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 and 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.

[2]

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

[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.

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[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.

[15]

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

[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.

[17]

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

[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.

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[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.

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.

[2]

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

[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.

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[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.

[15]

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

[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.

[17]

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

[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.

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[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.

[1]

Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, 2021, 29 (3) : 2457-2473. doi: 10.3934/era.2020124

[2]

Golamreza Zamani Eskandani, Hamid Vaezi. Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1469-1477. doi: 10.3934/dcds.2011.31.1469

[3]

Michelle Nourigat, Richard Varro. Conjectures for the existence of an idempotent in $\omega $-polynomial algebras. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1543-1551. doi: 10.3934/dcdss.2011.4.1543

[4]

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

[5]

A. A. Kirillov. Family algebras. Electronic Research Announcements, 2000, 6: 7-20.

[6]

M. P. de Oliveira. On 3-graded Lie algebras, Jordan pairs and the canonical kernel function. Electronic Research Announcements, 2003, 9: 142-151.

[7]

Yu-Lin Chang, Chin-Yu Yang. Some useful inequalities via trace function method in Euclidean Jordan algebras. Numerical Algebra, Control and Optimization, 2014, 4 (1) : 39-48. doi: 10.3934/naco.2014.4.39

[8]

Steffen Konig and Changchang Xi. Cellular algebras and quasi-hereditary algebras: a comparison. Electronic Research Announcements, 1999, 5: 71-75.

[9]

Stephen Doty and Anthony Giaquinto. Generators and relations for Schur algebras. Electronic Research Announcements, 2001, 7: 54-62.

[10]

Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39

[11]

Valentin Ovsienko, MichaeL Shapiro. Cluster algebras with Grassmann variables. Electronic Research Announcements, 2019, 26: 1-15. doi: 10.3934/era.2019.26.001

[12]

L. S. Grinblat. Theorems on sets not belonging to algebras. Electronic Research Announcements, 2004, 10: 51-57.

[13]

Adel Alahmadi, Hamed Alsulami, S.K. Jain, Efim Zelmanov. On matrix wreath products of algebras. Electronic Research Announcements, 2017, 24: 78-86. doi: 10.3934/era.2017.24.009

[14]

Jin-Yun Guo, Cong Xiao, Xiaojian Lu. On $ n $-slice algebras and related algebras. Electronic Research Archive, 2021, 29 (4) : 2687-2718. doi: 10.3934/era.2021009

[15]

Randall Dougherty and Thomas Jech. Left-distributive embedding algebras. Electronic Research Announcements, 1997, 3: 28-37.

[16]

G. Mashevitzky, B. Plotkin and E. Plotkin. Automorphisms of categories of free algebras of varieties. Electronic Research Announcements, 2002, 8: 1-10.

[17]

María José Beltrán, José Bonet, Carmen Fernández. Classical operators on the Hörmander algebras. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 637-652. doi: 10.3934/dcds.2015.35.637

[18]

A. Giambruno and M. Zaicev. Minimal varieties of algebras of exponential growth. Electronic Research Announcements, 2000, 6: 40-44.

[19]

Peigen Cao, Fang Li, Siyang Liu, Jie Pan. A conjecture on cluster automorphisms of cluster algebras. Electronic Research Archive, 2019, 27: 1-6. doi: 10.3934/era.2019006

[20]

Bernd Ammann, Robert Lauter and Victor Nistor. Algebras of pseudodifferential operators on complete manifolds. Electronic Research Announcements, 2003, 9: 80-87.

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (95)
  • HTML views (0)
  • Cited by (1)

[Back to Top]