View source for Baric algebra

''weighted algebra''

In 1939, in connection with the formalism of genetics, I.M.H. Etherington introduced the notion of baric algebra (cf. [[#References|[a1]]]; more commonly it is also called a weighted algebra). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b1101401.png" /> is a (commutative) [[Field|field]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b1101402.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b1101403.png" />-algebra, not necessarily commutative or associative, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b1101404.png" /> is a weighted algebra if there exists an algebra [[Morphism|morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b1101405.png" /> which is non-trivial. This means that one can write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b1101406.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b1101407.png" /> is a convenient element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b1101408.png" />. The morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b1101409.png" /> is called the weight function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014010.png" /> and one can regard a weighted algebra as a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014011.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014012.png" /> is an algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014013.png" /> the weight function. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014014.png" /> of weighted algebras is a morphism of algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014016.png" />. This gives a [[Category|category]], the category of weighted algebras. All constructions on weighted algebras are made in this category. For a finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014017.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014018.png" />, the following conditions are equivalent:

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 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" />.

It is easy to see that a [[Lie algebra|Lie algebra]] is not weighted; however, over any weighted algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014034.png" /> one can define a Lie algebra structure via the multiplication (bracket) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014035.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014036.png" />. A [[Jordan algebra|Jordan algebra]] may or may not be weighted; however, over any weighted algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014037.png" /> one can define a Jordan algebra structure via the multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014038.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014039.png" />. A weighted algebra is not necessarily associative (cf. [[Associative rings and algebras|Associative rings and algebras]]); however, over any weighted algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014040.png" /> with a unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014041.png" /> one can define an associative algebra structure via the multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014043.png" />. A [[Clifford algebra|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|Bernstein algebra]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M.H. Etherington,   "Genetic algebras"  ''Proc. Roy. Soc. Edinburgh'' , '''59'''  (1939)  pp. 242–258</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. McHale,   G.A. Ringwood,   "Haldane linearisation of baric algebras"  ''J. London Math. Soc. (2)'' , '''28'''  (1983)  pp. 17–26</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Micali,   Ph. Revoy,   "Sur les algèbres gamétiques"  ''Proc. Edinburgh Math. Soc'' , '''29'''  (1986)  pp. 187–197</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.K. Singh,   D.K. Singh,   "On baric algebras"  ''The Math. Education (2)'' , '''20'''  (1986)  pp. 54–55</TD></TR></table>