Skip to main content
SpringerLink
Log in

Binary Structures

  • Chapter
  • First Online:
Quantum Lie Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2150))

  • 1463 Accesses

Abstract

In this chapter, we consider binary generalizations of Lie algebras appeared in modern mathematics and mathematical physics. We consider recent developments and remaining problems on the subject. The chapter discusses Lie superalgebras, color Lie algebras, and Lie algebras in symmetric categories, free Lie τ-algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abe, E.: Hopf Algebras. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

  2. Andruskiewitsch, N., Graña, M.: Braided Hopf algebras over non abelian finite group. Bol. Acad. Nac. Cienc. Cordoba 63, 46–78 (1999)

    MATH  Google Scholar 

  3. 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)

    Google Scholar 

  4. Andruskiewitsch, N., Schneider, H.-J.: On the classification of finite-dimensional pointed Hopf algebras. Ann. Math. 171(2)(1), 375–417 (2010)

    Google Scholar 

  5. Andruskiewitsch, N., Heckenberger, I., Schneider, H.-J.: The Nichols algebra of a semisimple Yetter-Drinfeld module. Am. J. Math. 132(6), 1493–1547 (2010)

    MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ardizzoni, A.: On the combinatorial rank of a graded braided bialgebra. J. Pure Appl. Algebra 215(9), 2043–2054 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ardizzoni, A.: On primitively generated braided bialgebras. Algebr. Represent. Theor. 15(4), 639–673 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ardizzoni, A.: Universal enveloping algebras of the PBW type. Glasgow Math. J. 54, 9–26 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ardizzoni, A., Menini, C.: Associated graded algebras and coalgebras. Commun. Algebra 40(3), 862–896 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ardizzoni, A., Stumbo, F.: Quadratic lie algebras. Commun. Algebra 39(8), 2723–2751 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ardizzoni, A., Menini, C., Ştefan, D.: Braided bialgebras of Hecke type. J. Algebra 321, 847–865 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Google Scholar 

  15. Bahturin, Y., Fishman, D., Montgomery, S.: On the generalized Lie structure of associative algebras. Israel J. Math. 96, 27–48 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bahturin, Y., Mikhalev, A.A., Zaicev, M.: Infinite-Dimensional Lie Superalgebras. Handbook of Algebra, vol. 2, pp. 579–614. North-Holland, Amsterdam (2000)

    Google Scholar 

  17. Bahturin, Y., Fishman, D., Montgomery, S.: Bicharacters, twistings and Scheunert’s theorem for Hopf algebras. J. Algebra 236, 246–276 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bahturin, Y., Kochetov, M., Montgomery, S.: Polycharacters of cocommutative Hopf algebras. Can. Math. Bull. 45, 11–24 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Chapter  Google Scholar 

  21. Bautista, C.: A Poincaré-Birkhoff-Witt theorem for generalized Lie color algebras. J. Math. Phys. 39(7), 3828–3843 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Benkart, G., Witherspoon, S.: Representations of two-parameter quantum groups and Schur-Weyl duality. Hopf Algebras. Lecture Notes in Pure and Applied Mathematics, vol. 237, pp. 65–92. Dekker, New York (2004)

    Google Scholar 

  23. Benkart, G., Witherspoon, S.: Two-parameter quantum groups and Drinfel’d doubles. Algebr. Represent. Theor. 7(3), 261–286 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Benkart, G., Kang, S.-J., Melville, D.: Quantized enveloping algebras for Borcherds superalgebras. Trans. Am. Math. Soc. 350(8), 3297–3319 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Berezin, F.A.: Method of the Secondary Quantification. Nauka, Moscow (1965)

    Google Scholar 

  26. Berezin, F.A.: Mathematical foundations of the supersymmetric field theories. Nucl. Phys. 29(6), 1670–1687 (1979)

    MathSciNet  Google Scholar 

  27. Berezin, F.A.: Introduction to Superanalysis. Mathematical Physics and Applied Mathematics, vol. 9. D. Reidel, Dordrecht/Boston, MA (1987)

    Google Scholar 

  28. Berger, R.: The quantum Poincaré-Birkhoff-Witt theorem. Commun. Math. Phys. 143(2), 215–234 (1992)

    Article  MATH  Google Scholar 

  29. Bergeron, N., Gao, Y., Hu, N.: Drinfel’d doubles and Lusztig’s symmetries of two-parameter quantum groups. J. Algebra 301, 378–405 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Bergman, G.M.: The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  31. Boerner, H.: Representation of Groups with Special Consideration for the Needs of Modern Physics. North-Holland, New York, NY (1970)

    MATH  Google Scholar 

  32. Bokut, L.A.: Imbedding into simple associative algebras. Algebra Log. 15, 117–142 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  33. Bokut, L.A.: Associative Rings I. Library of the Department of Algebra and Logic, vol. 18. Novosibirsk State University, Novosibirsk (1977)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  35. Bokut, L.A., Chen, Y.: Gröbner–Shirshov bases and their calculation. Bull. Math. Sci. 4(3), 325–395 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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)

    Google Scholar 

  37. Bokut’, L.A., Kukin, G.P.: Algorithmic and Combinatorial Algebra. Mathematics and Its Applications, vol. 255. Kluwer, Dordrecht/Boston/London (1994)

    Google Scholar 

  38. Bokut, L.A., Vesnin, A.: Gröbner–Shirshov bases for some braid groups. J. Symb. Comput. 41(3–4), 357–371 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Borowiec, A., Kharchenko, V.K.: Algebraic approach to calculuses with partial derivatives. Sib. Adv. Math. 5(2), 10–37 (1995)

    MathSciNet  MATH  Google Scholar 

  40. 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. XIX

    Google Scholar 

  41. 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)

    Google Scholar 

  42. Braverman, A., Gaitsgory, D.: Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type. J. Algebra 181, 315–328 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  43. Buchberger, B.: An algorithm for finding a basis for the residue class ring of a zero-dimensional ideal. Ph.D. Thesis, University of Innsbruck (1965)

    Google Scholar 

  44. Buchberger, B.: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. Aequtiones Math. 4, 374–383 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  45. Cameron, P.J.: Permutation Groups, London Mathematical Society. Student Texts, vol. 45. University Press, Cambridge (1999)

    Google Scholar 

  46. Chaichian, M., Demichev, A.: Introduction to Quantum Groups. World Scientific, Singapore (1996)

    Book  MATH  Google Scholar 

  47. 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)

    Google Scholar 

  48. Chen, Y.Q., Shao, H.S., Shum, K.P.: On Rosso–Yamane theorem on PBW basis of U q (A N ). CUBO Math. J. 10, 171–194 (2008)

    MathSciNet  MATH  Google Scholar 

  49. Chuang, C.-L.: Identities with skew derivations. J. Algebra 224, 292–335 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  50. Cohen, M., Fishman, D., Westreich, S.: Schur’s double centralizer theorem for triangular Hopf algebras. Proc. Am. Math. Soc. 122(1), 19–29 (1994)

    MATH  Google Scholar 

  51. 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)

    MathSciNet  MATH  Google Scholar 

  52. Cohn, P.M.: The class of rings embeddable in skew fields. Bull. Lond. Math. Soc. 6, 147–148 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  53. Cohn, P.M.: Free Rings and Their Relations, 2nd edn. Academic Press, London (1985)

    MATH  Google Scholar 

  54. Connes, A.: Non-commutative Geometry. Academic Press, New York (1994)

    MATH  Google Scholar 

  55. Constantini, M., Varagnolo, M.: Quantum double and multiparameter quantum groups. Commun. Algebra 22(15), 6305–6321 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  56. Cotta-Ramusino, P., Rinaldi, M.: Multiparameter quantum groups related to link diagrams. Commun. Math. Phys. 142, 589–604 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  57. Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Interscience/Wiley, New York/London (1962)

    MATH  Google Scholar 

  58. De Concini, C., Kac, V.G., Procesi, C.: Quantum coadjoint action. J. Am. Math. Soc. 5, 151–189 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  59. Deng, B., Du, J.: Bases of quantized enveloping algebras. Pac. J. Math. 220(1), 33–48 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  60. Deng, B., Du, J.: Monomial bases for quantum affine \(\mathfrak{s}l_{n}\). Adv. Math. 191(2), 276–304 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  61. Diaconis, P.: Group Representations in Probability and Statistics. Lecture Notes Monograph Series, vol. 11. Institute of Mathematical Statistics, Hayward, CA (1988)

    Google Scholar 

  62. Diaz Sosa, M.L., Kharchenko, V.K.: Combinatorial rank of \(u_{q}(\mathfrak{s}\mathfrak{o}_{2n})\). J. Algebra 448, 48–73 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  63. Dito, G., Flato, M., Sternheimer, D., Takhtadjian, L.: Deformation quantization and Nambu mechanics. Commun. Math. Phys. 183, 1–22 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  64. Dixon, D., Mortimer, B.: Permutation Groups. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  65. Drinfeld, V.G.: Hopf algebras and the Yang–Baxter equation. Soviet Math. Dokl. 32, 254–258 (1985)

    MathSciNet  Google Scholar 

  66. Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematics, Berkeley, CA, vol. 1, pp. 798–820 (1986)

    MathSciNet  Google Scholar 

  67. 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)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  69. Etingof, P., Gelaki, S.: On cotriangular Hopf algebras. Am. J. Math. 123, 699–713 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  70. 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)

    MathSciNet  MATH  Google Scholar 

  71. Etingof, P., Schiffman, O.: Lectures on Quantum Groups. International Press Incorporated, Boston (1998)

    MATH  Google Scholar 

  72. FELIX – an assistant for algebraists. In: Watt, S.M. (ed.) ISSACi91, pp. 382–389. ACM Press, New York (1991)

    Google Scholar 

  73. Ferreira, V.O., Murakami, L.S.I., Paques, A.: A Hopf-Galois correspondence for free algebras. J. Algebra 276 407–416 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  74. Filippov, V.T.: n-Lie algebras. Sib. Math. J. 26(6), 126–140 (1985)

    Google Scholar 

  75. Finkelshtein, D.: On relations between commutators. Commun. Pure. Appl. Math. 8, 245–250 (1955)

    Article  MathSciNet  Google Scholar 

  76. Fischman, D., Montgomery, S.: A Schur double centralizer theorem for cotriangular Hopf algebras and generalized Lie algebras. J. Algebra 168, 594–614 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  77. 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)

    Chapter  Google Scholar 

  78. Flores de Chela, D., Green, J.A.: Quantum symmetric algebras II. J. Algebra 269, 610–631 (2003)

    Google Scholar 

  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)

    Google Scholar 

  80. Friedrichs, K.O.: Mathematical aspects of the quantum theory of fields V. Commun. Pure Applied Math. 6, 1–72 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  81. Frønsdal, C.: On the Classification of q-algebras. Lett. Math. Phys. 53, 105–120 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  82. Frønsdal, C., Galindo, A.: The ideals of free differential algebras. J. Algebra 222, 708–746 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  83. Gabber, O., Kac, V.G.: On defining relations of certain infinite-dimensional Lie algebras. Bull. Am. Math. Soc. 5, 185–189 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  84. Gavarini, F.: A PBW basis for Lusztig’s form of untwisted affine quantum groups. Commun. Algebra 27(2), 903–918 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  85. 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)

    MathSciNet  Google Scholar 

  86. Gomez, X., Majid, S.: Braided Lie algebras and bicovariant differential calculi over co-quasitriangular Hopf algebras. J. Algebra 261(2), 334–388 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  87. Graña, M., Heckenberger, I.: On a factorization of graded Hopf algebras using Lyndon words. J. Algebra 314(1), 324–343 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  88. 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)

    Google Scholar 

  89. Gurevich, D.: Algebraic aspects of the quantum Yang-Baxter equation. Leningr. Math. J. 2(4), 801–828 (1991)

    MATH  MathSciNet  Google Scholar 

  90. Heckenberger, I.: Weyl equivalence for rank 2 Nichols algebras of diagonal type. Ann. Univ. Ferrara Sez. VII (NS) 51(1), 281–289 (2005)

    Google Scholar 

  91. Heckenberger, I.: The Weyl groupoid of a Nichols algebra of diagonal type. Invent. Math. 164, 175–188 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  92. Heckenberger, I.: Classification of arithmetic root systems. Adv. Math. 220, 59–124 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  93. Heckenberger, I., Kolb, S.: Right coideal subalgebras of the Borel part of a quantized enveloping algebra. Int. Math. Res. Not. 2, 419–451 (2011)

    MathSciNet  MATH  Google Scholar 

  94. Heckenberger, I., Kolb, S.: Homogeneous right coideal subalgebras of quantized enveloping algebras. Bull. Lond. Math. Soc. 44(4), 837–848 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  95. Heckenberger, I., Schneider, H.-J.: Root systems and Weyl groupoids for Nichols algebras. Proc. Lond. Math. Soc. 101(3), 623–654 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  96. 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)

    Article  MathSciNet  MATH  Google Scholar 

  97. Heckenberger, I., Yamane, H.: Drinfel’d doubles and Shapovalov determinants. Rev. Un. Mat. Argentina 51(2), 107–146 (2010)

    MathSciNet  MATH  Google Scholar 

  98. Herstein, I.: Noncommutative Rings. Carus Mathematical Monographs, vol. 15, The Mathematical Association of America, USA (1968)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  101. Hopf, H.: Über die Topologeie der Gruppenmannigfaltigkeiten und ihre Verallgemeinerungen. Ann. Math. 42, 22–52 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  102. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Springer, New York-Heidelberg-Berlin (1972)

    Book  MATH  Google Scholar 

  103. Ion, B.: Relative PBW type theorems for symmetrically braided Hopf algebras. Commun. Algebra 39(7), 2508–2518 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  104. Jacobson, N.: Lie Algebras. Interscience, New York (1962)

    MATH  Google Scholar 

  105. Jantzen, J.C.: Lectures on Quantum Groups. Graduate Studies in Mathematics, vol. 6. American Mathematical Society, Providence, RI (1996)

    Google Scholar 

  106. Jimbo, M.: A q-difference analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–69 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  107. Joseph, A.: Quantum Groups and Their Primitive Ideals. Springer, Berlin, Heidelberg (1995)

    Book  MATH  Google Scholar 

  108. Joyal, A.: Foncteurs analytiques et espèces de structures. Combinatoire énumérative. Lecture Notes in Mathematics, vol. 1234, pp. 126–159. Springer, Berlin (1986)

    Google Scholar 

  109. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–87 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  110. Jumphreys, J.E.: Linear Algebraic Groups. Graduate Texts in Mathematics 21. Springer-Verlag, New York (1975)

    Google Scholar 

  111. Kac, G.I.: Ring groups and duality principle. Proc. Mosc. Math. Soc. 12, 259–301 (1963)

    MathSciNet  Google Scholar 

  112. Kac, V.G.: Simple irreducible graded Lie algebras of finite growth. Izvestija AN USSR (Ser. Mat.) 32, 1923–1967 (1968). English translation: Math. USSR-Izvestija, 2, 1271–1311 (1968)

    Google Scholar 

  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)

    Google Scholar 

  114. Kac, V.G.: A sketch of Lie superalgebra theory. Commun. Math. Phys. 53, 31–64 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  115. Kac, V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)

    Article  MATH  Google Scholar 

  116. Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  117. Kac, V.G.: Classification of infinite-dimensional simple linearly compact Lie superalgebras. Adv. Math. 139, 1–55 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  118. Kang, S.-J.: Quantum deformations of generalized Kac-Moody algebras and their modules. J. Algebra 175, 1041–1066 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  119. Kashiwara, M.: On crystal bases of the q-analog of universal enveloping algebras. Duke Math. J. 63(2), 465–516 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  120. Kassel, C.: Quantum Groups. Springer, New York (1995)

    Book  MATH  Google Scholar 

  121. Kharchenko, V.K.: Automorphisms and Derivations of Associative Rings. Mathematics and Its Applications (Soviet Series), vol. 69. Kluwer, Dordrecht/Boston/London (1991)

    Google Scholar 

  122. Kharchenko, V.K.: Noncommutative Galois Theory. Nauchnaja Kniga, Novosibirsk (1996)

    MATH  Google Scholar 

  123. 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)

    Google Scholar 

  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)

    Google Scholar 

  125. Kharchenko, V.K.: An existence condition for multilinear quantum operations. J. Algebra 217, 188–228 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  126. 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)

    Google Scholar 

  127. Kharchenko, V.K.: Skew primitive elements in Hopf algebras and related identities. J. Algebra 238(2), 534–559 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  128. Kharchenko, V.K.: A combinatorial approach to the quantification of Lie algebras. Pac. J. Math. 203(1), 191–233 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  129. Kharchenko, V.K.: Constants of coordinate differential calculi defined by Yang–Baxter operators. J. Algebra 267, 96–129 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  130. Kharchenko, V.K.: Multilinear quantum Lie operations. J. Math. Sci. (NY) 116(1), 3063–3073 (2003)

    Google Scholar 

  131. Kharchenko, V.K.: Braided version of Shirshov-Witt theorem. J. Algebra 294, 196–225 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  132. Kharchenko, V.K.: Connected braided Hopf algebras. J. Algebra 307, 24–48 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  133. Kharchenko, V.K.: PBW-bases of coideal subalgebras and a freeness theorem. Trans. Am. Math. Soc. 360(10), 5121–5143 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  134. Kharchenko, V.K.: Triangular decomposition of right coideal subalgebras, J. Algebra 324(11), 3048–3089 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  135. Kharchenko, V.K.: Right coideal subalgebras of U q +(so 2n+1). J. Eur. Math. Soc. 13(6), 1677–1735 (2011)

    MathSciNet  MATH  Google Scholar 

  136. Kharchenko, V.K.: Quantizations of Kac–Moody algebras. J. Pure Appl. Algebra 218(4), 666–683 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  137. Kharchenko, V.K., Andrede Álvarez, A.: On the combinatorial rank of Hopf algebras. Contemp. Math. 376, 299–308 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  138. 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)

    Article  MATH  Google Scholar 

  139. Kharchenko, V.K., Lara Sagahón, A.V.: Right coideal subalgebras in U q (sl n+1). J. Algebra 319(6), 2571–2625 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  140. 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); Corrigendum 35(1), 224–225 (2012)

    Google Scholar 

  141. Killing, W.: Die Zusammensetzung der steigen endlichen Transformationsgruppen, I. Math. Ann. 31, 252–290 (1886)

    Article  MathSciNet  MATH  Google Scholar 

  142. Klimik, A., Schmüdgen, K.: Quantum Groups and Their Representations. Springer, Berlin, Heidelberg (1997)

    Book  Google Scholar 

  143. Kochetov, M.: Generalized Lie algebras and cocycle twists. Commun. Algebra 36, 4032–4051 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  144. Kochetov, M., Radu, O.: Engel’s theorem for generalized Lie algebras. Algebr. Represent. Theor. 13(1), 69–77 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  145. 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)

    Article  MathSciNet  MATH  Google Scholar 

  146. Leclerc, B.: Dual canonical bases, quantum shuffles and q-characters. Math. Z. 246(4), 691–732 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  147. 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)

    MathSciNet  Google Scholar 

  148. Lothaire, M.: Combinatorics on Words. Encyclopedia of Mathematics and Its Applications, vol. 17. Addison-Wesley, Reading, MA (1983)

    Google Scholar 

  149. Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  150. Lothaire, M.: Applied Combinatorics on Words. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  151. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3(2), 447–498 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  152. Lusztig, G.: Finite-dimensional Hopf algebras arising from quantized enveloping algebras. J. Am. Math. Soc. 3(1), 257–296 (1990)

    MathSciNet  MATH  Google Scholar 

  153. Lusztig, G.: Introduction to Quantum Groups. Progress in Mathematics, vol. 110. Birkhauser, Boston (1993)

    Google Scholar 

  154. Lyndon, R.C.: On Burnside’s problem. Trans. Am. Math. Soc. 7, 202–215 (1954)

    MathSciNet  MATH  Google Scholar 

  155. Lyndon, R.C.: A theorem of Friedrichs. Mich. Math. J. 3(1), 27–29 (1955–1956)

    Google Scholar 

  156. Lyubashenko, V.V.: Hopf algebras and vector symmetries. Sov. Math. Surveys 41, 153–154 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  157. Magnus, W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  158. 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)

    Google Scholar 

  159. Majid, S.: Crossed products by braided groups and bosonization. J. Algebra 163, 165–190 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  160. Majid, S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  161. Mal’tcev, A.I.: On the immersion of an algebraic ring into a field. Math. Ann. 113, 686–691 (1937)

    Article  MathSciNet  Google Scholar 

  162. Mal’tcev, A.I.: On the inclusion of associative systems into groups, I. Mat. Sbornik 6(2), 331–336 (1939)

    Google Scholar 

  163. Mal’tcev, A.I.: On the inclusion of associative systems in groups, II. Mat. Sbornik 8(2), 251–263 (1940)

    Google Scholar 

  164. Mal’tcev, A.I.: Algebraic Systems. Nauka, Moscow (1965)

    Google Scholar 

  165. Mal’tcev, A.I.: Algorithms and Recursive Functions. Nauka, Moscow (1970)

    Google Scholar 

  166. Manin, Y.: Quantum Groups and Non-Commutative Geometry. CRM Publishing, Université de Montréal (1988)

    MATH  Google Scholar 

  167. Masuoka, A.: Formal groups and unipotent affine groups in non-categorical symmetry. J. Algebra 317, 226–249 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  168. Moody, R.V.: Lie algebras associated with generalized Cartan matrices. Bull. Am. Math. Soc. 73, 217–221 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  169. 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)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  173. Milinski, A.: Actions of pointed Hopf algebras on prime algebras. Commun. Algebra 23, 313–333 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  174. Milinski, A.: Operationen punktierter Hopfalgebren auf primen Algebren. Ph.D. thesis, München (1995)

    Google Scholar 

  175. Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. 81, 211–264 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  176. Montgomery, S.: Hopf Algebras and Their Actions on Rings. Conference Board of Mathematical Sciences, vol. 82. American Mathematical Society, Providence, RI (1993)

    Google Scholar 

  177. Mostovoy, J., Pérez Izquierdo, J.M.: Formal multiplications, bialgebras of distributions and non-associative Lie theory. Transform. Groups 15, 625–653 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  178. Mostovoy, J., Pérez-Izquierdo, J.M., Shestakov, I.P.: Hopf algebras in non-associative Lie theory. Bull. Math. Sci. 4(1), 129–173 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  179. Musson, I.M.: Lie Superalgebras and Enveloping Algebras. Graduate Studies in Mathematics, vol. 131. American Mathematical Society, Providence, RI (2012)

    Google Scholar 

  180. Nambu, Y.: Generalized Hamilton dynamics. Phys. Rev. D 7(8), 2405–2412 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  181. Nichols, W.: Bialgebras of type one. Commun. Algebra 6(15), 1521–1552 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  182. Nielsen, J.: A basis for subgroups of free groups. Math. Scand. 3, 31–43 (1955)

    MathSciNet  MATH  Google Scholar 

  183. 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)

    Google Scholar 

  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)

    Google Scholar 

  185. Pareigis, B.: On Lie algebras in the category of Yetter–Drinfeld modules. Appl. Categ. Struct. 6(2), 152–175 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  186. Pérez Izquierdo, J.M.: An envelope for Bol algebras. J. Algebra 284, 480–493 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  187. Pérez Izquierdo, J.M.: Algebras, hyperalgebras, nonassociative bialgebras and loops. Adv. Math. 208, 834–876 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  188. Pérez Izquierdo, J.M., Shestakov, I.P.: An envelope for Malcev algebras. J. Algebra 272, 379–393 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  189. Polishchuk, A.E., Positselsky, L.E.: Quadratic Algebras. University Lecture Series, vol. 37. American Mathematical Society, Providence, RI (2005)

    Google Scholar 

  190. Radford, D.E.: The structure of Hopf algebras with projection. J. Algebra 92, 322–347 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  191. Radford, D.E.: Hopf Algebras. Series on Knots and Everything, vol. 49. World Scientific, Singapore (2012)

    Google Scholar 

  192. Reineke, M.: Feigin’s map and monomial bases for quantized enveloping algebras. Math. Z. 237(3), 639–667 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  193. Reshetikhin, N.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett. Math. Phys. 20, 331–335 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  194. Ringel, C.M.: PBW-bases of quantum groups. J. Reine Angew. Math. 470, 51–88 (1996)

    MathSciNet  MATH  Google Scholar 

  195. 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)

    Article  MathSciNet  MATH  Google Scholar 

  196. Rosso, M.: Groupes quantiques et algèbres de battagesquantiques. C.R. Acad. Sci. Paris 320, 145–148 (1995)

    MathSciNet  MATH  Google Scholar 

  197. Rosso, M.: Quantum groups and quantum Shuffles. Invent. Math. 113(2) 399–416 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  198. Sabinin, L.V.: Smooth Quasigroups and Loops. Mathematics and Its Applications, vol. 429. Kluwer, Dordrecht/Boston/London (1999)

    Google Scholar 

  199. Sabinin, L.V.: Smooth quasigroups and loops: forty-five years of incredible growth. Comment. Math. Univ. Carol. 41, 377–400 (2000)

    MathSciNet  MATH  Google Scholar 

  200. Sagan, B.E.: The Symmetric Group. Graduate Texts in Mathematics, vol. 203. Springer, New York (2001)

    Google Scholar 

  201. Schauenberg, P.: On Coquasitriangular Hopf Algebras and the Quantum Yang–Baxter Equation. Algebra Berichte, vol. 67, Fischer, Munich (1992)

    Google Scholar 

  202. Schauenberg, P.: A characterization of the Borel-like subalgebras of quantum enveloping algebras. Commun. Algebra 24, 2811–2823 (1996)

    Article  MathSciNet  Google Scholar 

  203. Scheunert, M.: Generalized Lie algebras. J. Math. Phys. 20, 712–720 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  204. Scheunert, M.: The Theory of Lie Superalgebras. An Introduction. Lecture Notes in Mathematics, vol. 716. Springer, Berlin-Heidelberg-New York (1979)

    Google Scholar 

  205. Schreier, O.: Die Untergruppen der freien Gruppen. Abhandlungen Hamburg 5, 161–183 (1927)

    Article  MathSciNet  MATH  Google Scholar 

  206. Schützenberger, M.P., Sherman, S.: On a formal product over the conjugate classes in a free group. J. Math. Anal. Appl. 7, 482–488 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  207. Serre, J.-P.: Algébres de Lie Semi-simples Complexes. Benjamin, New York, Amsterdam (1966)

    MATH  Google Scholar 

  208. Shestakov, I.P., Umirbaev, U.U.: Free Akivis algebras, primitive elements, and hyperalgebras. J. Algebra 250, 533–548 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  209. Shirshov, A.I.: Subalgebras of free Lie algebras. Matem. Sb. 33(75), 441–452 (1953)

    MathSciNet  MATH  Google Scholar 

  210. Shirshov, A.I.: On some nonassociative nil-rings and algebraic algebras. Matem. Sbornik 41(83)(3), 381–394 (1957)

    Google Scholar 

  211. Shirshov, A.I.: On free Lie rings. Mat. Sb. 45(87)(2), 113–122 (1958)

    Google Scholar 

  212. Shirshov, A.I.: Some algorithmic problems for Lie algebras. Sib. Math. J. 3(2), 292–296 (1962)

    MATH  Google Scholar 

  213. Shneider, S., Sternberg, S.: Quantum Groups. International Press Incorporated, Boston (1993)

    Google Scholar 

  214. Shtern, A.S.: Free Lie superalgebras. Sib. Math. J. 27, 551–554 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  215. Skryabin, S.: Hopf Galois extensions, triangular structures, and Frobenius Lie algebras in prime characteristic. J. Algebra 277, 96–128 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  216. Stanley, R.P.: Some aspects of group acting on finite posets. J. Combin. Theory Ser. A 32, 132–161 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  217. Stover, R.: The equivalence of certain categories of twisted Lie algebras over a commutative ring. J. Pure Appl. Algebra 86, 289–326 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  218. Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence, RI (1996)

    Google Scholar 

  219. Sturmfels, B.: Solving Systems of Polynomial Equations. Conference Board of Mathematical Sciences, vol. 97. American Mathematical Society, Providence, RI (2002)

    Google Scholar 

  220. Sweedler, M.: Hopf Algebras. Benjamin, New York (1969)

    MATH  Google Scholar 

  221. 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)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  223. Takhtadjian, L.: On foundations of the generalized Nambu mechanics. Commun. Math. Phys. 160, 295–315 (1994)

    Article  Google Scholar 

  224. 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)

    Article  MathSciNet  MATH  Google Scholar 

  225. Ufer, S.: PBW bases for a class of braided Hopf algebras. J. Algebra 280(1), 84–119 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  226. Wess, J., Zumino, B.: Covariant differential calculus on the quantum hyperplane. Nucl. Phys. B Proc. Suppl. 18, 302–312 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  227. White, A.T.: Groups, Graphs, and Surfaces. North-Holland Mathematical Series, vol. 8. North-Holland, New York (1988)

    Google Scholar 

  228. Wielandt, H.: Finite Permutation Groups. Academic Press, New York (1964)

    MATH  Google Scholar 

  229. Witt, E.: Die Unterringe der freien Liescher Ringe. Math. Zeitschr. Bd. 64, 195–216 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  230. Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys. 122, 125–170 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  231. Yamaleev, R.M.: Elements of cubic quantum mechanics. Commun. JINR Dubna E2-88-147, 1–11 (1988)

    Google Scholar 

  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)

    Google Scholar 

  233. Yamaleev, R.M.: Model of multilinear Bose- and Fermi-like oscillator. Commun. JINR Dubna E2-92-66, 1–14 (1992)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  235. 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)

    Google Scholar 

  236. Zhevlakov, K.A., Slinko, A.M., Shestakov, I.P., Shirshov, A.I.: Rings that are Nearly Associative. Academic, New York (1982)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidad Nacional Autónoma de México, Cuautitlán Izcalli, Estado de México, Mexico

    Vladislav Kharchenko

Authors
  1. Vladislav Kharchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kharchenko, V. (2015). Binary Structures. In: Quantum Lie Theory. Lecture Notes in Mathematics, vol 2150. Springer, Cham. https://doi.org/10.1007/978-3-319-22704-7_7

Download citation

Publish with us

Policies and ethics

Search

Navigation