168
Views
25
CrossRef citations to date
0
Altmetric
Original Articles

PBW-Pairs of Varieties of Linear Algebras

&
Pages 667-687 | Received 12 Jun 2012, Published online: 18 Oct 2013

REFERENCES

  • Akivis , M. A. ( 1976 ). The local algebras of a multidimentional three-web . Sibirsk. Mat. Z. 17:5โ€“11 (Russian). English translation: Siberian Math. J. 17 : 3 โ€“ 8 .
  • Amitsur , S. A. ( 1955 ). The T-ideals of the free ring . J. London Math. Soc. 30 : 470 โ€“ 475 .
  • Artamonov , V. A. , Burgin , M. S. ( 1972 ). Some properties of subalgebras in varieties of linear ฮฉ-algebras . Matem. Sb. 87:67โ€“82 (Russian). English translation: Math. USSR-Sbornik 16 : 69 โ€“ 85 .
  • Artamonov , V. A. , Mikhalev , A. A. , Mikhalev , A. V. ( 2003 ). Combinatorial properties of free algebras of schreier varieties . In: Giambruno , A. , Regev , A. , Zaicev , M. , eds. Polynomial Identities and Combinatorial Methods . Marcel Dekker , pp. 47 โ€“ 99 .
  • Aust , C. ( 1974 ). Primitive elements and one relation algebras . Trans. Amer. Math. Soc. 193 : 375 โ€“ 387 .
  • Bloh , A. (1965). On a generalization of the concept of Lie algebra. Dokl. Akad. Nauk SSSR 165:471โ€“473 (Russian).
  • Braun , A. ( 1984 ). The nilpotency of the radical in a finitely generated PI-ring . J. Algebra 89 : 375 โ€“ 396 .
  • Burgin , M. S. ( 1969 ). The freeness theorem in some varieties of linear ฮฉ-algebras and ฮฉ-rings . Uspekhi Matem. Nauk 24:27โ€“38 (Russian). English translation: Russian Math. Surveys 24 : 25 โ€“ 35 .
  • Champagnier , C. , Chepovskii , A. A. , Mikhalev , A. A. , Mikhalev , A. V. ( 2009 ). Primitive elements of free nonassociative algebras . Fundam. Prikl. Mat. 13:171โ€“192 (Russian). English translation: J. Math. Sci. (N. Y.) 156 : 320 โ€“ 335 .
  • Chen , Y. , Li , Y. ( 2011 ). Some remarks on the Akivis algebras and the pre-Lie algebras . Czechoslovak Math. Z. 61 : 707 โ€“ 720 .
  • Chibrikov , E. ( 2011 ). On free Sabinin algebras . Commun. Algebra 39 : 4014 โ€“ 4035 .
  • Filippov , V. T. ( 1977 ). Lie center of binary Lie algebras . Algebra i Logika 16:213โ€“226 (Russian). English translation: Algebra and Logic 16 : 143 โ€“ 153 .
  • Filippov , V. T. ( 1982 ). Free Maltsev algebras and alternative algebras . Algebra i Logika 21:84โ€“107 (Russian). English translation: Algebra and Logic 21 : 58 โ€“ 73 .
  • Goldberg , V. V. ( 1990 ). Local differentiable quasigroups and webs . In: Chein , O. , Pflugfelder , H. O. , Smith , J. D. H. , eds. Quasigroups and Loops. Theory and Applications, Sigma Series in Pure Mathematics . Vol. 8 . Berlin : Heldermann Verlag , pp. 263 โ€“ 311 .
  • Hofmann , K. H. , Strambach , K. ( 1990 ). Topological and analytic loops , In: Chein , O. , Pflugfelder , H. O. , Smith , J. D. H. , eds. Quasigroups and Loops. Theory and Applications, Sigma Series in Pure Mathematics . Vol. 8 . Berlin : Heldermann Verlag , pp. 205 โ€“ 262 .
  • Jacobson , N. ( 1968 ). Structure and Representations of Jordan Algebras . American Mathematical Society Colloquium Publications . Vol. XXXIX . Providence , R.I. : American Mathematical Society .
  • Kemer , A. R. ( 1980 ). Capelli identities and nilpotency of the radical of finitely generated PI-algebras . Dokl. Akad. Nauk SSSR 255:793โ€“797 (Russian). English translation: Soviet Math. Dokl. 22 : 750 โ€“ 753 .
  • Kukin , G. P. ( 1970 ). Primitive elements of free Lie algebras . Algebra i Logika 9:458โ€“472 (Russian). English translation: Algebra and Logic 9 : 275 โ€“ 284 .
  • Kurosh , A. G. ( 1947 ). Nonassociative free algebras and free products of algebras . Mat. Sb. 20 : 239 โ€“ 262 (Russian) .
  • Kurosh , A. G. ( 1960 ). Free sums of multioperator algebras . Sibirsk. Mat. Zh. 24 : 62 โ€“ 77 (Russian) .
  • Kurosh , A. G. ( 1969 ). Multioperator rings and algebras . Uspekhi Matem. Nauk 24:3โ€“15 (Russian). English translation: Russian Math. Surveys 24 : 1 โ€“ 13 .
  • Lewin , J. ( 1968 ). On Schreier varieties of linear algebras . Trans. Amer. Math. Soc. 132 : 553 โ€“ 562 .
  • Makar-Limanov , L. G. ( 1985 ). Algebraically closed skew fields . J. Algebra 93 : 117 โ€“ 135 .
  • Magnus , W. ( 1930 ). รœber diskontinuierliche Gruppen mit einer definierden Relation (Der Freiheitssatz) . J. Reine Angew. Math. 163 : 141 โ€“ 165 .
  • Maneri , C. ( 1963 ). Simple (โˆ’1,1) rings with an idempotent . Proc. Amer. Math. Soc. 14 : 110 โ€“ 117 .
  • Medvedev , Yu. A. ( 1988 ). Free Jordan algebras . Algebra i Logika 27:172โ€“200, 244 (Russian). English translation: Algebra and Logic 27 : 110 โ€“ 127 .
  • Mikhalev , A. A. , Shpilrain , V. , Yu , J.-T. ( 2004 ). Combinatorial Methods: Free Groups, Polynomials, and Free Algebras . New York : Springer-Verlag .
  • Mikhalev , A. A. , Umirbaev , U. U. , Yu , J.-T. ( 2001 ). Automorphic orbits in free non-associative algebras . J. Algebra 243 : 198 โ€“ 223 .
  • Mikhalev , A. A. , Umirbaev , U. U. , Zolotykh , A. A. (1994). An example of a nonfree Lie algebra of cohomological dimension one. Uspekhi Mat. Nauk 49:203โ€“204 (Russian). English translation: Russian Math. Surveys 49:254.
  • Mikhalev , A. A. , Umirbaev , U. U. , Zolotykh , A. A. ( 1996 ). A Lie algebra with cohomological dimension one over a field of prime characteristic is not necessarily free . In: Fong , Y. , Knauer , U. , Mikhalev , A. V. , eds. First International Tainanโ€“Moscow Algebra Workshop , Berlin : Walter de Gruyter Publ. , pp. 257 โ€“ 264 .
  • Mikhalev , A. A. , Yu , J.-T. ( 2001 ). Stable equivalence problems for free algebras with the Nielsen-Schreier property . Intern. J. Algebra Comput. 11 : 779 โ€“ 786 .
  • Miheev , P. O. , Sabinin , L. V. ( 1987 ). On the infinitesimal theory of local analytic loops . Dokl. Akad. Nauk SSSR 297:801โ€“804 (Russian). English translation: Soviet Math. Dokl. 36 : 545 โ€“ 548 .
  • Mostovoy , J. , Pรฉrez-Izquierdo , J. M. ( 2001 ). Formal multiplications, bialgebras of distributions and non-associative Lie Theory . Transform. Groups 15 : 625 โ€“ 653 .
  • Pรฉrez-Izquierdo , J. M. ( 2007 ). Algebras, hyperalgebras, nonassociative bialgebras and loops . Adv. Math. 208 : 834 โ€“ 876 .
  • Pรฉrez-Izquierdo , J. M. , Shestakov , I. P. ( 2004 ). An envelope for Malcev algebras . J. Algebra 272 : 379 โ€“ 393 .
  • Razmyslov , Yu. P. ( 1973 ). On a problem of Kaplansky . Izv. Akad. Nauk SSSR, Ser. Mat. 37:483โ€“501 (Russian). English translation: Math. USSR Izv. 7 : 479 โ€“ 496 .
  • Razmyslov , Yu. P. ( 1974 ). The Jacobson radical of PI-algebras . Algebra i Logika 13:337โ€“360 (Russian). English translation: Algebra and Logic 13 : 192 โ€“ 204 .
  • Shestakov , I. P. ( 1999 ). Linear representability of Akivis algebras . Doklady Akad. Nauk 368:21โ€“23 (Russian). English translation: Doklady Mathematics 60 : 164 โ€“ 166 .
  • Shestakov , I. P. ( 1999 ). Every Akivis algebra is linear . Geometriae Dedicata 77 : 215 โ€“ 223 .
  • Shestakov , I. P. ( 1998/2000 ). Speciality and deformations of algebras. In: Bahturin, Yu., ed. Algebra. Proceedings of the International Algebraic Conference on Occasion of the 90th Birthday of A. G. Kurosh, Moscow, Russia, May 25โ€“30; Berlin, New York: de Gruyter, pp. 345โ€“356 .
  • Shestakov , I. P. , Umirbaev , U. U. ( 2002 ). Free Akivis algebras, primitive elements, and hyperalgebras . J. Algebra 250 : 533 โ€“ 548 .
  • Shirshov , A. I. ( 1953/2000 ). Subalgebras of free Lie algebras. it Mat. Sb. 33:441โ€“452 (Russian). English translation : In: Bokut , L. A. , Latyshev , V. , Shestakov , I. , Zelmanov , E. , eds. Selected Works of A. I. Shirshov . Basel : Birkhรคuser , pp. 4 โ€“ 13 .
  • Shirshov , A. I. ( 1954/2000 ). Subalgebras of free commutative and free anti-commutative algebras. Mat. Sb. 34:81โ€“88 (Russian). English translation: In: Bokut, L. A., Latyshev, V., Shestakov, I., Zelmanov, E., eds. Selected Works of A. I. Shirshov Basel: Birkhรคuser, pp. 19โ€“25 .
  • Shirshov , A. I. ( 1962/2009 ). Some algorithmic problems for Lie algebras. Sibirsk. Mat. Zh. 3:292โ€“296 (Russian). Eglish translations: SIGSAM Bull. 33:3โ€“6; In: Bokut, L. A., Latyshev, V., Shestakov, I., Zelmanov, E. eds. Selected Works of A. I. Shirshov Basel: Birkhรคuser, pp. 125โ€“130 .
  • Shirshov , A. I. ( 1962/2009 ). Some algorithmic problems for ฯต-algebras. Sibirsk. Mat. Z. 3:132โ€“137 (Russian). English translation: In: Bokut, L. A., Latyshev, V., Shestakov, I., Zelmanov, E. eds. Selected Works of A. I. Shirshov Basel: Birkhรคuser, pp. 119โ€“124 .
  • Umirbaev , U. U. ( 1994 ). On Schreier varieties of algebras . Algebra i Logika 33:317โ€“340 (Russian). English translation: Algebra and Logic 33 : 180 โ€“ 193 .
  • Witt , E. ( 1956 ). Die Unterringe der freien Lieschen Ringe . Math. Z. 64 : 195 โ€“ 216 .
  • Whitehead , J. H. C. ( 1936 ). On equivalent sets of elements in a free group . Ann. Math. 37 : 782 โ€“ 800 .
  • Yagzhev , A. V. (1980). Endomorphisms of free algebras. Sibirsk. Math. Zh. 21:181โ€“192 (Russian). English translation: Sib. Math. J. 21:133โ€“141.
  • Zhevlakov , K. A. , Slinko , A. M. , Shestakov , I. P. , Shirshov , A. I. ( 1978/1982 ). Rings that are Nearly Associative (in Russian), Nauka: Moscow. New York โ€“ London: English translation: Academic Press .
  • Zhukov , A. I. ( 1950 ). Reduced systems of defining relations in non-associative algebras . Mat. Sb. 27 : 267 โ€“ 280 (Russian) .
  • Communicated by V. A. Artamonov.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.