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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

H. Guzzo Jr., The Peirce decomposition for commutative train algebras, Comm. Algebra, 22 (1994), 5745-5757. doi: 10.1080/00927879408825160.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Ouattara, Sur les T-algèbres de Jordan, Linear Algebra Appl., 144 (1991), 11-21. doi: 10.1016/0024-3795(91)90056-3.  Google Scholar

[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.  Google Scholar

[10]

R. D. Schafer, "An Introduction to Nonassociative Algebras," Academic Press, New York, 1966. Google Scholar

[11]

S. Walcher, On Bernstein algebras which are train algebras, Proc. Edinb. Math. Soc. (2), 35 (1992), 159-166. doi: 10.1017/S0013091500005411.  Google Scholar

[12]

S. Walcher, Algebras which satisfy a train equation for the first three plenary powers, Arch. Math. (Basel), 56 (1991), 547-551.  Google Scholar

[13]

A. Wörz-Busekros, "Algebras in Genetics," Lecture Notes in Biomathematics, 36, Springer-Verlag, Berlin-New York, 1980. Google Scholar

show all references

References:
[1]

A. A. Albert, A theory of power-associative commutative algebras, Trans. Amer. Math. Soc., 69 (1950), 503-527.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

H. Guzzo Jr., The Peirce decomposition for commutative train algebras, Comm. Algebra, 22 (1994), 5745-5757. doi: 10.1080/00927879408825160.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Ouattara, Sur les T-algèbres de Jordan, Linear Algebra Appl., 144 (1991), 11-21. doi: 10.1016/0024-3795(91)90056-3.  Google Scholar

[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.  Google Scholar

[10]

R. D. Schafer, "An Introduction to Nonassociative Algebras," Academic Press, New York, 1966. Google Scholar

[11]

S. Walcher, On Bernstein algebras which are train algebras, Proc. Edinb. Math. Soc. (2), 35 (1992), 159-166. doi: 10.1017/S0013091500005411.  Google Scholar

[12]

S. Walcher, Algebras which satisfy a train equation for the first three plenary powers, Arch. Math. (Basel), 56 (1991), 547-551.  Google Scholar

[13]

A. Wörz-Busekros, "Algebras in Genetics," Lecture Notes in Biomathematics, 36, Springer-Verlag, Berlin-New York, 1980. Google Scholar

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