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.

References

[a1]	S. Bernstein, "Principe de stationarité et généralisation de la loi de Mendel" C.R. Acad. Sci. Paris , 177 (1923) pp. 581–584
[a2]	I.M.H. Etherington, "Genetic algebras" Proc. R. Soc. Edinburgh , 59 (1939) pp. 242–258
[a3]	H. Gonshor, "Contributions to genetic algebras" Proc. Edinburgh Math. Soc. (2) , 17 (1971) pp. 289–298
[a4]	P. Holgate, "Characterizations of genetic algebras" J. London Math. Soc. (2) , 6 (1972) pp. 169–174
[a5]	R.D. Schafer, "Structure of genetic algebras" Amer. J. Math. , 71 (1949) pp. 121–135
[a6]	S. Walcher, "Bernstein algebras which are Jordan algebras" Arch. Math. , 50 (1988) pp. 218–222
[a7]	A. Wörz-Busekros, "Algebras in genetics" , Lect. notes in biomath. , 36 , Springer (1980)