Difference between revisions of "Baric algebra"
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i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014019.png" /> is a weighted algebra; | i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014019.png" /> is a weighted algebra; | ||
| − | ii) there exists a finite basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014021.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014022.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014023.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014024.png" />) is the multiplication table of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014025.png" /> in this basis (the scalars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014026.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014027.png" />) are the structure | + | ii) there exists a finite basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014021.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014022.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014023.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014024.png" />) is the multiplication table of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014025.png" /> in this basis (the scalars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014026.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014027.png" />) are the [[structure constant]]s of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014028.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014029.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014030.png" />); |
iii) there exists a two-sided ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014032.png" />, of codimension one, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014033.png" />. | iii) there exists a two-sided ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014032.png" />, of codimension one, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014033.png" />. | ||
Latest revision as of 09:47, 11 April 2020
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 |
Baric algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baric_algebra&oldid=45325