Difference between revisions of "Genetic algebra"
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439701.png" /> be a non-associative, commutative algebra of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439702.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439703.png" />. | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439701.png" /> be a non-associative, commutative algebra of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439702.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439703.png" />. | ||
− | Let the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439704.png" /> be an algebraic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439705.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439706.png" /> be the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439707.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439708.png" /> (cf. also [[Extension of a field|Extension of a field]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439709.png" /> admit a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397011.png" />, with | + | Let the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439704.png" /> be an algebraic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439705.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439706.png" /> be the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439707.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439708.png" /> (cf. also [[Extension of a field|Extension of a field]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g0439709.png" /> admit a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397011.png" />, with [[structure constant]]s <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397012.png" />, defined by |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397013.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397013.png" /></td> </tr></table> | ||
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397019.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397021.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397022.png" />. | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397019.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397021.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397022.png" />. | ||
− | Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397023.png" /> is called a genetic algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397024.png" /> is called a canonical basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397025.png" />. The multiplication constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397027.png" />, are invariants of a genetic algebra; they are called the train roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397028.png" />. | + | Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397023.png" /> is called a genetic algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397024.png" /> is called a canonical basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397025.png" />. The multiplication constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397027.png" />, are invariants of a genetic algebra; they are called the ''train roots'' of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397028.png" />. |
An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397029.png" /> is called a [[baric algebra]] if there exists a non-trivial algebra homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397030.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397031.png" /> is called a weight homomorphism or simply a weight. Every genetic algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397032.png" /> is baric with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397033.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397036.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397037.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397038.png" />-dimensional ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397039.png" />. | An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397029.png" /> is called a [[baric algebra]] if there exists a non-trivial algebra homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397030.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397031.png" /> is called a weight homomorphism or simply a weight. Every genetic algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397032.png" /> is baric with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397033.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397036.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397037.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397038.png" />-dimensional ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043970/g04397039.png" />. |
Revision as of 18:22, 12 April 2020
Let be a non-associative, commutative algebra of dimension
over a field
.
Let the field be an algebraic extension of
, and let
be the extension of
over
(cf. also Extension of a field). Let
admit a basis
,
, with structure constants
, defined by
![]() |
which have the following properties:
,
for
,
;
,
for
,
;
.
Then is called a genetic algebra and
is called a canonical basis of
. The multiplication constants
,
, are invariants of a genetic algebra; they are called the train roots of
.
An algebra is called a baric algebra if there exists a non-trivial algebra homomorphism
;
is called a weight homomorphism or simply a weight. Every genetic algebra
is baric with
defined by
,
,
; and
is an
-dimensional ideal of
.
Let be the transformation algebra of the algebra
, i.e. the algebra generated by the (say) left transformations
,
,
, and the identity.
A non-associative, commutative algebra is a genetic algebra if and only if for every
,
, the coefficients of the characteristic polynomial are functions of
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 the weight
is uniquely determined if
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 -ploid) individuals which differ genetically at one or several loci. Let
be the genetically different gametes. The state of the population can be described by the vector
of frequencies of gametes,
![]() |
By random union of gametes and
, zygotes
are formed,
. In the absence of selection all zygotes have the same fertility. Let
be the relative frequency of gametes
,
, produced by a zygote
,
,
![]() | (a1) |
Let the segregation rates be symmetric, i.e.
![]() | (a2) |
Consider the elements as abstract elements which are free over the field
. In the vector space
a multiplication is defined by
![]() |
and its bilinear extension onto . Thereby
becomes a commutative algebra
, the gametic algebra. Actual populations correspond to elements
with
,
, and
. Random union of populations corresponds to multiplication of the corresponding elements in the algebra
. 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 is obtained from the gametic algebra
by duplication, i.e. as the symmetric tensor product of
with itself:
![]() | (a3) |
where
![]() |
The zygotic algebra describes the evolution of a population of diploid (-ploid) individuals under random mating.
A baric algebra with weight
is called a train algebra if the coefficients of the rank polynomial of all principal powers of
depend only on
, i.e. if this polynomial has the form
![]() | (a4) |
A baric algebra with weight
is called a special train algebra if
is nilpotent and the principal powers
,
, are ideals of
, 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 be a baric algebra with weight
. If all elements
of
satisfy the identity
![]() |
then is called a Bernstein algebra. Every Bernstein algebra possesses an idempotent
. The decomposition with respect to this idempotent reads
![]() |
where
![]() |
The integers and
are invariants of
, the pair
is called the type of the Bernstein algebra
, 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) |
Genetic algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genetic_algebra&oldid=41883