otdel. Mat. Inst. Steklov. 272. Vopr. Teor. Predst. Algebr. i Grupp. 7, 321–340 (2000)127.Kharchenko, V.K.: Skew primitive elements in Hopf algebras and related identities. J. Algebra 238(2), 534–559 (2001)MathSciNetMATHCrossRef128.Kharchenko, V.K.: A combinatorial approach to the quantification of Lie algebras. Pac. J. Math. 203(1), 191–233 (2002)MathSciNetMATHCrossRef129.Kharchenko, V.K.: Constants of coordinate differential calculi defined by Yang–Baxter operators. J. Algebra 267, 96–129 (2003)MathSciNetMATHCrossRef130.Kharchenko, V.K.: Multilinear quantum Lie operations. J. Math. Sci. (NY) 116(1), 3063–3073 (2003)131.Kharchenko, V.K.: Braided version of Shirshov-Witt theorem. J. Algebra 294, 196–225 (2005)MathSciNetMATHCrossRef132.Kharchenko, V.K.: Connected braided Hopf algebras. J. Algebra 307, 24–48 (2007)MathSciNetMATHCrossRef133.Kharchenko, V.K.: PBW-bases of coideal subalgebras and a freeness theorem. Trans. Am. Math. Soc. 360(10), 5121–5143 (2008)MathSciNetMATHCrossRef134.Kharchenko, V.K.: Triangular decomposition of right coideal subalgebras, J. Algebra 324(11), 3048–3089 (2010)MathSciNetMATHCrossRef135.Kharchenko, V.K.: Right coideal subalgebras of U
q +(so
2n+1). J. Eur. Math. Soc. 13(6), 1677–1735 (2011)MathSciNetMATH136.Kharchenko, V.K.: Quantizations of Kac–Moody algebras. J. Pure Appl. Algebra 218(4), 666–683 (2014)MathSciNetMATHCrossRef137.Kharchenko, V.K., Andrede Álvarez, A.: On the combinatorial rank of Hopf algebras. Contemp. Math. 376, 299–308 (2005)CrossRefMathSciNetMATH138.Kharchenko, V.K., Díaz Sosa, M.L.: Computing of the combinatorial rank of \(u_{q}(\mathfrak{s}o_{2n+1})\). Commun. Algebra 39(12), 4705–4718 (2011)MATHCrossRef139.Kharchenko, V.K., Lara Sagahón, A.V.: Right coideal subalgebras in U
q (sl
n+1). J. Algebra 319(6), 2571–2625 (2008)MathSciNetMATHCrossRef140.Kharchenko, V.K., Lara Sagahón, A.V., Garza Rivera, J.L.: Computing of the number of right coideal subalgebras of U
q (so
2n+1). J. Algebra 341(1), 279–296 (2011); Corrigen
dum 35(1), 224–225 (2012)141.Killing, W.: Die Zusammensetzung der steigen endlichen Transformationsgruppen, I. Math. Ann. 31, 252–290 (1886)MathSciNetCrossRefMATH142.Klimik, A., Schmüdgen, K.: Quantum Groups and Their Representations. Springer, Berlin, Heidelberg (1997)CrossRef143.Kochetov, M.: Generalized Lie algebras and cocycle twists. Commun. Algebra 36, 4032–4051 (2008)MathSciNetMATHCrossRef144.Kochetov, M., Ra
du, O.: Engel’s theorem for generalized Lie algebras. Algebr. Represent. Theor. 13(1), 69–77 (2010)MathSciNetMATHCrossRef145.Larson, R.G., Towber, J.: Two
dual classes of bialgebras related to the concept of “quantum groups” and “quantum Lie algebras”. Commun. Algebra 19(12), 3295–3345 (1991)MathSciNetMATHCrossRef146.Leclerc, B.: Dual canonical bases, quantum s
huffles and q-characters. Math. Z. 246(4), 691–732 (2004)MathSciNetMATHCrossRef147.Letzter, G.: Coideal subalgebras and quantum symmetric pairs. In: Montgomery, S., Schneider, H.-J. (eds.) New Directions in Hopf Algebras. Cambridge University Press, Cambridge, MSRI Publications, vol. 43, pp. 117–165 (2002)MathSciNet148.Lothaire, M.: Combinatorics on Words. Encyclopedia of Mathematics and Its Applications, vol. 17. Addison-Wesley, Reading, MA (1983)149.Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)MATHCrossRef150.Lothaire, M.: Applied Combinatorics on Words. Cambridge University Press, Cambridge (2005)MATHCrossRef151.Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3(2), 447–498 (1990)MathSciNetMATHCrossRef152.Lusztig, G.: Finite-dimensional Hopf algebras arising from quantized enveloping algebras. J. Am. Math. Soc. 3(1), 257–296 (1990)MathSciNetMATH153.Lusztig, G.: Intro
duction to Quantum Groups. Progress in Mathematics, vol. 110. Birkhauser, Boston (1993)154.Lyndon, R.C.: On Burnside’s problem. Trans. Am. Math. Soc. 7, 202–215 (1954)MathSciNetMATH155.Lyndon, R.C.: A theorem of Friedrichs. Mich. Math. J. 3(1), 27–29 (1955–1956)156.Lyubashenko, V.V.: Hopf algebras and vector symmetries. Sov. Math. Surveys 41, 153–154 (1986)MathSciNetMATHCrossRef157.Magnus, W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954)MathSciNetMATHCrossRef158.Majid, S.: Algebras and Hopf Algebras in Braided Categories. Advances in Hopf Algebras. Lecture Notes in Pure Applied Mathematics, vol. 158, pp. 55–105. Dekker, New York (1994)159.Majid, S.: Crossed pro
ducts by braided groups and bosonization. J. Algebra 163, 165–190 (1994)MathSciNetMATHCrossRef160.Majid, S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (1995)MATHCrossRef161.Mal’tcev, A.I.: On the immersion of an algebraic ring into a field. Math. Ann. 113, 686–691 (1937)MathSciNetCrossRef162.Mal’tcev, A.I.: On the inclusion of associative systems into groups, I. Mat. Sbornik 6(2), 331–336 (1939)163.Mal’tcev, A.I.: On the inclusion of associative systems in groups, II. Mat. Sbornik 8(2), 251–263 (1940)164.Mal’tcev, A.I.: Algebraic Systems. Nauka, Moscow (1965)165.Mal’tcev, A.I.: Algorithms and Recursive Functions. Nauka, Moscow (1970)166.Manin, Y.: Quantum Groups and Non-Commutative Geometry. CRM Publishing, Université de Montréal (1988)MATH167.Masuoka, A.: Formal groups and unipotent affine groups in non-categorical symmetry. J. Algebra 317, 226–249 (2007)MathSciNetMATHCrossRef168.Moody, R.V.: Lie algebras associated with generalized Cartan matrices. Bull. Am. Math. Soc. 73, 217–221 (1967)MathSciNetMATHCrossRef169.Miheev, P.O., Sabinin, L.V.: Quasigroups and differential geometry. In: Chein, O., Pflugfelder, H., Smith, J.D.H. (eds.) Quasigroups and Loops: Theory and Applications, pp. 357–430. Heldermann, Berlin (1990)170.Mikhalev, A.A.: Subalgebras of free color Lie superalgebras. Mat. Zametki 37(5), 653–661 (1985). English translation: Math. Notes 37, 356–360 (1985)171.Mikhalev, A.A.: Free color Lie superalgebras. Dokl. Akad. Nauk SSSR 286(3), 551–554 (1986). English translation: Soviet Math. Dokl. 33, 136–139 (1986)172.Mikhalev, A.A.: Subalgebras of free Lie p-Superalgebras. Mat. Zametki 43(2), 178–191 (1988). English translation: Math. Notes 43, 99–106 (1988)173.Milinski, A.: Actions of pointed Hopf algebras on prime algebras. Commun. Algebra 23, 313–333 (1995)MathSciNetMATHCrossRef174.Milinski, A.: Operationen punktierter Hopfalgebren auf primen Algebren. Ph.D. thesis, München (1995)175.Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. 81, 211–264 (1965)MathSciNetMATHCrossRef176.Montgomery, S.: Hopf Algebras and Their Actions on Rings. Conference Board of Mathematical Sciences, vol. 82. American Mathematical Society, Providence, RI (1993)177.Mostovoy, J., Pérez Izquierdo, J.M.: Formal multiplications, bialgebras of distributions and non-associative Lie theory. Transform. Groups 15, 625–653 (2010)MathSciNetMATHCrossRef178.Mostovoy, J., Pérez-Izquierdo, J.M., Shestakov, I.P.: Hopf algebras in non-associative Lie theory. Bull. Math. Sci. 4(1), 129–173 (2014)MathSciNetMATHCrossRef179.Musson, I.M.: Lie Superalgebras and Enveloping Algebras. Gra
duate Studies in Mathematics, vol. 131. American Mathematical Society, Providence, RI (2012)180.Nambu, Y.: Generalized Hamilton dynamics. Phys. Rev. D 7(8), 2405–2412 (1973)MathSciNetMATHCrossRef181.Nichols, W.: Bialgebras of type one. Commun. Algebra 6(15), 1521–1552 (1978)MathSciNetMATHCrossRef182.Nielsen, J.: A basis for subgroups of free groups. Math. Scand. 3, 31–43 (1955)MathSciNetMATH183.Pareigis, B.: Skew-primitive elements of quantum groups and braided Lie algebras. In: Caenepeel, S., Verschoren, A. (eds.) Rings, Hopf Algebras, and Brauer Groups (Antwerp/Brussels). Lecture Notes in Pure and Applied Mathematics, vol. 197, pp. 219–238. Marcel Dekker, Inc, New York (1998)184.Pareigis, B.: On Lie algebras in braided categories. In: Budzyński, R., Pusz, W., Zakrzewski, S. (eds.) Quantum Groups and Quantum Spaces, vol. 40, pp. 139–158. Banach Center Publications, Warszawa (1997)185.Pareigis, B.: On Lie algebras in the category of Yetter–Drinfeld mo
dules. Appl. Categ. Struct. 6(2), 152–175 (1998)MathSciNetCrossRefMATH186.Pérez Izquierdo, J.M.: An envelope for Bol algebras. J. Algebra 284, 480–493 (2005)MathSciNetMATHCrossRef187.Pérez Izquierdo, J.M.: Algebras, hyperalgebras, nonassociative bialgebras and loops. Adv. Math. 208, 834–876 (2007)MathSciNetMATHCrossRef188.Pérez Izquierdo, J.M., Shestakov, I.P.: An envelope for Malcev algebras. J. Algebra 272, 379–393 (2004)MathSciNetMATHCrossRef189.Polishc
huk, A.E., Positselsky, L.E.: Quadratic Algebras. University Lecture Series, vol. 37. American Mathematical Society, Providence, RI (2005)190.Radford, D.E.: The structure of Hopf algebras with projection. J. Algebra 92, 322–347 (1985)MathSciNetMATHCrossRef191.Radford, D.E.: Hopf Algebras. Series on Knots and Everything, vol. 49. World Scientific, Singapore (2012)192.Reineke, M.:
Feigin’s map and monomial bases for quantized enveloping algebras. Math. Z. 237(3), 639–667 (2001)MathSciNetMATHCrossRef193.Reshetikhin, N.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett. Math. Phys. 20, 331–335 (1990)MathSciNetMATHCrossRef194.Ringel, C.M.: PBW-bases of quantum groups. J. Reine Angew. Math. 470, 51–88 (1996)MathSciNetMATH195.Rosso, M.: An analogue of the Poincare–Birkhoff–Witt theorem and the universal R-matrix of U
q (sl(N + 1)). Commun. Math. Phys. 124, 307–318 (1989)MathSciNetMATHCrossRef196.Rosso, M.: Groupes quantiques et algèbres de battagesquantiques. C.R. Acad. Sci. Paris 320, 145–148 (1995)MathSciNetMATH197.Rosso, M.: Quantum groups and quantum S
huffles. Invent. Math. 113(2) 399–416 (1998)MathSciNetCrossRefMATH198.Sabinin, L.V.: Smooth Quasigroups and Loops. Mathematics and Its Applications, vol. 429. Kluwer, Dordrecht/Boston/London (1999)199.Sabinin, L.V.: Smooth quasigroups and loops: forty-five years of incredible growth. Comment. Math. Univ. Carol. 41, 377–400 (2000)MathSciNetMATH200.Sagan, B.E.: The Symmetric Group. Gra
duate Texts in Mathematics, vol. 203. Springer, New York (2001)201.Schauenberg, P.: On Coquasitriangular Hopf Algebras and the Quantum Yang–Baxter Equation. Algebra Berichte, vol. 67, Fischer, Munich (1992)202.Schauenberg, P.: A characterization of the Borel-like subalgebras of quantum enveloping algebras. Commun. Algebra 24, 2811–2823 (1996)CrossRefMathSciNet203.Scheunert, M.: Generalized Lie algebras. J. Math. Phys. 20, 712–720 (1979)MathSciNetMATHCrossRef204.Scheunert, M.: The Theory of Lie Superalgebras. An Intro
duction. Lecture Notes in Mathematics, vol. 716. Springer, Berlin-Heidelberg-New York (1979)205.Schreier, O.: Die Untergruppen der freien Gruppen. Abhandlungen Hamburg 5, 161–183 (1927)MathSciNetMATHCrossRef206.Schützenberger, M.P., Sherman, S.: On a formal pro
duct over the conjugate classes in a free group. J. Math. Anal. Appl. 7, 482–488 (1963)MathSciNetMATHCrossRef207.Serre, J.-P.: Algébres de Lie Semi-simples Complexes. Benjamin, New York, Amsterdam (1966)MATH208.Shestakov, I.P., Umirbaev, U.U.: Free Akivis algebras, primitive elements, and hyperalgebras. J. Algebra 250, 533–548 (2002)MathSciNetMATHCrossRef209.Shirshov, A.I.: Subalgebras of free Lie algebras. Matem. Sb. 33(75), 441–452 (1953)MathSciNetMATH210.Shirshov, A.I.: On some nonassociative nil-rings and algebraic algebras. Matem. Sbornik 41(83)(3), 381–394 (1957)211.Shirshov, A.I.: On free Lie rings. Mat. Sb. 45(87)(2), 113–122 (1958)212.Shirshov, A.I.: Some algorithmic problems for Lie algebras. Sib. Math. J. 3(2), 292–296 (1962)MATH213.Shneider, S., Sternberg, S.: Quantum Groups. International Press Incorporated, Boston (1993)214.Shtern, A.S.: Free Lie superalgebras. Sib. Math. J. 27, 551–554 (1986)MathSciNetCrossRefMATH215.Skryabin, S.: Hopf Galois extensions, triangular structures, and Frobenius Lie algebras in prime characteristic. J. Algebra 277, 96–128 (2004)MathSciNetMATHCrossRef216.Stanley, R.P.: Some aspects of group acting on finite posets. J. Combin. Theory Ser. A 32, 132–161 (1982)MathSciNetMATHCrossRef217.Stover, R.: The equivalence of certain categories of twisted Lie algebras over a commutative ring. J. Pure Appl. Algebra 86, 289–326 (1993)MathSciNetMATHCrossRef218.Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence, RI (1996)219.Sturmfels, B.: Solving Systems of Polynomial Equations. Conference Board of Mathematical Sciences, vol. 97. American Mathematical Society, Providence, RI (2002)220.Sweedler, M.: Hopf Algebras. Benjamin, New York (1969)MATH221.Takeuchi, M:. Survey of braided Hopf algebras. In: New Trends in Hopf Algebra Theory. Contemporary Mathematics, vol. 267, pp. 301–324. American Mathematical Society, Providence, RI (2000)222.Takeuchi, M.: A survey on Nichols algebras. In: De la Peña, J.A., Vallejo, E., Atakishieyev, N. (eds.) Algebraic Structures and Their Representations. Contemporary Mathematics, vol. 376, pp. 105–117. American Mathematical Society, Providence, RI (2005)CrossRef223.Takhtadjian, L.: On foundations of the generalized Nambu mechanics. Commun. Math. Phys. 160, 295–315 (1994)CrossRef224.Towber, J.: Multiparameter quantum forms of the enveloping algebra \(U_{gl_{n}}\) related to the Faddeev-Reshetikhin-Takhtajan U(R) constructions. J. Knot Theory Ramif. 4(5), 263–317 (1995)MathSciNetMATHCrossRef225.Ufer, S.: PBW bases for a class of braided Hopf algebras. J. Algebra 280(1), 84–119 (2004)MathSciNetMATHCrossRef226.Wess, J., Zumino, B.: Covariant differential calculus on the quantum hyperplane. Nucl. Phys. B Proc. Suppl. 18, 302–312 (1991)MathSciNetCrossRefMATH227.White, A.T.: Groups, Graphs, and Surfaces. North-Holland Mathematical Series, vol. 8. North-Holland, New York (1988)228.Wielandt, H.: Finite Permutation Groups. Academic Press, New York (1964)MATH229.Witt, E.: Die Unterringe der freien Liescher Ringe. Math. Zeitschr. Bd. 64, 195–216 (1956)MathSciNetMATHCrossRef230.Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys. 122, 125–170 (1989)MathSciNetMATHCrossRef231.Yamaleev, R.M.: Elements of cubic quantum mechanics. Commun. JINR Dubna E2-88-147, 1–11 (1988)232.Yamaleev, R.M.: Model of multilinear oscillator in a space of non-integer quantum numbers. Commun. JINR Dubna E2-88-871, 1–10 (1988)233.Yamaleev, R.M.: Model of multilinear Bose- and Fermi-like oscillator. Commun. JINR Dubna E2-92-66, 1–14 (1992)234.Yamane, H.: A Poincaré-Birkhoff-Witt theorem for quantized universal enveloping algebras of type A
N . Publ. RIMS. Kyoto Univ. 25, 503–520 (1989)MathSciNetMATHCrossRef235.Yanai, T.: Galois correspondence theorem for Hopf algebra actions. In: Algebraic Structures and Their Representations. Contemporary Mathematics, vol. 376, pp. 393–411. American Mathematical Society, Providence, RI (2005)236.Zhevlakov, K.A., Slinko, A.M., Shestakov, I.P., Shirshov, A.I.: Rings that are Nearly Associative. Academic, New York (1982)MATH About this Chapter Title Poincaré-Birkhoff-Witt Basis Book Title Quantum Lie Theory Book Subtitle A Multilinear Approach Pages pp 71-97 Copyright 2015 DOI 10.1007/978-3-319-22704-7_2 Print ISBN 978-3-319-22703-0 Online ISBN 978-3-319-22704-7 Series Title Lecture Notes in Mathematics Series Volume 2150 Series ISSN 0075-8434 Publisher Springer International Publishing Copyright Holder Springer International Publishing Switzerland Additional Links About this Book Topics Associative Rings and Algebras Non-associative Rings and Algebras Group Theory and Generalizations Quantum Physics eBook Packages Mathematics and Statistics Authors Vladislav Kharchenko (14) Author Affiliations 14. Universidad Nacional Autónoma de México, Cuautitlán Izcalli, Estado de México, Mexico Continue reading... To view the rest of this content please follow the download PDF link above.