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 multiplication 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
baric
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
,
,
Let the
segregation rates
be symmetric, i.e.
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:
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
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.