Citations From References: 1 From Reviews: 0 Previous Up Next MR2362673 (2008i:17042) 17D92 17A05 17D05 Mallol, Cristi´an (RCH-FRN-EM); Varro, Richard (F-MONT3-MI) Train alg`ebresalternatives: relation entre nilindice et degr´e.(French. English summary) [Alternative train algebras: relation between nilindex and degree] Comm. Algebra 35 (2007), no. 11, 3603{3610. In this paper train algebras are considered. These algebras appear in a biological context, and so they are baric algebras. A baric algebra A over a field K is a K-algebra with a weight homomorphism !: A ! K, that is, a nonzero algebra homomorphism. In general, algebras appearing in genetics are non-associative, but commutative. A train algebra of degree n + 1 is a baric algebra (A; !) satisfying n n+1 X n−k+1 k x = αk!(x) x ; k=1 for every x 2 A. k k+1 k Here, x denotes the left principal power of x; that is, x = Lxx, where Lx: A ! A denotes the usual left multiplication. In a previous paper, the first author studied associative train algebras, proving that the only train identities of degree 3 are x3 = !(x)x2 and x3 = 2!(x)x2 − !(x)2x and the ones of degree 4 are x4 = !(x)x3, x4 = 2!(x)x3 − !(x)2x2, x4 = 3!(x)x3 − 3!(x)2x2 + !(x)3x. In the present paper the associativity assumption is removed. Instead, the authors consider power-associativity (an algebra is power associative if the subalgebra generated by one element is associative). So the main result of the paper proves that, assuming char K 6= 0, the only polynomials associated to power-associative train algebras are of the type xp(x − 1)n−p, with 0 ≤ p ≤ n − 1. The existence of a nonzero idempotent element is also proved. If the algebra considered is alternative (recall that an alternative algebra can be characterized as an algebra in which every two elements generate an associative algebra), then using the previously proved existence of a nonzero idempotent and the associated Peirce decomposition, A = A00 + A01 + A10 + A11, the following result can be proved: p n−p Theorem. If A is a train algebra of polynomial x (x − 1) , then U00 is nil of index ≤ p + 1 and U11 is nil of nilindex ≤ n − p. And, conversely, if at least one among the subspaces Uij is zero, U00 and U11 are nil of nilindex r; s respectively, then A is a train algebra of degree r + s. Consuelo Mart´ınez References 1. Costa, R. (1990). Principal train algebras of rank 3 and dimension ≤ 5. Proc. Roy. Soc. Edinburgh 33:61{70. MR1038765 (90m:17043) 2. Costa, R. (1991). On train algebras of rank 3. Lin. Alg. Appl. 148:1{12. MR1090748 (92a:17054) 3. Etherington, I. M. H. (1941). Non-associative algebra and the symbolism of genetics. Proc. Roy. Soc. Edinburgh. B 61:24{32. MR0003557 (2,237e) 4. Gonshor, H. (1971). Contributions to Genetic algebras. Proc. Edinburgh Math. Soc. 17:289{298. MR0302218 (46 #1371) 5. Gonshor, H. (1973). Contributions to Genetic algebras II. Proc. Edinburgh Math. Soc. 18:273{279. MR0325173 (48 #3522) 6. Gonz´alez,S., Guti´errez,J. C., Mart´ınez,C. (1996). On regular Bernstein algebras. Linear Algebra Appl 241/243:389{400. MR1400450 (97d:17019) 7. Gonz´alez,S., L´opez-D´ıaz,C., Mart´ınez,C. (1999). Bernstein superalgebras of low dimension. Comm. Algebra 27(9):4477{4492. MR1705881 (2000i:17050) 8. Guti´errezFern´andez,J. C. (2000). Principal and plenary train algebras. Comm. Algebra 28:653{667. MR1736753 (2000m:17036) 9. Guzzo, Jr. H. (1994). The Peirce decomposition for commutative train algebras. Comm. Algebra 22:5745{5757. MR1298748 (95h:17042) 10. Guzzo, Jr. H., Vicente, P. (1997). Train algebras of rank n which are Bernstein and Power-associative algebras. Nova. J. Math. Game Theory and Algebra 6(2{3):103{ 112. MR1457227 (98g:17033) 11. L´opez-S´anchez, J., Rodr´ıguez,E. (1996). On train algebras of rank 4. Comm. Algebra 24:439{445. MR1421199 (97i:17023) 12. Lyubich, Ju. I. (1992). Mathematical Structures in Population Genetics. Berlin- Heidelberg: Springer-Verlag. MR1224676 (95f:92018) 13. Mallol, C. (2002). Sur quelques train alg`ebresassociatives non commutatives. Alg. Groups Geom. 19(1):395{408. MR1949203 (2003i:17032) 14. Mallol, C., Varro, R. (2002). Alg`ebresde mutation et train alg´ebres. East-West J. Math. 4(1):77{85. MR1995940 (2004m:17051) 15. Mallol, C., Varro, R., Benavides, R. (2003). Gam´etisationd alg`ebrespond´er´ees. Journal of Algebra 261:1{18. MR1967153 (2004a:17038) 16. Lynn Reed, M. (1997). Algebras structures of genetic inheritance. Bull. Amer. Math. Soc. (New Series) 34(2):107{131. MR1414973 (98e:17043) 17. Shafer, R. D. (1966). An Introduction to Nonassociatives Algebras. New York: Academic Press. MR0210757 (35 #1643) 18. W¨orz-Busekros,A. (1980). Algebras in Genetics. Lecture Notes in Biomathematics, Vol. 36. New York: Springer-Verlag. MR0599179 (82e:92033) Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors. c Copyright American Mathematical Society 2008, 2015.