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On baric algebras with prescribed automorphisms. (English) Zbl 0586.17014
The automorphism group of the gametic algebra for a $$2m$$-ploid population with $$n+1$$ alleles is isomorphic to $$A(n)$$ the affine group of $$\mathbb R^n$$. This result is obtained as a special case from a study of automorphisms of slightly more general algebras. The author defines a certain class of algebras called $$\alpha$$-algebras. Roughly speaking, an $$\alpha$$-algebra is a baric algebra in which certain specified linear operators are automorphisms. The definition of these operators is too technical to be given here.
The author classifies the $$\alpha$$-algebras for $$m\leq 5$$. In a later section he has some results also in the case where $$m>5$$. Finally, he claims to have results dealing with derivations of $$\alpha$$-algebras which he says will be published elsewhere.

##### MSC:
 17D92 Genetic algebras
Full Text:
##### References:
 [1] Etherington, I.M.H., Genetic algebras, Proc. roy. soc. Edinburgh, 59, 242-258, (1939) · JFM 66.1209.01 [2] Etherington, I.M.H.; Etherington, I.M.H., Commutative train algebras of ranks 2 and 3, J. London math. soc., J. London math. soc., 20, 238-149, (1945) · Zbl 0063.01291 [3] Etherington, I.M.H., Special train algebras, Quart. J. math. Oxford, 12, 1-8, (1941) · JFM 67.0093.04 [4] Schafer, R.D., Structure of genetic algebras, Amer. J. math., 71, 121-135, (1949) · Zbl 0034.02004 [5] Etherington, I.M.H., Non-commutative train algebras of rank 2 and 3, Proc. London math. soc., 52, 2, 241-252, (1951) · Zbl 0042.03401 [6] Gonshor, H., Special train algebras arising in genetics, Proc. Edinburgh math. soc., 12, 2, 41-53, (1960) · Zbl 0249.17003 [7] Gonshor, H., Special train algebras arising in genetics II, Proc. Edinburgh math. soc., 14, 2, 333-338, (1965) · Zbl 0139.03102 [8] Holgate, P.; Holgate, P., Genetic algebras associated with polyploidy, Proc. Edinburgh math. soc., Proc. Edinburgh math. soc., 17, 2, 120-9, (1970) · Zbl 0144.27202 [9] Gonshor, H., Contributions to genetic algebras, Proc. Edinburgh math. soc., 17, 2, 289-298, (1971) · Zbl 0247.92002 [10] Gonshor, H., Contributions to genetic algebras II, Proc. Edinburgh math. soc., 18, 2, 273-279, (1973) · Zbl 0272.92012 [11] Heuch, I., Genetic algebras considered as elements in a vector space, SIAM J. appl. math., 35, 695-703, (1978) · Zbl 0392.17013 [12] Wörz-Busekros, A., Algebras in genetics, Lecture notes in biomathematics, (1980), Springer New York, No. 36 · Zbl 0431.92017 [13] Costa, R., On the derivations of gametic algebras for polyploidy with multiple alleles, Bol. soc. brasil. mat., 13, 2, 69-81, (1982) · Zbl 0575.17013
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