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

Train algebras of degree 2 and exponent 3

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.
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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