# Bernstein algebra

Around 1900, S.N. Bernstein (cf. [a3], [a4], [a5]) worked on an important problem concerning the laws of formal genetics. This problem is known today as the Bernstein problem. Following Yu.I. Lyubich (cf. [a10]), this problem can be expressed as follows. The state of a population in a given generation is described by a vector in $\mathbf R ^ {n}$ whose coordinates satisfy $x _ {i} \geq 0$( $i = 1 \dots n$) and $\sum _ {i = 1 } ^ {n} x _ {i} = 1$. The set $S$ of all states is a simplex in $\mathbf R ^ {n}$ and the vertices $e _ {i}$( $i = 1 \dots n$) of $S$ are the different types of individuals in the population. If $\gamma _ {ijk }$ is the probability that an individual $e _ {k}$ appears in the next generation from parents of types $e _ {i}$ and $e _ {j}$, then $\sum _ {k = 1 } ^ {n} \gamma _ {ijk } = 1$( $i,j = 1 \dots n$) and $\gamma _ {ijk } = \gamma _ {jik }$( $i,j,k = 1 \dots n$). In absence of selection and under random hypothesis, the state of the population in the next generation can be written, in terms of coordinates, as $x _ {k} ^ \prime = \sum _ {i,j = 1 } ^ {n} \gamma _ {ijk } x _ {i} x _ {j}$( $k = 1 \dots n$). These relations define a quadratic operator $V : S \rightarrow S$ called the evolutionary quadratic operator. The Bernstein stationarity principle says that $V ^ {2} = V$ and the Bernstein problem aims at describing all quadratic operators satisfying this principle. Bernstein solved his problem for $n = 3$ and much progress was achieved recently (cf. [a6], [a8]) in this direction. The Bernstein problem can be translated in terms of algebra structure. In fact, over $\mathbf R ^ {n}$ an algebra structure can be defined via the operator $V$ by
$$xy = { \frac{1}{2} } ( V ( x + y ) - V ( x ) - V ( y ) )$$
for all $x, y \in \mathbf R ^ {n}$, and if $\omega : {\mathbf R ^ {n} } \rightarrow \mathbf R$ is the mapping defined by $x = ( x _ {1} \dots x _ {n} ) \mapsto \sum _ {i = 1 } ^ {n} x _ {i}$, then $V ^ {2} = V$ if and only if $( x ^ {2} ) ^ {2} = \omega ( x ) ^ {2} x ^ {2}$ for all $x \in \mathbf R ^ {n}$. Moreover, $\omega ( xy ) = \omega ( x ) \omega ( y )$ for all $x, y \in \mathbf R ^ {n}$. Of course, to define this multiplication over the whole space $\mathbf R ^ {n}$ starting from the simplex $S$, one has to make convenient extensions of this multiplication by bilinearity. Now, in general, if $K$ is a (commutative) field and $A$ is a commutative $K$- algebra, then a weighted algebra $( A, \omega )$ over $K$ is said to be a Bernstein algebra if $( x ^ {2} ) ^ {2} = \omega ( x ) ^ {2} x ^ {2}$ for all $x \in A$( cf. [a2]). In recent years (1990s), the theory of Bernstein algebras has been substantially improved. V.M. Abraham (cf. [a1]) suggests the construction of a generalized Bernstein algebra. In this perspective, for an element $x \in A$, where $( A, \omega )$ is a weighted algebra, the plenary powers $x ^ {[ m ] }$ of $x$ are defined by $x ^ {[ 1 ] } = x$ and $x ^ {[ m -1 ] } x ^ {[ m -1 ] } = x ^ {[ m ] }$ for all integer $m \geq 2$. The plenary powers can be interpreted by saying that they represent random mating between discrete non-overlapping generations. $( A, \omega )$ is called an $n$ th order Bernstein algebra if $x ^ {[ n + 2 ] } = \omega ( x ) ^ {2 ^ {n} } x ^ {[ n + 1 ] }$ for all $x \in A$, where $n \geq 1$ is the smallest such integer (cf. [a11]). Second-order Bernstein algebras are simply called Bernstein algebras and first-order Bernstein algebras are also called gametic diploid algebras. The interpretation of the equation $x ^ {[ n + 2 ] } = x ^ {[ n + 1 ] }$( $x \in A$ such that $\omega ( x ) = 1$) is that equilibrium in the population is reached after exactly $n$ generations of intermixing. For genetic properties of Bernstein algebras, see [a7] and [a12].