Baric algebra

As Baric algebra refers to a linear algebra with a nontrivial weight function ( English baric , of Greek βάρος Baros , German , heavy, overweight ' ). Baric algebras are a generalization of the genetic algebras considered in theoretical biology .

definition

A (not necessarily associative) algebra over a field is called a Baric algebra if there is a nontrivial algebra homomorphism . is called weight function, means weight of . ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle w\colon A\longrightarrow K}$ ${\displaystyle w}$ ${\displaystyle w(x)}$ ${\displaystyle x\in A}$

The term Baric Algebra was introduced by IMH Etherington in 1939 when studying genetic algebra. From presentation theoretical point of view, Baric algebra is an algebra with a nontrivial representation over their Skalarkörper. Non-associative algebras generally have no matrix representation at all, the simplest form of which is a representation over the scalar body.

Characterizations

• A non-associative algebra is a Baric algebra if and only if there is an ideal such that ${\displaystyle k}$ ${\displaystyle A}$ ${\displaystyle I\subset A}$ ${\displaystyle A/I\cong k}$
• A non-associative -dimensional algebra is exactly then a Baric algebra, if they have a genetic basis has, that is, between the base members , a relationship with coefficients for which is true: . ${\displaystyle n}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle A}$ ${\displaystyle u_{1},u_{2},\dots ,u_{k}}$ ${\displaystyle u_{i}\cdot u_{j}=\sum _{k=1}^{n}\gamma _{ijk}u_{k}}$ ${\displaystyle \gamma _{ijk}\in \mathbb {R} }$ ${\displaystyle \sum _{k=1}^{n}\gamma _{ijk}=1,\;\gamma _{ijk}\geq 0}$
• A non-associative dimensional algebra is exactly then Baric algebra if there is a -dimensional Ideal is, applies to: . ${\displaystyle n}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle A}$ ${\displaystyle (n-1)}$ ${\displaystyle N\subset A}$ ${\displaystyle A^{\rm {2}}\nsubseteq N}$

Examples

• ${\displaystyle \mathbb {R} ^{3}}$with the vector product as multiplication forms a non-associative algebra. This is not a Baric algebra, because there is no ideal of dimension 2 in it, but one that would be needed so that the quotient would be too isomorphic. More generally it can be shown that semi-simple Lie algebras are not Baric algebras. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$
• ${\displaystyle \mathbb {R} ^{2}}$with two basis vectors on which a multiplication is explained as follows: ${\displaystyle e_{1},e_{2}}$

${\displaystyle e_{1}\cdot e_{1}=e_{2},\;e_{1}\cdot e_{2}=e_{1},\;e_{2}\cdot e_{1}=e_{2},\;e_{2}\cdot e_{2}=e_{1}}$.

This gives a genetic basis and defines a Baric algebra; the multiplication is not associative:

${\displaystyle (e_{1}\cdot e_{2})\cdot e_{2}\neq e_{1}\cdot (e_{2}\cdot e_{2})}$.

A non-trivial weight function is . ${\displaystyle w(\alpha _{1}e_{1}+\alpha _{2}e_{2})=\alpha _{1}+\alpha _{2}}$

• Gametic algebra G of simple Mendelian inheritance:

${\displaystyle \mathbb {R} ^{2}}$with two basis vectors and the following multiplication table: ${\displaystyle a_{1},a_{2}}$

.	${\displaystyle a_{1}}$	${\displaystyle a_{2}}$
${\displaystyle a_{1}}$	${\displaystyle a_{1}}$	${\displaystyle {1 \over 2}a_{1}+{1 \over 2}a_{2}}$
${\displaystyle a_{2}}$	${\displaystyle {1 \over 2}a_{1}+{1 \over 2}a_{2}}$	${\displaystyle a_{2}}$

is a Baric algebra with a weight function . ${\displaystyle w(\alpha _{1}a_{1}+\alpha _{2}a_{2})=\alpha _{1}+\alpha _{2}}$

literature

• Rudolf Lidl, Johann Wiesenbauer: Ring theory and applications: Fundamentals and application examples in coding theory and in genetics . Akademische Verlagsgesellschaft, Wiesbaden 1980, ISBN 3-400-00371-9
• Angelika Wörz-Busekros: Algebras in Genetics . Springer-Verlag, Berlin / Heidelberg / New York 1980, ISBN 3-540-09978-6 .
• IMH Etherington: Genetic Algebras . In: Proc. Roy. Soc. Edinburgh , 59, 1939, pp. 242-258