Genetic Volterra algebras and their derivations

TL;DR

This paper investigates the structure of genetic Volterra algebras, characterizing associative cases and derivations, and establishing conditions for trivial derivations in non-associative cases, especially in three dimensions.

Contribution

It provides a complete description of associative genetic Volterra algebras and characterizes derivations, including a classification of three-dimensional cases and conditions for trivial derivations.

Findings

All derivations are trivial in associative cases.

A sufficient condition for trivial derivations in non-associative cases.

All local derivations in three-dimensional cases are actual derivations.

Abstract

The present paper is devoted to genetic Volterra algebras. We first study characters of such algebras. We fully describe associative genetic Volterra algebras, in this case all derivations are trivial. In general setting, i.e. when the algebra is not associative, we provide a sufficient condition to get trivial derivation on generic Volterra algebras. Furthermore, we describe all derivations of three dimensional generic Volterra algebras, which allowed us to prove that any local derivation is a derivation of the algebra.

Equations

$$x_k^{(1)}=\sum _{i,j=1}^m{p}_{ij,k}{x}_i^{(0)}{x}_j^{(0)},k\in \{1,\dots ,m\}.$$

$$S^{m-1}=\left\{\text{x}=(x_i)\in {\mathbb{R}}^m:x_i\ge 0,\forall i\in I,\sum _{i=1}^m{x}_i=1\right\}.$$

$$V:x_k^{\prime }=\sum _{i,j=1}^m{p}_{ij,k}{x}_i{x}_j,k=1,2,\dots ,m$$

$$p_{ij,k}\ge 0,p_{ij,k}=p_{ji,k},\sum _{k=1}^m{p}_{ij,k}=1,i,j,k\in \{1,2,\dots ,m\}$$

$$p_{ij,k}=0\text{mbox}ifk\notin \{i,j\},\text{for all}i,j,k\in I.$$

$$p_{ii,i}=1\text{mbox}and{p}_{ij,i}+p_{ij,j}=1,\text{for all}i,j\in I,(i\ne j).$$

$$(\mathbf{x}\circ \mathbf{y}{)}_k=\sum _{i,j=1}^m{p}_{ij,k}{x}_i{y}_j$$

$${\mathbf{e}}_i\circ {\mathbf{e}}_j=p_{ij,i}{\mathbf{e}}_i+p_{ij,j}{\mathbf{e}}_j,$$

$$a_i=h({\mathbf{e}}_i)=h({\mathbf{e}}_i\circ {\mathbf{e}}_i)=h^2({\mathbf{e}}_i)=a_i^2,$$

$$h_E(\mathbf{x})=\sum _{i\in E}{x}_i.$$

$$h_E({\mathbf{e}}_i\circ {\mathbf{e}}_j)=h_E(p_{ij,i}{\mathbf{e}}_i+p_{ij,j}{\mathbf{e}}_j)=p_{ij,i}{h}_E({\mathbf{e}}_i)+p_{ij,j}{h}_E({\mathbf{e}}_j).$$

$$0=h_E({\mathbf{e}}_i)h_E({\mathbf{e}}_j)=h_E({\mathbf{e}}_i\circ {\mathbf{e}}_j)=p_{ij,j}$$

$$h_E({\mathbf{e}}_i\circ {\mathbf{e}}_j)=p_{ij,i}+p_{ij,j}=1=h_E({\mathbf{e}}_i)h_E({\mathbf{e}}_j).$$

$$h_E({\mathbf{e}}_i\circ {\mathbf{e}}_j)=0=h_E({\mathbf{e}}_i)h_E({\mathbf{e}}_j).$$

$$h_E({\mathbf{e}}_i\circ {\mathbf{e}}_j)=p_{ij,j}=0=h_E({\mathbf{e}}_i)h_E({\mathbf{e}}_j).$$

$$p_{jk,j}{p}_{ij,i}+p_{jk,k}{p}_{ik,i}=p_{ij,i}{p}_{ik,i}$$

$$p_{ij,i}{p}_{ik,k}+p_{ij,j}{p}_{jk,k}=p_{jk,k}{p}_{ik,k}.$$

$$p_{jk,j}=1-p_{jk,k},p_{ij,i}=1-p_{ij,j},p_{ik,i}=1-p_{ik,k}.$$

$$0$$

$${\mathbf{e}}_i\circ ({\mathbf{e}}_j\circ {\mathbf{e}}_k)=({\mathbf{e}}_i\circ {\mathbf{e}}_j)\circ {\mathbf{e}}_k,\text{for all}i,j,k\in I.$$

$${\mathbf{e}}_i\circ ({\mathbf{e}}_j\circ {\mathbf{e}}_k)$$

$${\mathbf{e}}_i\circ {\mathbf{e}}_j=({\mathbf{e}}_i\circ {\mathbf{e}}_j)\circ {\mathbf{e}}_j.$$

$$({\mathbf{e}}_i\circ {\mathbf{e}}_j)\circ {\mathbf{e}}_j$$

$$p_{ij,i}(1-p_{ij,i})=0$$

$${\mathbf{e}}_i\circ ({\mathbf{e}}_j\circ {\mathbf{e}}_k)=({\mathbf{e}}_i\circ {\mathbf{e}}_j)\circ {\mathbf{e}}_k.$$

$${\mathbf{e}}_i\circ ({\mathbf{e}}_j\circ {\mathbf{e}}_k)$$

$$({\mathbf{e}}_i\circ {\mathbf{e}}_j)\circ {\mathbf{e}}_k=p_{ij,i}(p_{ik,i}{\mathbf{e}}_i+p_{ik,k}{\mathbf{e}}_k)+p_{ij,j}(p_{jk,j}{\mathbf{e}}_j+p_{jk,k}{\mathbf{e}}_k).$$

$$p_{jk,j}{p}_{ij,i}+p_{jk,k}{p}_{ik,i}=p_{ij,i}{p}_{ik,i},$$

$$p_{ij,i}{p}_{ik,k}+p_{ij,j}{p}_{jk,k}=p_{jk,k}{p}_{ik,k}.$$

$$V({\mathbf{x}})_{k}=x_{k}\bigg{(}1+\sum_{i=1}^{m}a_{ik}x_{i}\bigg{)},\ \ k=1,\dots,m,$$

$$({\mathbf{e}}_j\circ {\mathbf{e}}_k)\circ {\mathbf{e}}_i={\mathbf{e}}_j\circ {\mathbf{e}}_i={\mathbf{e}}_j\circ ({\mathbf{e}}_k\circ {\mathbf{e}}_i),$$

Full text

1. Introduction

There exist several classes of non-associative algebras (baric, evolution, Bernstein, train, stochastic, etc.), whose investigation has provided a number of significant contributions to theoretical population genetics [21, 23]. Such classes have been defined different times by several authors, and all algebras belonging to these classes are generally called genetic. In [4] it was introduced the formal language of abstract algebra to the study of the genetics. Note that problems of population genetics can be traced back to Bernstein’s work [1, 2] where evolution operators were studied. Such kind of operators are mostly described by quadratic stochastic operators (see also [14]).

A quadratic stochastic operator is used to present the time evolution of species in biology [13, 22], which arises as follows. Consider a population consisting of $m$ species (or traits) which we denote by $I=\{1,2,\cdots,m\}$. Let $x^{(0)}=\left(x_{1}^{(0)},\cdots,x_{m}^{(0)}\right)$ be a probability distribution of species at an initial state and $p_{ij,k}$ be a probability that individuals in the $i^{th}$ and $j^{th}$ species (traits) interbreed to produce an individual from $k^{th}$ species (trait). Then a probability distribution $x^{(1)}=\left(x_{1}^{(1)},\cdots,x_{m}^{(1)}\right)$ of the spices (traits) in the first generation can be found as a total probability, i.e.,

[TABLE]

This means that the correspondence $x^{(0)}\to x^{(1)}$ defines a mapping $V$ called the evolution operator. This mapping, for a free population, is a quadratic mapping of the simplex of all probability distribution on $I$. Therefore, such an operator is also called quadratic stochastic operator (QSO). In other words, a QSO describes a distribution of the next generation if the distribution of the current generation was given. The fascinating applications of QSO to population genetics were given in [9, 14]. In [7, 16], it was given along self-contained exposition of the recent achievements and open problems in the theory of the QSO.

Note that each QSO defines an algebraic structure on the vector space $\mathbb{R}^{m}$ containing the simplex (see next section for definitions). Such an algebra is called genetic algebra. Several works are devoted (see[14, 21]) to certain properties of these algebras. We point out that the algebras that arise in genetics (via gametic, zygotic, or copular algebras) have very interesting structures. They are generally commutative but nonassociative, yet they are not necessarily Lie, Jordan, or alternative algebras. In addition, many of the algebraic properties of these structures have genetic significance. For example, a more modern use of the genetic algebra theory to self fertilization can be found in [11]. Therefore, it is the interplay between the purely mathematical structure and the corresponding genetic properties that makes this subject so fascinating. We refer to [23] for the comprehensive reference.

In population genetics, it is important to study dynamics of so-called Volterra operators [19]. The dynamics of such kind of operators have been investigated in [6]. However, genetic algebras associated to these operators were not completely studied yet. Therefore, in the present paper, we are going systematically investigate these kind of algebras (see Sections 3 and 4). On the other hand, there has recently been much work on the subject of derivations of genetic algebras (for example [8, 15, 17]). Certain interpretations of the derivations have been discussed in [12]. Moreover, in these investigations, derivations of genetic Volterra algebras were not studied. In this paper, we fully describe associative genetic Volterra algebras, in the later case, all derivations are trivial. In section 5, we consider a general setting, i.e. the algebra is not necessarily associative. In this case, we provide a sufficient condition to get a trivial derivation on generic Volterra algebra. Furthermore, we describe all derivations of three dimensional generic Volterra algebra, which allowed us to prove that any local derivation is a derivation of the algebra.

2. Preliminaries

This section is devoted to some necessary notations.

Let $I=\{1,\dots,m\}$. By $\{{\mathbf{e}}_{i}\}_{i\in I}$ we denote the standard basis in $\mathbb{R}^{m}$, i.e. ${\mathbf{e}}_{i}=(\delta_{i1},\dots,\delta_{im})$, where $\delta$ is the Kronecker’s Delta. Throughout this paper, we consider the simplex:

[TABLE]

A quadratic stochastic operator (QSO) is a mapping of the simplex $S^{m-1}$ into itself of the form

[TABLE]

where $P_{ij,k}$ are heredity coefficients, which satisfy the following conditions:

[TABLE]

A QSO $V$ defined by (2.2) is called Volterra operator [6] if one has

[TABLE]

From (2.3) and (2.4) we infer that

[TABLE]

Remark 2.1**.**

Note that it is obvious that the biological behavior of condition (2.4) is that the offspring repeats one of its parents’ genotype (see [6, 7]).

Let $V$ be a QSO and suppose that $\mathbf{x},\mathbf{y}\in\mathbb{R}^{m}$ are arbitrary vectors, we introduce a multiplication rule (see [10]) on $\mathbb{R}^{m}$ by

[TABLE]

where $\mathbf{x}=(x_{1},\ldots,x_{m}),\mathbf{y}=(y_{1},\ldots,y_{m})\in\mathbb{R}^{m}$.

The pair $(\mathbb{R}^{m},\circ)$ is called genetic algebra. We note the this algebra is commutative, i.e. $\mathbf{x}\circ\mathbf{y}=\mathbf{y}\circ\mathbf{x}$. Certain algebraic properties of such kind of algebras were investigated in [10, 14, 23]. In general, the genetic algebra is not necessarily to be associative. In [5, 20] associativity of low dimensional genetic algebras have been studied. If $V$ is a Volterra QSO, then the associated genetic algebra is called genetic Volterra algebra.

Remark 2.2**.**

Let $A$ be a Volterra algebra generated by heredity coefficients $\{p_{ij,k}\}$. Then from (2.5) and (2.6) we immediately find

• (a)

for every $i,j\in I$ ($i\neq j$) one has

[TABLE]

• (b)

${\mathbf{e}}_{i}^{2}={\mathbf{e}}_{i}$* for every $i\in I$.*

Theorem 2.3**.**

[14]** Let $A$ be an algebra over $\mathbb{R}$. If it has a genetic realization with respect to the natural basis ${{\mathbf{e}}_{i},...,{\mathbf{e}}_{m}}$, then $A$ is a (non associative) Banach algebra with respect to the norm $\|{\mathbf{x}}\|=\sum\limits_{i=1}^{m}|x_{i}|$ for ${\mathbf{x}}=\sum\limits_{i=1}^{m}x_{i}{\mathbf{e}}_{i}\in A$.

Recall that a derivation on algebra $(A,\circ)$ is a linear mapping $D:A\to A$ such that $D(u\circ v)=D(u)\circ v+u\circ D(v)$ for all $u,v\in A$. It is clear that $D\equiv 0$ is also a derivation, and such derivation is called trivial one.

3. Characters of genetic Volterra algebras

In this section, we characterize all characters of genetic Volterra algebras.

Let $A$ be a genetic Volterra algebra. We recall that a character of $A$ is a linear functional $h$ from $A$ to $\mathbb{R}$ with $h({\mathbf{x}}\circ{\mathbf{y}})=h({\mathbf{x}})h({\mathbf{y}})$ for all ${\mathbf{x}},{\mathbf{y}}\in A$.

Lemma 3.1**.**

Let $A$ be an $m$-dimensional genetic Volterra algebra. If $h({\mathbf{x}})=\sum\limits_{i=1}^{m}a_{i}x_{i}$ is a character of $A$, then $a_{i}\in\{0,1\}$.

Proof.

Let, as before, $\{{\mathbf{e}}_{i}\}$ be vertices of the simplex $S^{m-1}$. It is clear that $h({\mathbf{e}}_{i})=a_{i}$. Moreover, one can see that each vector ${\mathbf{e}}_{i}$ is an idempotent of the algebra, i.e. ${\mathbf{e}}^{2}_{i}={\mathbf{e}}_{i}$. This implies that

[TABLE]

which means $a_{i}(a_{i}-1)=0$. Hence, $a_{i}\in\{0,1\}$. ∎

The proved Lemma 3.1 implies that for every character $h$ of a genetic Volterra algebra, one can find a subset $E\subset I(=\{1,2,\dots,m\})$ such that $h=h_{E}$, where

[TABLE]

Theorem 3.2**.**

Let $A$ be a genetic Volterra algebra and $E\subset I$. Then the following conditions are equivalent:

• (i)

The functional $h_{E}$ is a character;

• (ii)

For all $i\in I\setminus E$, $j\in E$ one has $p_{ij,j}=0$.

Proof.

(i)$\Rightarrow$(ii). Assume that $h_{E}$ is a character. Due to (2.7) one finds

[TABLE]

Then for $i\in I\setminus E$, $j\in E$ from (3.1) one finds $h_{E}({\mathbf{e}}_{i})=0$ and $h_{E}({\mathbf{e}}_{j})=1$. Hence, one gets

[TABLE]

which is the required assertion.

(ii)$\Rightarrow$(i). First note that $h$ is character if and only if one has $h({\mathbf{e}}_{i})h({\mathbf{e}}_{j})=h({\mathbf{e}}_{i}\circ{\mathbf{e}}_{i})$ for all $i,j\in I$.

To prove the required assertion we consider several cases.

Case 1. Assume that $i,j\in E$. Then from (3.1) we have $h_{E}({\mathbf{e}}_{i})=h_{E}({\mathbf{e}}_{j})=1$, and hence from (3.2) one gets

[TABLE]

Case 2. Assume that $i,j\in I\setminus E$. Then $h_{E}({\mathbf{e}}_{i})=h_{E}({\mathbf{e}}_{j})=0$. From (3.2) we have

[TABLE]

Case 3. Let $i\in I\setminus E$, $j\in E$. Then due to our assumption one has $p_{ij,j}=0$. On the other hand, we find $h_{E}({\mathbf{e}}_{i})=0$ and $h_{E}({\mathbf{e}}_{j})=1$. Again from (3.2) one gets

[TABLE]

This completes the proof. ∎

4. Associativity of genetic Volterra algebras

In this section, we find necessary and sufficient conditions for the associativity of generic Volterra algebras in terms of the heredity coefficients.

Before formulation our main result, we prove an auxiliary fact.

Lemma 4.1**.**

Let $\{p_{ij,k}\}$ be the heredity coefficient of a Volterra algebra. Then for any $i,j,k$ with $(i-j)(j-k)(k-i)\neq 0$ the equality

[TABLE]

is equivalent to

[TABLE]

Proof.

For any $i,j,k$ with $(i-j)(j-k)(k-i)\neq 0$ due to Volterra condition, we have

[TABLE]

Therefore, from (4.1) we have

[TABLE]

which is (4.2). The proof is complete. ∎

Now we are ready to formulate a main result of this section.

Theorem 4.2**.**

Let $A$ be a genetic Volterra algebra. Then the following conditions are equivalent:

• (i)

$A$* is associative;*

• (ii)

one has

• (a)

$p_{ij,i}\in\{0,1\}$* for any $i,j\in I$;*

• (b)

$p_{jk,j}p_{ij,i}+p_{jk,k}p_{ik,i}=p_{ij,i}p_{ik,i}$* for all $i,j,k\in I$ with $(i-j)(j-k)(k-i)\neq 0$.*

Proof.

(ii)$\Rightarrow$(i). Assume that (a) and (b) conditions are satisfied. To prove the associativity, it is enough to establish the associativity on basis elements ${\mathbf{e}}_{1}$, ${\mathbf{e}}_{2},\dots,{\mathbf{e}}_{m}$, i.e.

[TABLE]

Due to the commutativity of the algebra, to show the last equality, it is sufficient to prove (4.3) for the cases: $k=j$ and $(i-j)(j-k)(k-i)\neq 0$, respectively.

The case $k=j$ immediately follows from (a). Therefore, we assume $(i-j)(j-k)(k-i)\neq 0$. Then using (2.7) with (b) one finds

[TABLE]

This means that $A$ is associative.

(i)$\Rightarrow$(ii). Now we suppose that $A$ is associative, i.e. (4.3) holds.

First, we assume that $k=j$. Then due to ${\mathbf{e}}^{2}_{j}={\mathbf{e}}_{j}$ we find

[TABLE]

Due to (2.7) one gets

[TABLE]

Now substituting (2.7), (4.5) into (4.4), and equalizing appropriate coefficients on basis elements, we obtain

[TABLE]

which implies $p_{ij,i}\in\{0,1\}$.

Let us assume that $(i-j)(j-k)(k-i)\neq 0$ and consider the equality

[TABLE]

Keeping in mind (2.7) from the left side of (4.6) we get

[TABLE]

Similarly, the right hand side of (4.6) reduces to

[TABLE]

Now substituting (4.7),(4.8) into (4.6), and equalizing appropriate coefficients on basis elements, we obtain the system of equations

[TABLE]

The last equalities, due to Lemma 4.1, are equivalent. This completes the proof.∎

Remark 4.3**.**

We note that associativity conditions for genetic Volterra algebras have been found in low dimensional setting in [5, 18].

Recall that if $V$ is a Volterra operator on $S^{m-1}$, then it can be represented as follows (see [6]):

[TABLE]

where $|a_{ik}|\leq 1$ and $a_{ik}=-a_{ki}$. Hence, this representation allows us for each Volterra operator to assign a skew-symmetric matrix $(a_{ij})$. This correspondence is a one-to-one linear transformation. One concludes that the set of Volterra operators is convex, and its extremal elements are characterized by $|a_{ik}|=1$ ($k\neq i$). Therefore, a Volterra algebra $A$, associated with $V$, is called extremal if it corresponds to some extremal Volterra operator.

According to the mentioned correspondence, each Volterra operator defines a skew-symmetric matrix, which defines some graph [6]. Namely, we suppose that $a_{ki}\not=0$ at $i\not=k.$ Let us consider a full graph $G_{m}$ consisting of $m$ vertices: $1,2,\dots,m$. Define a tournament $T_{m},$ as a graph consisting of $m$ vertices labeled by $1,2,\dots,m$ corresponding to a skew-symmetrical matrix $(a_{ik})$ according to the following rule: there is an arrow from $k$ to $i$ if $a_{ki}<0,$ and a reverse arrow otherwise. Hence, to every Volterra algebra, we associate the constructed tournament. A number of properties of genetic Volterra algebras can be obtained by means of the theory of tournaments.

Recall that a tournament is said to be strong if it is possible to go from any vertex to any other vertex according to directions on the edges. A strong subtournament of the tournament $T_{m}$ ( $m\geq 3$) composed of three vertices is called a cyclic triple.

It is interesting (independent of interest) to get associativity condition of the Volterra algebra in terms of the corresponding tournament. We have the following

Corollary 4.4**.**

Let $A$ be a Volterra algebra. Then $A$ is associative iff it is an extremal, and the corresponding tournament doesn’t have cyclic triple.

Proof.

Assume that $A$ is associative. Then due to Theorem 4.2 (a) we immediately find the extremity of $A$. Now we suppose that the corresponding tournament has a cyclic triple, i.e. there are three vertices $i,j,k$ which form a cyclic triple. Without loss of generality (since $i,j,k$ are cyclic tripe), we may choose $p_{ij,i}=1$, $p_{jk,j}=1$ and $p_{ki,k}=1$. But this contradicts to Theorem 4.2 (b), i.e. $p_{jk,j}p_{ij,i}+p_{jk,k}p_{ik,i}=1\neq 0=p_{ij,i}p_{ik,i}$.

Now assume that $A$ is extremal, and the corresponding tournament doesn’t have cyclic triple. Let us establish that $A$ is associative.

Due to our assumptions, for any three basis elements ${\mathbf{e}}_{i},{\mathbf{e}}_{j},{\mathbf{e}}_{k}$ we have one of the following possibilities:

1. case 1.

${\mathbf{e}}_{i}\circ{\mathbf{e}}_{j}={\mathbf{e}}_{j}$, ${\mathbf{e}}_{k}\circ{\mathbf{e}}_{j}={\mathbf{e}}_{j}$ and ${\mathbf{e}}_{k}\circ{\mathbf{e}}_{i}={\mathbf{e}}_{i}$; 2. case 2.

${\mathbf{e}}_{i}\circ{\mathbf{e}}_{j}={\mathbf{e}}_{j}$, ${\mathbf{e}}_{k}\circ{\mathbf{e}}_{j}={\mathbf{e}}_{j}$ and ${\mathbf{e}}_{k}\circ{\mathbf{e}}_{i}={\mathbf{e}}_{k}$; 3. case 3.

${\mathbf{e}}_{i}\circ{\mathbf{e}}_{j}={\mathbf{e}}_{i}$, ${\mathbf{e}}_{k}\circ{\mathbf{e}}_{j}={\mathbf{e}}_{j}$ and ${\mathbf{e}}_{k}\circ{\mathbf{e}}_{i}={\mathbf{e}}_{i}$; 4. case 4.

${\mathbf{e}}_{i}\circ{\mathbf{e}}_{j}={\mathbf{e}}_{i}$, ${\mathbf{e}}_{k}\circ{\mathbf{e}}_{j}={\mathbf{e}}_{k}$ and ${\mathbf{e}}_{k}\circ{\mathbf{e}}_{i}={\mathbf{e}}_{i}$; 5. case 5.

${\mathbf{e}}_{i}\circ{\mathbf{e}}_{j}=e_{j}$, ${\mathbf{e}}_{k}\circ{\mathbf{e}}_{j}={\mathbf{e}}_{k}$ and ${\mathbf{e}}_{k}\circ{\mathbf{e}}_{i}={\mathbf{e}}_{k}$; 6. case 6.

${\mathbf{e}}_{i}\circ{\mathbf{e}}_{j}={\mathbf{e}}_{i}$, $e_{k}\circ{\mathbf{e}}_{j}={\mathbf{e}}_{k}$ and ${\mathbf{e}}_{k}\circ{\mathbf{e}}_{i}={\mathbf{e}}_{k}$.

For the case 1, we will show the associativity of the vectors ${\mathbf{e}}_{i},{\mathbf{e}}_{j},{\mathbf{e}}_{k}$ (the other cases can be proceeded by the same argument). Indeed, we have

[TABLE]

these equalities yield the assertion. The proof is complete. ∎

Definition 4.5**.**

A tournament is called transitive if it does not have any cyclic triple.

Remark 4.6**.**

[3]** There is only one transitive tournament of a given order $m$ (up to isomorphism).

Recall that two tournaments are said to be isomorphic, if there exists a permutation of the vertices which transforms one tournament into the other.

Theorem 4.7**.**

[20]** If the tournaments of two extremal Volterra algebras are isomorphic, then the corresponding algebras are isomorphic as well.

Theorem 4.8**.**

Any associative genetic Volterra algebra is isomorphic to the algebra with structural coefficients

[TABLE]

Proof.

It is clear that the corresponding tournament is extremal and transitive, hence due to Corollary 4.4 the Volterra algebra is associative. According to Remark 4.6 and Theorem 4.7 we infer the required assertion. ∎

5. Derivations of genetic Volterra algebras

It is interesting to find all derivations of given algebra. The well-known Kadison’s Theorem states that all derivations of associative and commutative algebras are trivial. Therefore, under conditions of Theorem 4.2 any derivation of genetic Volterra algebra is trivial. In this section, we are going to describe derivations of genetic Volterra algebras.

Let $A$ be a genetic Volterra algebra generated by the heredity coefficients $\{p_{ij,k}\}$. We put

[TABLE]

Any derivation $d$ of $A$ has the following form

[TABLE]

for some matrix $(d_{ij})$.

Lemma 5.1**.**

Let $A$ be a genetic Volterra algebra, and $d$ be its derivation given by (5.1). If $j\neq i$ and $j\notin I_{i}$, then $d_{ij}=d_{ji}=0$

Proof.

Due to our denotation and the definition of the algebra, one can see that if $j\notin I_{i}$ then $i\notin I_{j}$. Therefore, it is enough to establish $d_{ij}=0$.

From the definition of the derivation, we immediately find

[TABLE]

From (2.7) and (5.1) it follows that

[TABLE]

Now inserting (5.3) and (5.1) into (5.2) and equalizing the corresponding coefficients on basis elements, we obtain

[TABLE]

which implies that $d_{ij}=0$ if $j\notin I_{i}$. This completes the proof. ∎

Corollary 5.2**.**

Let $A$ be a genetic Volterra algebra. If one has

[TABLE]

for all $i,j\in I$, then any derivations of $A$ is trivial.

Proof.

Due to $p_{ij,j}\neq\frac{1}{2}$ from Lemma 5.1 we obtain $d_{ij}=0$ if $i\neq j$. This means that every derivation has the following form $D({\mathbf{e}}_{i})=d_{ii}{\mathbf{e}}_{i}$. From the definition of the derivation one has $D({\mathbf{e}}_{i})=2{\mathbf{e}}_{i}\circ D({\mathbf{e}}_{i})$. This, due to $2{\mathbf{e}}_{i}\circ D({\mathbf{e}}_{i})=2d_{ii}({\mathbf{e}}_{i}\circ{\mathbf{e}}_{i})=2d_{ii}{\mathbf{e}}_{i}$, yields $d_{ii}=0$. The proof is complete. ∎

Now we are going to describe all nontrivial derivations of three dimensional genetic Volterra algebras.

Theorem 5.3**.**

Let $A$ be a three dimensional genetic Volterra algebra. The algebra has a nontrivial derivation if and only if there exist $i,j,k$ with $\{i,j,k\}=\{1,2,3\}$ such that $p_{ij,i}=\frac{1}{2}$, $p_{ik,i}=p_{jk,j}$.

Proof.

”if” part. According to Corollary 5.2 we have that $I_{j}$ is not empty for some $j\in\{1,2,3\}$. Without loss of generality we may assume that $0\neq|I_{1}|\geq|I_{2}|\geq|I_{3}|$. Then it is possible that either $|I_{1}|=1$ or 2. Therefore, we have three main possible cases:

• (a)

$|I_{1}|=1$, $|I_{2}|=1$ and $|I_{3}|=0$;

• (b)

$|I_{1}|=2$, $|I_{2}|=1$ and $|I_{3}|=1$;

• (c)

$|I_{1}|=2$, $|I_{2}|=2$ and $|I_{3}|=2$.

Note that other cases can be reduced to the listed ones.

In the considered setting, the derivation has a form

[TABLE]

Now let us investigate above listed cases one by one.

Case (a). Assume that $|I_{1}|=1$, $|I_{2}|=1$ and $|I_{3}|=0$. Then we have

[TABLE]

and $p_{23,2}=1-p_{23,3}\neq\frac{1}{2}$. Hence, due to Lemma 5.1 one finds $d_{13}=d_{31}=d_{23}=d_{32}=0$. Using the same argument as in the proof of Corollary 5.2 one can show that $d_{33}=0$. Hence, we obtain $D({\mathbf{e}}_{3})=0$.

In the considered case, we have $D({\mathbf{e}}_{1})=d_{11}{\mathbf{e}}_{1}+d_{12}{\mathbf{e}}_{2}$. Keeping in mind the equality

[TABLE]

from $D({\mathbf{e}}_{1})=2{\mathbf{e}}_{1}\circ D({\mathbf{e}}_{1})$ one finds $d_{11}=-d_{12}$. Let us denote $a:=d_{11}$.

Using the same argument with ${\mathbf{e}}_{2}$, we get $d_{21}=-d_{22}$. Hence, we denote $b:=d_{21}$.

It is clear that $D({\mathbf{e}}_{2}\circ{\mathbf{e}}_{3})={\mathbf{e}}_{3}\circ D({\mathbf{e}}_{2})+D({\mathbf{e}}_{3})\circ{\mathbf{e}}_{2}$. Therefore, due to

[TABLE]

and

[TABLE]

we obtain $b(p_{13,3}-p_{23,3})=0$.

Now using the same argument with $D({\mathbf{e}}_{1}\circ{\mathbf{e}}_{3})={\mathbf{e}}_{3}\circ D({\mathbf{e}}_{1})+D({\mathbf{e}}_{3})\circ{\mathbf{e}}_{1}$ one finds $a(p_{13,3}-p_{23,3})=0$. Since at least one of $a$ and $b$ is non zero, hence we have $p_{13,3}=p_{23,3}$. This means that $1-p_{13,3}=1-p_{23,3}$, so $p_{13,1}=p_{23,2}$.

Case (b). Let us suppose that $|I_{1}|=2$, $|I_{2}|=1$ and $|I_{3}|=1$. Then we have $p_{12,1}=p_{12,2}=\frac{1}{2}$, $p_{13,1}=p_{13,3}=\frac{1}{2}$ and $p_{23,2}=1-p_{23,3}\neq\frac{1}{2}$. So, from Lemma 5.1 one concludes that $d_{23}=d_{32}=0$.

From $D({\mathbf{e}}_{3})=d_{31}{\mathbf{e}}_{1}+d_{33}{\mathbf{e}}_{3}$ and

[TABLE]

with $D({\mathbf{e}}_{3})=2{\mathbf{e}}_{3}\circ D({\mathbf{e}}_{3})$ we obtain $d_{31}=-d_{33}$. Hence, we denote $a:=d_{31}$.

Using the same argument with ${\mathbf{e}}_{2}$, one gets $d_{21}=-d_{22}$, so one denotes $b:=d_{21}$.

Inserting the equalities $D({\mathbf{e}}_{1})=d_{11}{\mathbf{e}}_{1}+d_{12}{\mathbf{e}}_{2}+d_{13}{\mathbf{e}}_{3}$ and

[TABLE]

into $D({\mathbf{e}}_{1})=2{\mathbf{e}}_{1}\circ D({\mathbf{e}}_{1})$, and equalizing corresponding coefficients at ${\mathbf{e}}_{1}$ we find $d_{11}+d_{12}=-d_{13}$. So, we denote $c:=d_{11}$ and $d:=d_{12}$.

It is clear that

[TABLE]

On the other hand, we have

[TABLE]

Similarly, one finds

[TABLE]

Now inserting the last equalities into (5.5), we obtain

[TABLE]

[TABLE]

From $3\notin I_{2}$ one gets $a=b=0$. Hence, we have

[TABLE]

We know that $D({\mathbf{e}}_{1}\circ{\mathbf{e}}_{3})={\mathbf{e}}_{3}\circ D({\mathbf{e}}_{1})+D({\mathbf{e}}_{3})\circ{\mathbf{e}}_{1}$. Hence from

[TABLE]

and

[TABLE]

one finds $d(1/2-p_{23,2})=0$. From $3\notin I_{2}$ it follows that $d=0$. This implies $D({\mathbf{e}}_{1})=c{\mathbf{e}}_{1}-c{\mathbf{e}}_{3}$, $D({\mathbf{e}}_{2})=0$ and $D({\mathbf{e}}_{3})=0$.

Similarly, from $D({\mathbf{e}}_{1}\circ{\mathbf{e}}_{2})={\mathbf{e}}_{2}\circ D({\mathbf{e}}_{1})+D({\mathbf{e}}_{2})\circ{\mathbf{e}}_{1}$ with

[TABLE]

and

[TABLE]

we obtain $c(1/2-p_{23,2})=0$. Taking into account $3\notin I_{2}$, one finds $c=0$. Hence, $D({\mathbf{e}}_{1})=0$, $D({\mathbf{e}}_{2})=0$ and $D({\mathbf{e}}_{3})=0$. This means that the algebra $A$ has only trivial derivation, which contradicts to our assumption.

Case (c). In this case, we have $|I_{1}|=2$, $|I_{2}|=2$ and $|I_{3}|=2$, which yield $p_{12,1}=p_{12,2}=p_{13,1}=p_{13,3}=p_{23,2}=p_{23,3}=\frac{1}{2}$. This implies the required assertion.

”only if” part. Assume that $p_{12,1}=\frac{1}{2}$, $p_{13,1}=p_{23,2}$. We are going to show that the existence of non trivial derivation of $A$. There are two possibilities:

1. (A)

$p_{12,1}=\frac{1}{2}$, $p_{13,1}=p_{23,2}\neq\frac{1}{2}$; 2. (B)

$p_{12,1}=p_{13,1}=p_{23,2}=\frac{1}{2}$.

In case (A) we define a linear mapping $D$ as follows:

[TABLE]

where $a,b\in\mathbb{R}$. One can check that the defined mapping is a derivation.

In case (B) we define a linear mapping $D$ as follows:

[TABLE]

where $a,b,c,d\in\mathbb{R}$. One can check that the defined mapping is a derivation. This completes the proof. ∎

Lemma 5.4**.**

Let $A$ be a three dimensional genetic Volterra algebra such that $p_{12,1}=\frac{1}{2}.$ Then the following statements hold:

• (A)

If $p_{13,1}=p_{23,2}\neq\frac{1}{2},$ then each derivation of $A$ is of the form

[TABLE]

• (B)

If $p_{13,1}=p_{23,2}=\frac{1}{2},$ then each derivation $D$ of $A$ is defined by

[TABLE]

where $\sum\limits_{j=1}^{3}d_{ij}=0$ for $i=1,2,3.$

Proof.

(A). Let $D({\mathbf{e}}_{i})=\sum_{j=1}^{3}d_{ij}{\mathbf{e}}_{j}$ ($i=1,2,3$) be a derivation. From the condition (A) we infer that $d_{13}=d_{31}=d_{23}=d_{32}=d_{22}=0$. The equality

[TABLE]

implies $d_{33}=0$. Using the relations $D({\mathbf{e}}_{1})=2{\mathbf{e}}_{1}\circ D({\mathbf{e}}_{1})$ and $D({\mathbf{e}}_{2})=2{\mathbf{e}}_{2}\circ D({\mathbf{e}}_{2})$ we get that $d_{11}=-d_{12}=:a$ and $d_{21}=-d_{22}=:b$. One can check that the defined mapping is indeed a derivation.

(B) Let $D({\mathbf{e}}_{i})=\sum_{j=1}^{3}d_{ij}{\mathbf{e}}_{j}$($i=1,2,3$) be a derivation. Then using the given condition, we have

[TABLE]

which yields $\sum_{j=1}^{3}d_{ij}=0$ for $i=1,2,3.$

Now let us establish that the reverse. Namely, we show that the mapping given by (5.8) with $\sum_{j=1}^{3}d_{ij}=0$ for $i=1,2,3$ is a derivation. Indeed, we have

[TABLE]

This completes the proof.

∎

Now we want to apply Lemma 5.4 to describe all local derivations of the three dimensional genetic Volterra algebra. Recall that a linear map $\Delta:A\to A$ is called local derivation if for any $x\in A$ there exists a derivation $D_{x}:A\to A$ such that $\Delta(x)=D_{x}(x)$.

Theorem 5.5**.**

Let $A$ be a three dimensional genetic Volterra algebra. Then any local derivation is a derivation.

Proof.

Due to Lemma 5.4 a derivation on $A$ exists if one of the following cases hold:

1. (A)

$p_{12,1}=\frac{1}{2}$, $p_{13,1}=p_{23,2}\neq\frac{1}{2}$; 2. (B)

$p_{12,1}=p_{13,1}=p_{23,2}=\frac{1}{2}$.

Let us consider the mentioned cases one by one.

Assume that case (A) holds. Then due to Lemma 5.4 any derivation of $A$ has a form given by (5.6).

Suppose that $\Delta$ is a local derivation of $A$. Then from the definitions of local derivation, one finds

[TABLE]

By means of (5.6) one infers that

[TABLE]

Therefore, from (5.9) and (5.10) we find

[TABLE]

which according to Lemma 5.4 means that $\Delta$ is a derivation.

Now assume that case (B) holds. Then for a local derivation $\Delta$ we have (5.9). From (5.7) one finds

[TABLE]

This again by Lemma 5.4 yields that $\Delta$ is a derivation.∎

From this theorem we can formulate the following

Conjecture 5.6**.**

Let $A$ be an $n$-dimensional genetic Volterra algebra. Then a category of local derivations of $A$ coincides with the category of derivations of $A$.

Acknowledgments

The authors are grateful to an anonymous referee whose valuable comments and remarks improved the presentation of this paper.