December  2011, 4(6): 1359-1370. doi: 10.3934/dcdss.2011.4.1359

Derivations in power-associative algebras

1. 

Université Polytechnique de Bobo-Dioulasso, 01 BP 1091 Bobo-Dioulasso 01, Burkina Faso

2. 

Université de Koudougou, BP 376 Koudougou, Burkina Faso

3. 

Université Montpellier 2, Place Eugène Bataillon, 34095 Montpellier Cedex, France

4. 

Université de Ouagadougou, 03 BP 7021 Ouagadougou, Burkina Faso

Received  March 2009 Revised  November 2009 Published  December 2010

In this paper we investigate derivations of a commutative power-associative algebra. Particular cases of stable and partially stable algebras are inspected. Some attention is paid to the Jordan case. Further results are given. Especially, we show that the core of a $n^{th}$-order Bernstein algebra which is power-associative is a Jordan algebra.
Citation: Joseph Bayara, André Conseibo, Artibano Micali, Moussa Ouattara. Derivations in power-associative algebras. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1359-1370. doi: 10.3934/dcdss.2011.4.1359
References:
[1]

M. T. Alcalde, C. Burgueño, A. Labra and A. Micali, Sur les algèbres de Bernstein, Proc. Lond. Math. Soc., III. Ser., 58 (1989), 51-68.  Google Scholar

[2]

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

[3]

J. Bayara, A. Conseibo, M. Ouattara and 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]

I. M. H. Etherington, Genetic algebras, Proc. R. Soc. Edinb., 59 (1939), 242-258.  Google Scholar

[5]

M. A. García-Muñiz and S. González, Weighted, Bernstein and Jordan algebras, Comm. Algebra, 26 (1998), 913-930. doi: 10.1080/00927879808826173.  Google Scholar

[6]

M. A. García-Muñiz and C. Martínez, Derivations in second order Bernstein algebras, in "Nonassociative Algebra and its Applications" (eds. Costa, Roberto and al.), Proceedings of the fourth international conference, São Paulo, Brazil. New York, NY: Marcel Dekker; Lect. Notes Pure Appl. Math., 211 (2000), 105-124.  Google Scholar

[7]

H. Gonshor, Derivations in genetic algebras, Comm. Algebra, 16 (1988), 1525-1542. doi: 10.1080/00927879808823643.  Google Scholar

[8]

H. Guzzo Jr. and P. Vicente, Derivations in $n$th-order Bernstein algebras, Int. J. Math. Game Theory Algebra, 12 (2002), 171-185.  Google Scholar

[9]

H. Guzzo Jr. and P. Vicente, Derivatives in $n$th-order Bernstein algebras. II, Algebras Groups Geom., 19 (2002), 423-444.  Google Scholar

[10]

P. Holgate, The interpretation of derivations in genetic algebras, Linear Algebra Appl., 85 (1987), 75-79. doi: 10.1016/0024-3795(87)90209-6.  Google Scholar

[11]

N. Jacobson, "Structure and Representations of Jordan Algebras," Amer. Math. Soc. Colloquium Pulications 39, 1968. Google Scholar

[12]

L. A. Kokoris, Simple power-associative algebras of degree two, Ann. of Math. (3), 64 (1956), 544-550. doi: 10.2307/1969601.  Google Scholar

[13]

L. A. Kokoris, New results on power-associative algebras, Ann. of Math. (3), 77 (1954), 363-373.  Google Scholar

[14]

C. Mallol, A. Micali and M. Ouattara, Sur les algèbres de Bernstein IV, Linear Algebra Appl., 158 (1991), 1-26. doi: 10.1016/0024-3795(91)90048-2.  Google Scholar

[15]

A. Micali and M. Ouattara, Sur la dupliquée d'une algèbre. II, Bull. Soc. Math. Belg., Sér. A, 43 (1991), 113-125.  Google Scholar

[16]

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

[17]

J. M. Osborn, Varieties of algebras, Adv. Math., 8 (1972), 163-369. doi: 10.1016/0001-8708(72)90003-5.  Google Scholar

[18]

M. L. Reed, Algebraic structure of genetic inheritance, Bull. Amer. Math. Soc., 34 (1997), 107-130. doi: 10.1090/S0273-0979-97-00712-X.  Google Scholar

[19]

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

[20]

J. Tits, A theorem on generic norms of strictly power associative algebras, Proc. Amer. Math. Soc., 15 (1964), 35-36. doi: 10.1090/S0002-9939-1964-0158912-0.  Google Scholar

[21]

D. A. Towers and K. Bowman, On power associative Bernstein algebras of arbitrary order, Algebras, Groups and Geometries, 13 (1996), 295-322.  Google Scholar

[22]

S. Walcher, Bernstein algebras which are Jordan algebras, Arch. Math., 50 (1988), 218-222. doi: 10.1007/BF01187737.  Google Scholar

[23]

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

[24]

K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov and A. I. Shirshov, "Rings that are Nearly Associative," Academic Press, New York, 1982. Google Scholar

show all references

References:
[1]

M. T. Alcalde, C. Burgueño, A. Labra and A. Micali, Sur les algèbres de Bernstein, Proc. Lond. Math. Soc., III. Ser., 58 (1989), 51-68.  Google Scholar

[2]

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

[3]

J. Bayara, A. Conseibo, M. Ouattara and 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]

I. M. H. Etherington, Genetic algebras, Proc. R. Soc. Edinb., 59 (1939), 242-258.  Google Scholar

[5]

M. A. García-Muñiz and S. González, Weighted, Bernstein and Jordan algebras, Comm. Algebra, 26 (1998), 913-930. doi: 10.1080/00927879808826173.  Google Scholar

[6]

M. A. García-Muñiz and C. Martínez, Derivations in second order Bernstein algebras, in "Nonassociative Algebra and its Applications" (eds. Costa, Roberto and al.), Proceedings of the fourth international conference, São Paulo, Brazil. New York, NY: Marcel Dekker; Lect. Notes Pure Appl. Math., 211 (2000), 105-124.  Google Scholar

[7]

H. Gonshor, Derivations in genetic algebras, Comm. Algebra, 16 (1988), 1525-1542. doi: 10.1080/00927879808823643.  Google Scholar

[8]

H. Guzzo Jr. and P. Vicente, Derivations in $n$th-order Bernstein algebras, Int. J. Math. Game Theory Algebra, 12 (2002), 171-185.  Google Scholar

[9]

H. Guzzo Jr. and P. Vicente, Derivatives in $n$th-order Bernstein algebras. II, Algebras Groups Geom., 19 (2002), 423-444.  Google Scholar

[10]

P. Holgate, The interpretation of derivations in genetic algebras, Linear Algebra Appl., 85 (1987), 75-79. doi: 10.1016/0024-3795(87)90209-6.  Google Scholar

[11]

N. Jacobson, "Structure and Representations of Jordan Algebras," Amer. Math. Soc. Colloquium Pulications 39, 1968. Google Scholar

[12]

L. A. Kokoris, Simple power-associative algebras of degree two, Ann. of Math. (3), 64 (1956), 544-550. doi: 10.2307/1969601.  Google Scholar

[13]

L. A. Kokoris, New results on power-associative algebras, Ann. of Math. (3), 77 (1954), 363-373.  Google Scholar

[14]

C. Mallol, A. Micali and M. Ouattara, Sur les algèbres de Bernstein IV, Linear Algebra Appl., 158 (1991), 1-26. doi: 10.1016/0024-3795(91)90048-2.  Google Scholar

[15]

A. Micali and M. Ouattara, Sur la dupliquée d'une algèbre. II, Bull. Soc. Math. Belg., Sér. A, 43 (1991), 113-125.  Google Scholar

[16]

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

[17]

J. M. Osborn, Varieties of algebras, Adv. Math., 8 (1972), 163-369. doi: 10.1016/0001-8708(72)90003-5.  Google Scholar

[18]

M. L. Reed, Algebraic structure of genetic inheritance, Bull. Amer. Math. Soc., 34 (1997), 107-130. doi: 10.1090/S0273-0979-97-00712-X.  Google Scholar

[19]

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

[20]

J. Tits, A theorem on generic norms of strictly power associative algebras, Proc. Amer. Math. Soc., 15 (1964), 35-36. doi: 10.1090/S0002-9939-1964-0158912-0.  Google Scholar

[21]

D. A. Towers and K. Bowman, On power associative Bernstein algebras of arbitrary order, Algebras, Groups and Geometries, 13 (1996), 295-322.  Google Scholar

[22]

S. Walcher, Bernstein algebras which are Jordan algebras, Arch. Math., 50 (1988), 218-222. doi: 10.1007/BF01187737.  Google Scholar

[23]

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

[24]

K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov and A. I. Shirshov, "Rings that are Nearly Associative," Academic Press, New York, 1982. Google Scholar

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