## On the train algebras of degree four: structures and classifications. (Sur les train algèbres de degré quatre: structures et classifications.)(English)Zbl 1193.17017

The paper deals with train algebras of degree 4. A baric algebra $$(A,\omega)$$ is a train algebra of degree 4 if its elements satisfy an identity of the form $x^4=\sum_{k=1}^{3}\alpha_k \omega(x)^{3-k}x^k.$ The study of train algebras of degree 4 can be reduced, as proved in C. Mallol and R. Varro [J. Algebra 261, 1–18 (1985; Zbl 1134.17312)], to the study of those whose polynomial identity is of the form $x^4=\varepsilon x^3+\delta x^2+(1-\varepsilon-\delta)x$ with $$\varepsilon\in\{0,2\}$$.
The paper is mainly concerned with train algebras of degree 4 such that $$\varepsilon\in\{0,2\}$$ and $$\delta\neq -\frac54,\;\frac34,\;\frac74$$ (Type 1) and such that $$\varepsilon=0$$ and $$\delta=\frac34$$ (Type 2). The main interest of these two types is that they have idempotents and, consequently, we have the Peirce decomposition. The existence of this decomposition leads to structure theorems for these two types of algebras and also, for algebras of dimension $$\leq 4$$, to a complete classification up to isomorphism.

### MSC:

 17D92 Genetic algebras

Zbl 1134.17312
Full Text:

### References:

  Etherington I. M. H., Proc. Roy. Soc. Edinburgh 59 pp 242– (1939) · Zbl 0027.29402  DOI: 10.1112/jlms/s1-15.2.136 · Zbl 0027.15502  Etherington I. M. H., Proc. Roy. Soc. Edinburgh B 61 pp 24– (1941)  DOI: 10.1080/00927870008826850 · Zbl 0962.17022  DOI: 10.1080/00927879408825160 · Zbl 0810.17025  Jacobson N., Lie Algebras (1962)  Lopez-Sanchez J., Comm. Alg. 24 pp 439– (1996)  DOI: 10.1090/S0273-0979-97-00712-X · Zbl 0876.17040  Lyubich Ju. I., Mathematical Structures in Population Genetics (1992)  Mallol C., East-West J. of Math. 4 pp 77– (2002)  DOI: 10.1016/S0021-8693(02)00681-6 · Zbl 1134.17312  DOI: 10.1080/00927870500261454 · Zbl 1094.17001  DOI: 10.1112/jlms/s2-28.1.17 · Zbl 0515.17010  Worz-Busekros A., Algebras in Genetics 36 (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.