Train algebras of degree 2 and exponent 3

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December 2011, 4(6): 1371-1386. doi: 10.3934/dcdss.2011.4.1371

Train algebras of degree 2 and exponent 3

Joseph Bayara ^1, , André Conseibo ^2, , Moussa Ouattara ^3, and Artibano Micali ^4,

1.	Université Polytechnique de Bobo-Dioulasso, 01 BP 1091 Bobo-Dioulasso 01
2.	Université de Koudougou, BP 376 Koudougou
3.	Université de Ouagadougou, 03 BP 7021 Ouagadougou
4.	Université Montpellier 2, Place Eugène Bataillon, 34095 Montpellier Cedex

Received February 2009 Revised October 2009 Published December 2010

Abstract
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In this paper we investigate the structure of weighted algebras satisfying the equation $(x^3)^2 = \omega(x)^3x^3$, a class of algebras properly containing the class of Bernstein algebras. We give the classification of these algebras in dimension three. Some results about the structure of algebras satisfying the more general equation $(x^n)^2 = \omega(x)^nx^n$, for $n\geq 2$, are also obtained.

Keywords: Peirce decomposition, Jordan algebra, Power-associative, Bernstein algebra, Train algebra, Idempotent..

Mathematics Subject Classification: Primary: 17D92; Secondary: 17A3.

Citation: Joseph Bayara, André Conseibo, Moussa Ouattara, Artibano Micali. Train algebras of degree 2 and exponent 3. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1371-1386. doi: 10.3934/dcdss.2011.4.1371

References:

[1]	A. A. Albert, A theory of power-associative commutative algebras, Trans. Amer. Math. Soc., 69 (1950), 503-527.
[2]	I. Basso, R. Costa, J. Carlos Gutiérrez and H. Guzzo Jr., Cubic algebras of exponent $2$: Basic properties, Int. J. Math. Game Theory and Algebra, 9 (1999), 245-258.
[3]	J. Bayara, A. Conseibo and M. Ouattara et F. Zitan, Power-associative algebras that are train algebras, J. Algebra, 324 (2010), 1159-1176. doi: 10.1016/j.jalgebra.2010.06.012.
[4]	H. Guzzo Jr. and P. Vicente, Train algebras of rank $n$ which are Bernstein or power-associative algebras, Nova J. Math. Game Theory Algebra, 6 (1997), 103-112.
[5]	H. Guzzo Jr., The Peirce decomposition for commutative train algebras, Comm. Algebra, 22 (1994), 5745-5757. doi: 10.1080/00927879408825160.
[6]	J. Lopez Sanchez and E. Rodriguez S. Maria, On train algebras of rank $4$, Comm. Algebra, 24 (1996), 4439-4445. doi: 10.1080/00927879608825825.
[7]	A. Micali and M. Ouattara, Structure des algèbres de Bernstein, Linear Algebra Appl., 218 (1995), 77-88. doi: 10.1016/0024-3795(93)00159-W.
[8]	M. Ouattara, Sur les T-algèbres de Jordan, Linear Algebra Appl., 144 (1991), 11-21. doi: 10.1016/0024-3795(91)90056-3.
[9]	M. Ouattara, Sur une classe d'algèbres à puissances associatives, Linear Algebra Appl., 235 (1996), 47-62. doi: 10.1016/0024-3795(94)00113-8.
[10]	R. D. Schafer, "An Introduction to Nonassociative Algebras," Academic Press, New York, 1966.
[11]	S. Walcher, On Bernstein algebras which are train algebras, Proc. Edinb. Math. Soc. (2), 35 (1992), 159-166. doi: 10.1017/S0013091500005411.
[12]	S. Walcher, Algebras which satisfy a train equation for the first three plenary powers, Arch. Math. (Basel), 56 (1991), 547-551.
[13]	A. Wörz-Busekros, "Algebras in Genetics," Lecture Notes in Biomathematics, 36, Springer-Verlag, Berlin-New York, 1980.

show all references

References:

[1]	A. A. Albert, A theory of power-associative commutative algebras, Trans. Amer. Math. Soc., 69 (1950), 503-527.
[2]	I. Basso, R. Costa, J. Carlos Gutiérrez and H. Guzzo Jr., Cubic algebras of exponent $2$: Basic properties, Int. J. Math. Game Theory and Algebra, 9 (1999), 245-258.
[3]	J. Bayara, A. Conseibo and M. Ouattara et F. Zitan, Power-associative algebras that are train algebras, J. Algebra, 324 (2010), 1159-1176. doi: 10.1016/j.jalgebra.2010.06.012.
[4]	H. Guzzo Jr. and P. Vicente, Train algebras of rank $n$ which are Bernstein or power-associative algebras, Nova J. Math. Game Theory Algebra, 6 (1997), 103-112.
[5]	H. Guzzo Jr., The Peirce decomposition for commutative train algebras, Comm. Algebra, 22 (1994), 5745-5757. doi: 10.1080/00927879408825160.
[6]	J. Lopez Sanchez and E. Rodriguez S. Maria, On train algebras of rank $4$, Comm. Algebra, 24 (1996), 4439-4445. doi: 10.1080/00927879608825825.
[7]	A. Micali and M. Ouattara, Structure des algèbres de Bernstein, Linear Algebra Appl., 218 (1995), 77-88. doi: 10.1016/0024-3795(93)00159-W.
[8]	M. Ouattara, Sur les T-algèbres de Jordan, Linear Algebra Appl., 144 (1991), 11-21. doi: 10.1016/0024-3795(91)90056-3.
[9]	M. Ouattara, Sur une classe d'algèbres à puissances associatives, Linear Algebra Appl., 235 (1996), 47-62. doi: 10.1016/0024-3795(94)00113-8.
[10]	R. D. Schafer, "An Introduction to Nonassociative Algebras," Academic Press, New York, 1966.
[11]	S. Walcher, On Bernstein algebras which are train algebras, Proc. Edinb. Math. Soc. (2), 35 (1992), 159-166. doi: 10.1017/S0013091500005411.
[12]	S. Walcher, Algebras which satisfy a train equation for the first three plenary powers, Arch. Math. (Basel), 56 (1991), 547-551.
[13]	A. Wörz-Busekros, "Algebras in Genetics," Lecture Notes in Biomathematics, 36, Springer-Verlag, Berlin-New York, 1980.

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