# Genetic algebra

Let $A$ be a non-associative, commutative algebra of dimension $n + 1$ over a field $K$.

Let the field $L$ be an algebraic extension of $K$, and let $A _ {L}$ be the extension of $A$ over $L$( cf. also Extension of a field). Let $A _ {L}$ admit a basis $\{ c _ {0} , c _ {1} \dots c _ {n} \}$, $c _ {0} \in A$, with structure constants $\lambda _ {ijk}$, defined by

$$c _ {i} c _ {j} = \ \sum _ {k = 0 } ^ { n } \lambda _ {ijk} c _ {k} ,\ \ i, j = 0 \dots n,$$

which have the following properties:

$\lambda _ {000} = 1$,

$\lambda _ {0jk} = 0$ for $k < j$, $j = 1 \dots n$; $k = 0 \dots n$,

$\lambda _ {ijk} = 0$ for $k \leq \max \{ i, j \}$, $i, j = 1 \dots n$; $k = 0 \dots n$.

Then $A$ is called a genetic algebra and $\{ c _ {0} \dots c _ {n} \}$ is called a canonical basis of $A$. The multiplication constants $\lambda _ {0ii}$, $i = 0 \dots n$, are invariants of a genetic algebra; they are called the train roots of $A$.

An algebra $A$ is called a baric algebra if there exists a non-trivial algebra homomorphism $\omega : A \rightarrow K$; $\omega$ is called a weight homomorphism or simply a weight. Every genetic algebra $A$ is baric with $\omega : A \rightarrow K$ defined by $\omega ( c _ {0} ) = 1$, $\omega ( c _ {i} ) = 0$, $i = 1 \dots n$; and $\mathop{\rm ker} \omega$ is an $n$- dimensional ideal of $A$.

Let $T ( A)$ be the transformation algebra of the algebra $A$, i.e. the algebra generated by the (say) left transformations $L _ {a} : A \rightarrow A$, $x \mapsto ax$, $a \in A$, and the identity.

A non-associative, commutative algebra $A$ is a genetic algebra if and only if for every $T \in T ( A)$, $T = f ( L _ {a _ {1} } \dots L _ {a _ {k} } )$, the coefficients of the characteristic polynomial are functions of $\omega ( a _ {1} ) \dots \omega ( a _ {k} )$ only.

Historically, genetic algebras were first defined by this property (R.D. Schafer [a5]). H. Gonshor [a3] proved the equivalence with the first definition above. P. Holgate [a4] proved that in a baric algebra $A$ the weight $\omega$ is uniquely determined if $\mathop{\rm ker} \omega$ is a nil ideal.

Algebras in genetics originate from the work of I.M.H. Etherington [a2], who put the Mendelian laws into an algebraic form. Consider an infinitely large, random mating population of diploid (or $2r$- ploid) individuals which differ genetically at one or several loci. Let $a _ {0} \dots a _ {n}$ be the genetically different gametes. The state of the population can be described by the vector $( \alpha _ {0} \dots \alpha _ {n} )$ of frequencies of gametes,

$$0 \leq \alpha _ {i} \leq 1,\ \ i = 0 \dots n; \ \ \sum _ {i = 0 } ^ { n } \alpha _ {i} = 1.$$

By random union of gametes $a _ {i}$ and $a _ {j}$, zygotes $a _ {i} a _ {j}$ are formed, $i, j = 0 \dots n$. In the absence of selection all zygotes have the same fertility. Let $\gamma _ {ijk}$ be the relative frequency of gametes $a _ {k}$, $k = 0 \dots n$, produced by a zygote $a _ {i} a _ {j}$, $i, j = 0 \dots n$,

$$\tag{a1 } \left . \begin{array}{c} 0 \leq \gamma _ {ijk} \leq 1,\ \ i, j, k = 0 \dots n; \\ \sum _ {k = 0 } ^ { n } \gamma _ {ijk} = 1,\ \ i, j = 0 \dots n. \end{array} \right \}$$

Let the segregation rates $\gamma _ {ijk}$ be symmetric, i.e.

$$\tag{a2 } \gamma _ {ijk} = \gamma _ {jik} ,\ \ i, j, k = 0 \dots n.$$

Consider the elements $a _ {0} \dots a _ {n}$ as abstract elements which are free over the field $\mathbf R$. In the vector space $V = \{ {\sum _ {i = 0 } ^ {n} \alpha _ {i} a _ {i} } : {\alpha _ {i} \in \mathbf R , i = 0 \dots n } \}$ a multiplication is defined by

$$a _ {i} a _ {j} = \ \sum _ {k = 0 } ^ { n } \gamma _ {ijk} a _ {k} ,\ \ i, j = 0 \dots n,$$

and its bilinear extension onto $V \times V$. Thereby $V$ becomes a commutative algebra $G$, the gametic algebra. Actual populations correspond to elements $a = \sum _ {i = 0 } ^ {n} \alpha _ {i} a _ {i} \in G$ with $0 \leq \alpha _ {i} \leq 1$, $i = 0 \dots n$, and $\sum _ {i = 0 } ^ {n} \alpha _ {i} = 1$. Random union of populations corresponds to multiplication of the corresponding elements in the algebra $G$. Under rather general assumptions (including mutation, crossing over, polyploidy) gametic algebras are genetic algebras. Examples can be found in [a2] or [a7].

The zygotic algebra $Z$ is obtained from the gametic algebra $G$ by duplication, i.e. as the symmetric tensor product of $G$ with itself:

$$\tag{a3 } Z = G \otimes G/J,$$

where

$$J : = \left \{ { \sum _ {i \in I } ( x _ {i} \otimes y _ {i} - y _ {i} \otimes x _ {i} ) } : { x _ {i} , y _ {i} \in G,\ i \in I,\ | I | < \infty } \right \} .$$

The zygotic algebra describes the evolution of a population of diploid ( $2r$- ploid) individuals under random mating.

A baric algebra $A$ with weight $\omega$ is called a train algebra if the coefficients of the rank polynomial of all principal powers of $x$ depend only on $\omega ( x)$, i.e. if this polynomial has the form

$$\tag{a4 } x ^ {r} + \beta _ {1} \omega ( x) x ^ {r - 1 } + \dots + \beta _ {r - 1 } \omega ^ {r - 1 } ( x) x = 0.$$

A baric algebra $A$ with weight $\omega$ is called a special train algebra if $N = \mathop{\rm ker} \omega$ is nilpotent and the principal powers $N ^ {i}$, $i \in N$, are ideals of $A$, cf. [a2]. Etherington [a2] proved that every special train algebra is a train algebra. Schafer [a5] showed that every special train algebra is a genetic algebra and that every genetic algebra is a train algebra. Further characterizations of these algebras can be found in [a7], Chapts. 3, 4.

Let $A$ be a baric algebra with weight $\omega$. If all elements $x$ of $A$ satisfy the identity

$$x ^ {2} x ^ {2} = \ \omega ^ {2} ( x) x ^ {2} ,$$

then $A$ is called a Bernstein algebra. Every Bernstein algebra possesses an idempotent $e$. The decomposition with respect to this idempotent reads

$$A = E \oplus U \oplus V,$$

where

$$E = e \cdot K,\ \ \left . U = \mathop{\rm Im} L _ {e} \right | _ {N} ,\ \ V = \mathop{\rm ker} L _ {e} .$$

The integers $p = \mathop{\rm dim} U$ and $q = \mathop{\rm dim} V$ are invariants of $A$, the pair $( p + 1 , q)$ is called the type of the Bernstein algebra $A$, cf. [a7], Chapt. 9. In [a6] necessary and sufficient conditions have been given for a Bernstein algebra to be a Jordan algebra.

Bernstein algebras were introduced by S. Bernstein [a1] as a generalization of the Hardy–Weinberg law, which states that a randomly mating population is in equilibrium after one generation.

How to Cite This Entry:
Genetic algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genetic_algebra&oldid=47078
This article was adapted from an original article by A. WÃ¶rz-Busekros (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article