Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Derivations and invariant forms of Jordan and alternative tori

Authors: Erhard Neher and Yoji Yoshii
Journal: Trans. Amer. Math. Soc. 355 (2003), 1079-1108
MSC (2000): Primary 17C10; Secondary 17B60, 17B70, 17C60
Published electronically: November 1, 2002
MathSciNet review: 1938747
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Jordan and alternative tori are the coordinate algebras of extended affine Lie algebras of types ${A}_1$ and ${A}_2$. In this paper we show that the derivation algebra of a Jordan torus is a semidirect product of the ideal of inner derivations and the subalgebra of central derivations. In the course of proving this result, we investigate derivations of the more general class of division graded Jordan and alternative algebras. We also describe invariant forms of these algebras.

References [Enhancements On Off] (What's this?)

  • 1. B. Allison, S. Azam, S. Berman, Y. Gao, and A. Pianzola, Extended affine Lie algebras and their root systems, Memoirs Amer. Math. Soc., vol. 603, Amer. Math. Soc., Providence, RI, 1997. MR 97i:17015
  • 2. Bruce N. Allison and Yun Gao, The root system and the core of an extended affine Lie algebra, Selecta Math. (N.S.) 7 (2001), no. 2, 149-212. MR 2002g:17041
  • 3. G. Benkart, Derivations and invariant forms of Lie algebras graded by finite root systems, Canad. J. Math. 50 (1998), 225-241. MR 99d:17013
  • 4. S. Berman, Y. Gao, and Y. Krylyuk, Quantum tori and the structure of elliptic quasi-simple Lie algebras, J. Funct. Anal. 135 (1996), 339-389. MR 97b:17007
  • 5. S. Berman, Y. Gao, Y. Krylyuk, and E. Neher, The alternative torus and the structure of elliptic quasi-simple Lie algebras of type $A_2$, Trans. Amer. Math. Soc. 347 (1995), 4315-4363. MR 96b:17009
  • 6. M. Boulagouaz, The graded and tame extensions, Lecture Notes in Pure and Appl. Math., Commutative Ring Theory, vol. 153, Marcel Dekker, New York, 1994. MR 94k:12010
  • 7. D. Z. Dokovic and K. Zhao, Derivations, isomorphisms, and second cohomology of generalized Witt algebras, Trans. Amer. Math. Soc. 350(2) (1998), 643-664. MR 98d:17031
  • 8. A. Duff, Derivations, invariant forms and central extensions of orthosymplectic Lie superalgebras, Ph.D. dissertation, University of Ottawa, Department of Mathematics and Statistics, 2002.
  • 9. A. Dzhumadil'daev, Cohomologies and deformations of right-symmetric algebras, J. Math. Sci. (New York) 93 (1999), 836-876. MR 2000e:17002
  • 10. -, Minimal identities for right-symmetric algebras, J. Algebra 225 (2000), 201-230. MR 2001b:17002
  • 11. R. Farnsteiner, Derivations and central extensions of finitely generated graded Lie algebras, J. Algebra 118 (1988), 33-45. MR 89h:17018
  • 12. K. R. Goodearl and E. S. Letzter, Quantum $n$-space as a quotient of classical $n$-space, Trans. Amer. Math. Soc. 352 (2000), 5855-5876. MR 2001e:16061
  • 13. I. N. Herstein, Topics in ring theory, Chicago Lectures in Mathematics, The University of Chicago Press, 1969. MR 42:6018
  • 14. N. Jacobson, Structure and representations of Jordan algebras, Amer. Math. Soc. Colloq. Publ., vol. 39, American Mathematical Society, Providence, RI, 1968. MR 40:4330
  • 15. -, Exceptional Lie algebras, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1971. MR 44:1707
  • 16. N. Kawamoto, Generalizations of Witt algebras over a field of characteristic zero, Hiroshima Math. J. 16 (1986), 417-426. MR 88d:17017
  • 17. M. Koecher, Imbedding of Jordan algebras into Lie algebras I, Amer. J. Math. 89 (1967), 787-816. MR 35:5480
  • 18. T. Y. Lam, A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 1991. MR 92f:16001
  • 19. O. Loos, Spiegelungsräume und homogene symmetrische Räume, Math. Z. 99 (1967), 141-170. MR 35:3608
  • 20. K. McCrimmon, Alternative algebras, (unpublished book).
  • 21. -, Zelmanov's prime theorem for quadratic Jordan algebras, J. Algebra 76 (1982), 297-326. MR 83h:17019
  • 22. -, Jordan centroids, Comm. Algebra 27, no. 2 (1999), 933-954. MR 2000b:17041
  • 23. C. Nastasescu and F. van Oystaeyen, Graded ring theory, North-Holland Mathematical Library, vol. 28, North-Holland, 1982. MR 84i:16002
  • 24. J. M. Osborn and D. S. Passman, Derivations of skew polynomial rings, J. Algebra 176 (1995), 417-448. MR 97i:16030
  • 25. D. S. Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press, Boston, 1989. MR 90g:16002
  • 26. -, Simple Lie algebras of Witt type, J. Algebra 206 (1998), 682-692. MR 99j:17012
  • 27. S. Tan, TTK algebras and vertex operator representations, J. Algebra 211 (1999), 298-342. MR 2000f:17035
  • 28. X. Xu, New generalized simple Lie algebras of Cartan type over a field with characteristic zero, J. Algebra 224 (2000), 23-58. MR 2001b:17021
  • 29. Y. Yoshii, Classification of division $\mathbf{Z}^{\mathbf{n}}$-graded alternative algebras, to appear in J. Algebra.
  • 30. -, Quantum tori with graded involution, to appear in Canad. Math. Bull.
  • 31. -, Coordinate algebras of extended affine Lie algebras of type $A_1$, J. Algebra 234 (2000), 128-168. MR 2001i:17031
  • 32. -Root-graded Lie algebras with compatible grading, Comm. Algebra 29 (2001), no. 8, 3365-3391. MR 2002g:17044
  • 33. K. A. Zhevlakov, A. M. Slinko, J. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative, Academic Press, New York, 1982. MR 83i:17001

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 17C10, 17B60, 17B70, 17C60

Retrieve articles in all journals with MSC (2000): 17C10, 17B60, 17B70, 17C60

Additional Information

Erhard Neher
Affiliation: Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada

Yoji Yoshii
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Address at time of publication: Department of Mathematics, Van Vleck Hall, University of Wisconsin-Madison, Madison, Wisconsin 53706

Received by editor(s): April 3, 2001
Received by editor(s) in revised form: January 31, 2002
Published electronically: November 1, 2002
Additional Notes: The research of the first author was partially supported by an NSERC (Canada) research grant
The research of the second author was supported by a Fields Postdoctoral Fellowship (Fall 2000) and a PIMS Postdoctoral Fellowship (2001)
Dedicated: Dedicated to Holger Petersson
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society