Derivations and invariant forms of Jordan and alternative tori
Authors:
Erhard Neher and Yoji Yoshii
Journal:
Trans. Amer. Math. Soc. 355 (2003), 10791108
MSC (2000):
Primary 17C10; Secondary 17B60, 17B70, 17C60
Published electronically:
November 1, 2002
MathSciNet review:
1938747
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Jordan and alternative tori are the coordinate algebras of extended affine Lie algebras of types and . 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.
 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, 149212. MR 2002g:17041
 3.
G. Benkart, Derivations and invariant forms of Lie algebras graded by finite root systems, Canad. J. Math. 50 (1998), 225241. MR 99d:17013
 4.
S. Berman, Y. Gao, and Y. Krylyuk, Quantum tori and the structure of elliptic quasisimple Lie algebras, J. Funct. Anal. 135 (1996), 339389. MR 97b:17007
 5.
S. Berman, Y. Gao, Y. Krylyuk, and E. Neher, The alternative torus and the structure of elliptic quasisimple Lie algebras of type , Trans. Amer. Math. Soc. 347 (1995), 43154363. 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), 643664. 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 rightsymmetric algebras, J. Math. Sci. (New York) 93 (1999), 836876. MR 2000e:17002
 10.
, Minimal identities for rightsymmetric algebras, J. Algebra 225 (2000), 201230. MR 2001b:17002
 11.
R. Farnsteiner, Derivations and central extensions of finitely generated graded Lie algebras, J. Algebra 118 (1988), 3345. MR 89h:17018
 12.
K. R. Goodearl and E. S. Letzter, Quantum space as a quotient of classical space, Trans. Amer. Math. Soc. 352 (2000), 58555876. 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), 417426. MR 88d:17017
 17.
M. Koecher, Imbedding of Jordan algebras into Lie algebras I, Amer. J. Math. 89 (1967), 787816. MR 35:5480
 18.
T. Y. Lam, A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131, SpringerVerlag, New York, 1991. MR 92f:16001
 19.
O. Loos, Spiegelungsräume und homogene symmetrische Räume, Math. Z. 99 (1967), 141170. MR 35:3608
 20.
K. McCrimmon, Alternative algebras, (unpublished book).
 21.
, Zelmanov's prime theorem for quadratic Jordan algebras, J. Algebra 76 (1982), 297326. MR 83h:17019
 22.
, Jordan centroids, Comm. Algebra 27, no. 2 (1999), 933954. MR 2000b:17041
 23.
C. Nastasescu and F. van Oystaeyen, Graded ring theory, NorthHolland Mathematical Library, vol. 28, NorthHolland, 1982. MR 84i:16002
 24.
J. M. Osborn and D. S. Passman, Derivations of skew polynomial rings, J. Algebra 176 (1995), 417448. 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), 682692. MR 99j:17012
 27.
S. Tan, TTK algebras and vertex operator representations, J. Algebra 211 (1999), 298342. MR 2000f:17035
 28.
X. Xu, New generalized simple Lie algebras of Cartan type over a field with characteristic zero, J. Algebra 224 (2000), 2358. MR 2001b:17021
 29.
Y. Yoshii, Classification of division 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 , J. Algebra 234 (2000), 128168. MR 2001i:17031
 32.
Rootgraded Lie algebras with compatible grading, Comm. Algebra 29 (2001), no. 8, 33653391. 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
 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, 149212. MR 2002g:17041
 3.
 G. Benkart, Derivations and invariant forms of Lie algebras graded by finite root systems, Canad. J. Math. 50 (1998), 225241. MR 99d:17013
 4.
 S. Berman, Y. Gao, and Y. Krylyuk, Quantum tori and the structure of elliptic quasisimple Lie algebras, J. Funct. Anal. 135 (1996), 339389. MR 97b:17007
 5.
 S. Berman, Y. Gao, Y. Krylyuk, and E. Neher, The alternative torus and the structure of elliptic quasisimple Lie algebras of type , Trans. Amer. Math. Soc. 347 (1995), 43154363. 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), 643664. 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 rightsymmetric algebras, J. Math. Sci. (New York) 93 (1999), 836876. MR 2000e:17002
 10.
 , Minimal identities for rightsymmetric algebras, J. Algebra 225 (2000), 201230. MR 2001b:17002
 11.
 R. Farnsteiner, Derivations and central extensions of finitely generated graded Lie algebras, J. Algebra 118 (1988), 3345. MR 89h:17018
 12.
 K. R. Goodearl and E. S. Letzter, Quantum space as a quotient of classical space, Trans. Amer. Math. Soc. 352 (2000), 58555876. 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), 417426. MR 88d:17017
 17.
 M. Koecher, Imbedding of Jordan algebras into Lie algebras I, Amer. J. Math. 89 (1967), 787816. MR 35:5480
 18.
 T. Y. Lam, A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131, SpringerVerlag, New York, 1991. MR 92f:16001
 19.
 O. Loos, Spiegelungsräume und homogene symmetrische Räume, Math. Z. 99 (1967), 141170. MR 35:3608
 20.
 K. McCrimmon, Alternative algebras, (unpublished book).
 21.
 , Zelmanov's prime theorem for quadratic Jordan algebras, J. Algebra 76 (1982), 297326. MR 83h:17019
 22.
 , Jordan centroids, Comm. Algebra 27, no. 2 (1999), 933954. MR 2000b:17041
 23.
 C. Nastasescu and F. van Oystaeyen, Graded ring theory, NorthHolland Mathematical Library, vol. 28, NorthHolland, 1982. MR 84i:16002
 24.
 J. M. Osborn and D. S. Passman, Derivations of skew polynomial rings, J. Algebra 176 (1995), 417448. 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), 682692. MR 99j:17012
 27.
 S. Tan, TTK algebras and vertex operator representations, J. Algebra 211 (1999), 298342. MR 2000f:17035
 28.
 X. Xu, New generalized simple Lie algebras of Cartan type over a field with characteristic zero, J. Algebra 224 (2000), 2358. MR 2001b:17021
 29.
 Y. Yoshii, Classification of division 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 , J. Algebra 234 (2000), 128168. MR 2001i:17031
 32.
 Rootgraded Lie algebras with compatible grading, Comm. Algebra 29 (2001), no. 8, 33653391. 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
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Additional Information
Erhard Neher
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
Email:
neher@uottawa.ca
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 WisconsinMadison, Madison, Wisconsin 53706
Email:
yoshii@math.ualberta.ca, yoshii@math.wisc.edu
DOI:
http://dx.doi.org/10.1090/S0002994702030131
PII:
S 00029947(02)030131
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
