The Freiheitssatz and automorphisms for free brace algebras

References

• Bokut, L. A. (1972). Insolvability of the word problem for Lie algebras, and subalgebras of finitely presented lie algebras. Izv. Akad. Nauk SSSR Ser. Mat. 36(6):1173–1219.
   Google Scholar
• Chapoton, F. (2002). Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces. J. Pure Appl. Algebra 168(1):1–18. DOI:10.1016/S0022-4049(01)00052-4.
   Web of Science ®Google Scholar
• Chen, Y. Q., Li, Y., Tang, Q. Y. (2017). Gröbner-Shirshov bases for some Lie algebras. Sib. Math. J. 58(1):176–182. DOI:10.1134/S0037446617010220.
   Web of Science ®Google Scholar
• Cohn, P. M. (1962). Free Rings, Lecture Notes. New Haven, Connecticut: Yale University.
   Google Scholar
• Cohn, P. M. (1974). Progress in free associative algebras. Isr. J. Math. 19(1-2):109–151. DOI:10.1007/BF02756628.
   Web of Science ®Google Scholar
• Connes, A., Kreimer, D. (1998). Hopf algebras, Renormalization and noncommutative geometry. Comm. Math. Phys. 199(1):203–242. DOI:10.1007/s002200050499.
   Web of Science ®Google Scholar
• (1). Czerniakiewicz, A. J. (1971). Automorphisms of a free associative algebra of rank 2.I. Trans. Amer. Math. Soc. 160:393–401. DOI:10.1090/S0002-9947-1971-0280549-1. (2). Czerniakiewicz, A. J. (1972). Automorphisms of a free associative algebra of rank 2.II. Trans. Amer. Math. Soc. 171:309–315. DOI:10.1090/S0002-9947-1972-0310021-2.
   Google Scholar
• Foissy, L. (2002). Les algèbres de Hopf des arbres enracinés, I. Bull. Sci. Math. 126(3):193–239. DOI:10.1016/S0007-4497(02)01108-9.
   Google Scholar
• Foissy, L. (2002). Les algèbres de Hopf des arbres enracinés, II. Bull. Sci. Math. 126:1249–1288. DOI:10.1016/S0007-4497(02)01113-2.
   Google Scholar
• Foissy, L. (2010). Free brace algebras are free pre-Lie algebras. Commun. Algebra 38(9):3358–3369. DOI:10.1080/00927870903115000.
   Web of Science ®Google Scholar
• Jung, H. W. E. (1942). Über ganze birationale Transformationen der Ebene. J. Reine Angew. Math. 184:161–174.
   Google Scholar
• Kolesnikov, P. S., Makar-Limanov, L., Shestakov, I. P. (2014). The Freiheitssatz for generic Poisson algebras. SIGMA 10(115):15.
   Google Scholar
• Kozybaev, D. (2007). On the structure of universal multiplicative algebras of free right-symmetric algebras. Vestnik KazNU 3(54):3–9.
   Google Scholar
• Kozybaev, D., Makar-Limanov, L., Umirbaev, U. (2008). The Freiheitssatz and automorphisms of free right-symmetric algebras. Asian-Eur. J. Math. 01(02):243–254. DOI:10.1142/S1793557108000230.
   Google Scholar
• Kukin, G. P. (1977). On the word problem for Lie algebras, Sibirsk. Mat. Zh. 18(5):1194–1197.
   Web of Science ®Google Scholar
• Li, Y., Mo, Q., Zhao, X. (2018). Gröbner-Shirshov bases for brace algebras. Commun. Algebra 46(11):4577–4589. DOI:10.1080/00927872.2018.1448846.
   Web of Science ®Google Scholar
• Magnus, W. (1930). Über discontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz). J. Reine Angew. Math. 163:141–165.
   Google Scholar
• Makar-Limanov, L. (1970). The automorphisms of the free algebra of two generators, Funksional. Anal. i Prilozhen. 4(3):107–108.
   Google Scholar
• Makar-Limanov, L. (1985). Algebraically closed skew fields. J. Algebra 93(1):117–135. DOI:10.1016/0021-8693(85)90177-2.
   Web of Science ®Google Scholar
• Makar-Limanov, L., Turusbekova, U., Umirbaev, U. (2009). Automorphisms and derivations of free Poisson algebras in two variables. J. Algebra 322(9):3318–3330. DOI:10.1016/j.jalgebra.2008.01.005.
   Web of Science ®Google Scholar
• Makar-Limanov, L., Umirbaev, U. (2011a). The Freiheitssatz for Poisson algebras. J. Algebra 328(1):495–503. DOI:10.1016/j.jalgebra.2010.08.015.
   Web of Science ®Google Scholar
• Makar-Limanov, L., Umirbaev, U. (2011b). The Freiheitssatz for Novikov algebras. TWMS J. Pure Appl. Math. 2:228–235.
   Google Scholar
• Mikhalev, A. A., Shestakov, I. P. (2014). PBW-pairs of varieties of linear algebras. Comm. Algebra 42(2):667–687. DOI:10.1080/00927872.2012.720867.
   Web of Science ®Google Scholar
• Ronco, M. (2001). A Milnor-Moore theorem for dendriform Hopf algebras. C. R. Acad. Sci. Paris Sér. I Math. 332(2):109–114. DOI:10.1016/S0764-4442(00)01778-X.
   Google Scholar
• Ronco, M. (2002). Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras. J. Algebra 254(1):152–172. DOI:10.1016/S0021-8693(02)00097-2.
   Web of Science ®Google Scholar
• Segal, D. (1994). Free left-symmetric algebras and an analogue of the Poincaré-Birkhoff-Witt theorem. J. Algebra 164(3):750–772. DOI:10.1006/jabr.1994.1088.
   Web of Science ®Google Scholar
• Shirshov, A. I. (1962). Some algorithmic problems for ε-algebras. Sibirsk. Mat. Z 3:132–137.
   Google Scholar
• Shirshov, A. I. (1962). Some algorithmic problems for Lie algebras. Sibirsk. Mat. Z. 3(2):292–296 (in Russian). SIGSAM Bull. 33(2):3–6 (1999, in English). DOI:10.1145/334714.334715.
   Google Scholar
• van der Kulk, W. (1953). On polynomial rings in two variables. Nieuw Archief voor Wiskunde 1(3):33–41.
   Google Scholar
• Zhukov, A. I. (1950). Reduced systems of defining relations in non-associative algebras. Mat. Sb. 27:267–280.
   Google Scholar