Namespaces
Variants
Actions

Baric algebra

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

weighted algebra

In 1939, in connection with the formalism of genetics, I.M.H. Etherington introduced the notion of baric algebra (cf. [a1]; more commonly it is also called a weighted algebra). If is a (commutative) field and a -algebra, not necessarily commutative or associative, one says that is a weighted algebra if there exists an algebra morphism which is non-trivial. This means that one can write , where is a convenient element of . The morphism is called the weight function of and one can regard a weighted algebra as a pair where is an algebra and the weight function. A morphism of weighted algebras is a morphism of algebras such that . This gives a category, the category of weighted algebras. All constructions on weighted algebras are made in this category. For a finite-dimensional -algebra , the following conditions are equivalent:

i) is a weighted algebra;

ii) there exists a finite basis of over such that if () is the multiplication table of in this basis (the scalars () are the structure constants of ), then ();

iii) there exists a two-sided ideal of , of codimension one, such that .

It is easy to see that a Lie algebra is not weighted; however, over any weighted algebra one can define a Lie algebra structure via the multiplication (bracket) for all . A Jordan algebra may or may not be weighted; however, over any weighted algebra one can define a Jordan algebra structure via the multiplication for all . A weighted algebra is not necessarily associative (cf. Associative rings and algebras); however, over any weighted algebra with a unit one can define an associative algebra structure via the multiplication for all . A Clifford algebra is not weighted; in particular, the algebra of complex numbers and the algebra of quaternions are not weighted. All Bernstein algebras are weighted (cf. Bernstein algebra).

References

[a1] I.M.H. Etherington, "Genetic algebras" Proc. Roy. Soc. Edinburgh , 59 (1939) pp. 242–258
[a2] D. McHale, G.A. Ringwood, "Haldane linearisation of baric algebras" J. London Math. Soc. (2) , 28 (1983) pp. 17–26
[a3] A. Micali, Ph. Revoy, "Sur les algèbres gamétiques" Proc. Edinburgh Math. Soc , 29 (1986) pp. 187–197
[a4] M.K. Singh, D.K. Singh, "On baric algebras" The Math. Education (2) , 20 (1986) pp. 54–55
How to Cite This Entry:
Baric algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baric_algebra&oldid=45325
This article was adapted from an original article by A. Micali (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article