Lotka–Volterra coalgebras

Abstract

We study Lotka–Volterra coalgebras, a new family of nonassociative coalgebras emerging from population genetics. We focus on their defining algebraic properties and deal with characterizing the existence of counital and character maps. We also provide a classification of their basis elements into (co)algebraically persistent and transient generators resulting in a semidirect sum decomposition of Lotka–Volterra coalgebras.

Keywords:

• Lotka–Volterra coalgebra
• baric coalgebra
• semidirect sum decomposition

1. Introduction

Lotka–Volterra algebras were introduced by Itoh [1] as a nonassociative framework for describing ternary interactions in competition systems. Despite being aimed by random models mirroring the kinetic theory of gases, a fact remarked by Itoh in [1], these algebras were later shown to include, as particular examples, normal Bernstein algebras. Bernstein algebras became essential in the study of quadratic stochastic operators in population genetics [2].

Given a field $\mathbb{K}$ of characteristic not 2, a commutative $\mathbb{K}$-algebra $\mathcal{A}$ is Lotka–Volterra if it admits a basis $\mathcal{B}=\{e_1,\dots ,e_n\}$ such that $e_i{e}_j=e_j{e}_i=(\frac{1}{2}+a_{ij})e_i+(\frac{1}{2}+a_{ji})e_j$, for all i, j = 1, …, n, where $\mathbf{A}=(a_{ij}{)}_{i,j=1}^n$ is a skew-symmetric matrix. We refer to [3, 4] and references therein for the algebraic treatment of Lotka–Volterra algebras. Alternatively, Lotka–Volterra algebras can be defined as follows:

Definition 1.1

A $\mathbb{K}$-algebra $\mathcal{A}$ is Lotka–Volterra (for short, an LV-algebra) if it is endowed with a basis $\mathcal{B}=\{e_1,e_2,\dots ,e_n\}$ and multiplication $e_i{e}_j=\sum _{k=1}^n{\beta }_{ij}^k{e}_k$, such that for any 1 ≤ i, j, k ≤ n:

• $k\notin \{i,j\}⟹{\beta }_{ij}^k=0,$

• ${\beta }_{ij}^k={\beta }_{ji}^k$,

• $\sum _{k=1}^n{\beta }_{ij}^k=1$.

From the genetic viewpoint Lotka–Volterra algebras reproduce the forward evolution of populations where offspring necessarily reproduce the genetic type of one of the progenitors, assuming also panmixia, that is, considering that no progenitor has any specific weight on offspring type. As a result, LV-algebras are, by definition, commutative. Notice however that nonnegativity of the multiplication constants has been later not required [3] as it happens to be the case for (nonassociative) algebras with genetic realization [5]. This was not however the case in [1], where the requirement $|a_{ij}|\le \frac{1}{2}$ on the skew-symmetric matrix entries ensured the genetic realization of Itoh's Lotka–Volterra algebras.

Lotka–Volterra algebras are a family of genetic algebras different from the recently studied evolution algebras [6], although both families of algebras, Lotka–Volterra and evolution, are defined by square, not cubic, structure matrices. The simplest example of Lotka–Volterra algebra is the gametic algebra for simple Mendelian inheritance [7, p.135], also called an elementary algebra by Heuch and Holgate, given by the zero matrix A = 0_n×n

Our aim here is, retaining those properties characterizing the genetic population under consideration, i.e. offspring reproducing some progenitor type and panmixia, to consider Tian and Li's approach based on the search for new algebraic structures focused on describing the backward inheritance within genetic systems [8]. As a result, Lotka–Voterra coalgebras are introduced, providing a new framework for the treatment of such populations.

Lotka–Volterra coalgebras constitute therefore a new family of noncoassociative coalgebras whose defining conditions reflect the backward dynamics of the transference of genetic inheritance within the considered populations. Even though here we are mainly concerned with the study of their algebraic structure, the connection between Lotka–Volterra coalgebras and the theory of dynamical systems will also become apparent.

After this introduction, the paper is organized as follows. Section two provides the definition of Lotka–Volterra coalgebras, together with examples, and also focuses on their matrix realization. In the third section, we cope with the duality of Lotka–Volterra structures. A well-known result states the correspondence, through dual vector spaces, between algebraic and coalgebraic structures defined on finite-dimensional vector spaces. Here we examine up to what extent the duality between Lotka–Volterra algebras and coalgebras is retained, to continue in Section four tackling with the algebraic description of Lotka–Volterra coalgebras.

Section five is devoted to a rather classical problem in algebraic genetics [7, p.103], namely here, discussing which (coalgebra) constructions retain the property of being Lotka–Volterra. To finish the algebraic description of Lotka–Volterra coalgebras we achieve in this work, Section six focuses on their algebraic structure. Distinguishing generators of Lotka–Volterra coalgebras (i.e. their natural basis elements) into group-like and nongroup-like elements provides a semidirect sum decomposition (see [6, p.45]) of Lotka–Volterra coalgebras into a subcoalgebra spanned by algebraic persistent elements and a coideal spanned by algebraic transient elements. This classification of the coalgebra natural basis elements brings into a linkage between Lotka–Volterra coalgebras and dynamical systems, to be exploited in a forthcoming work.

2. Lotka Volterrra coalgebras

Throughout the paper, unless otherwise stated, $\mathbb{K}$ will be a field of characteristic not 2, and all objects will be defined over finite-dimensional $\mathbb{K}$-vector spaces. We refer the reader to [9] for basic notions on coalgebras.

A coalgebra $(\mathcal{C},\mathrm{\Delta })$ is a $\mathbb{K}$-vector space $\mathcal{C}$ endowed with a linear map (comultiplication) $\mathrm{\Delta }:\mathcal{C}⟶\mathcal{C}\otimes \mathcal{C}$. Given a basis $\mathcal{B}=\{e_1,e_2,\dots ,e_n\}$ of the $\mathbb{K}$-vector space $\mathcal{C}$, we write $\mathrm{\Delta }(e_k)=\sum _{i,j=1}^n{\beta }_{ij}^k{e}_i\otimes e_j,k=1,\dots ,n.$A subcoalgebra D of a coalgebra $\mathcal{C}$ is a subspace D of $\mathcal{C}$ such that Δ(D)⊆ D ⊗ D. A vector subspace I of $\mathcal{C}$ is a left (resp. right) coideal of $\mathcal{C}$ if $\mathrm{\Delta }(I)\subseteq \mathcal{C}\otimes I$ (resp. $\mathrm{\Delta }(I)\subseteq I\otimes \mathcal{C}$) and a coideal if $\mathrm{\Delta }(I)\subseteq I\otimes \mathcal{C}+\mathcal{C}\otimes I$. Subcoalgebras are one-sided coideals and, one-sided coideals are coideals. Intersections and sums of subcoalgebras (resp. left or right coideals) are again subcoalgebras (resp. left or right coideals). Sums of coideals are also still coideals, but this does not apply to intersections.

The comultiplication structure constants can be rearranged into a cubic (n × n × n) matrix $\mathbf{P}=(p_{ijk}{)}_{i,j,k=1}^n\in {\mathbb{K}}^{n\times n\times n}$, given by $p_{ijk}={\beta }_{ij}^k$. Cubic matrices are usually unfolded by their frontal slices as n × n² rectangular matrices P = (P_::1|P_::2| · · · |P_::n), where, for any k = 1, …, n, P_::k is the n × n matrix with (i, j)-th entry p_ijk, i, j = 1, …, n. Thus the k-th frontal slice P_::k of P contains the comultiplication constants arising from Δ(e_k).

Definition 2.1

A coalgebra $(\mathcal{C},\mathrm{\Delta })$ is Lotka–Volterra (for short, an LV-coalgebra) if it admits a basis $\mathcal{B}=\{e_1,e_2,\dots ,e_n\}$, satisfying for all i, j, k = 1, …, n:

• $k\notin \{i,j\}⟹{\beta }_{ij}^k=0$;

• ${\beta }_{ij}^k={\beta }_{ji}^k$;

• $\sum _{i,j=1}^n{\beta }_{ij}^k=1$;

where $\{{\beta }_{ij}^k{\}}_{i,j,k=1}^n$ is a set of scalars such that $\mathrm{\Delta }(e_k)=\sum _{i,j=1}^n{\beta }_{ij}^k{e}_i\otimes e_j,k=1,\dots ,n.$

Remark 2.1

Any such a basis $\mathcal{B}$ will be called a natural basis. All bases considered in the paper will be assumed to be natural bases. Because of LCV-1 and LCV-2 we have that $\mathrm{\Delta }(e_k)={\beta }_{kk}^k{e}_k\otimes e_k+\sum _{i=1,i\ne k}^n{\beta }_{ik}^k(e_i\otimes e_k+e_k\otimes e_i),k=1,\dots ,n.$Moreover, the above relation and LCV-3 imply ${\beta }_{kk}^k+2\left(\sum _{i=1,i\ne k}^n{\beta }_{ik}^k\right)=1,\text{for all}k=1,\dots ,n,$or, equivalently, $\sum _{i=1,i\ne k}^n{\beta }_{ik}^k=\frac{1}{2}(1-{\beta }_{kk}^k).$

Definition 2.2

A real (i.e. $\mathbb{K}=\mathbb{R}$) LV-coalgebra $(\mathcal{C},\mathrm{\Delta })$, with natural basis $\mathcal{B}$, has genetic realization if:

• ${\beta }_{ij}^k\ge 0$ for all i, j, k = 1, …, n.

Example 2.3

Let G be a finite group of order n. Then the group coalgebra $\mathcal{C}=\mathbb{K}G$ is an LV-coalgebra with natural basis $\mathcal{B}=\{e_g\mid g\in G\}$ and ${\beta }_{ij}^k={\delta }_{ki}{\delta }_{kj}$, for all i, j, k = 1, …, n.

Definition 2.4

An LV-coalgebra $\mathcal{C}$ will be said to be trivial if ${\beta }_{ij}^k={\delta }_{ki}{\delta }_{kj}$, for all i, j, k = 1, …, n., i.e. if $\mathcal{C}$ is a group coalgebra.

Example 2.5

Tian and Li [8, p. 248] considered the case of a randomly mating population of diploid individuals differing in a locus alleles $\mathcal{B}=\{e_1,e_2,\dots ,e_n\}$. The n-dimensional $\mathbb{K}$-vector space $\mathcal{C}$ with basis $\mathcal{B}$ and comultiplication given by ${\beta }_{ij}^k=\frac{1}{2n}({\delta }_{ki}+{\delta }_{kj})$, for all i, j, k = 1, …, n, is an LV-coalgebra. Following [7, Example 1.1, p. 8], we refer to $\mathcal{C}$ to be the n-dimensional gametic coalgebra for simple Mendelian inheritance.

The class of LV-coalgebras is a family of (genetic) coalgebras not containing the zygotic coalgebra or the gametic coalgebra of autotetraploids [10]. Moreover, an evolution coalgebra $\mathcal{C}$ is Lotka–Volterra if and only if $\mathcal{C}$ is a trivial LV-coalgebra, that is, a group coalgebra [11].

Displayed by its frontal slices, the cubic matrix associated to an LV-coalgebra has the following form: $\mathbf{P}=\left(\begin{array}{cccccccccccccc}{\beta }_{11}^1& {\beta }_{12}^1& \cdots & {\beta }_{1n}^1& 0& {\beta }_{12}^2& \cdots & 0& \cdots & \cdots & 0& 0& \cdots & {\beta }_{1n}^n\\ {\beta }_{21}^1& 0& \cdots & 0& {\beta }_{21}^2& {\beta }_{22}^2& \cdots & {\beta }_{2n}^2& \cdots & \cdots & 0& 0& \cdots & {\beta }_{2n}^n\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮& \cdots & \cdots & ⋮& ⋮& \ddots & ⋮\\ {\beta }_{n1}^1& 0& \cdots & 0& 0& {\beta }_{n2}^2& \cdots & 0& \cdots & \cdots & {\beta }_{n1}^n& {\beta }_{n2}^n& \cdots & {\beta }_{nn}^n\end{array}\right),$which, by LVC-2, is (1,2)-symmetrical with equal accompanying matrices [10, Lemma 2].

Proposition 2.6

Let $\mathcal{C}$ be an LV-coalgebra. Then, with respect to a natural basis $\mathcal{B}$, the accompanying matrices of $\mathcal{C}$ (i.e. those of its associated cubic matrix) are ${\mathbf{P}}_{(i)}={\mathbf{P}}_{(j)}=\left(\begin{array}{cccc}\sum _{i=1}^n{\beta }_{i1}^1& {\beta }_{12}^2& \cdots & {\beta }_{1n}^n\\ {\beta }_{12}^1& \sum _{i=1}^n{\beta }_{i2}^2& \cdots & {\beta }_{2n}^n\\ ⋮& ⋮& \ddots & ⋮\\ {\beta }_{1n}^1& {\beta }_{n2}^2& \cdots & \sum _{i=1}^n{\beta }_{in}^n\end{array}\right).$

Proof.

See [10, 3.3].

Mimicking the algebra setting, LV-coalgebras are proposed to model backward dynamics in genetic systems when offspring necessarily reproduce the genetic type of one progenitor (LVC-1). Panmixia, i.e. considering that paternal or maternal type have no specific weight on offspring type, is also assumed, resulting in assumption LVC-2. Finally, for real LV-coalgebras, having genetic realization amounts to: ${\beta }_{ik}^k={\beta }_{ki}^k=\{\begin{array}{ll}{\displaystyle P(\text{one progenitor of type }i\mid \text{offspring of type }k),}& i\ne k;\\ {\displaystyle P(\text{both progenitors of type}i=k\mid \text{offspring of type }k),}& i=k.\end{array}$

3. Duality

As remarked in the introduction, when posing the problem of introducing coalgebras as an alternative mathematical framework for modelling backward genetic inheritance, Tian and Li [8] stressed on the fact that looking for new coalgebraic structures should imply more than just dualizing already existing genetic-like algebras. In this section, it is shown that, as previously defined, LV-coalgebras give rise to a new family of algebraic structures playing their own role in the mathematical study of population genetics.

The dual vector space ${\mathcal{C}}^{\ast }=Ho{m}_{\mathbb{K}}(\mathcal{C},\mathbb{K})$ of any coalgebra $(\mathcal{C},\mathrm{\Delta })$ is endowed with an algebra structure with multiplication induced by Δ*. Conversely, for any finite dimensional algebra $\mathcal{A}$ the isomorphism $(\mathcal{A}\otimes \mathcal{A}{)}^{\ast }\cong {\mathcal{A}}^{\ast }\otimes {\mathcal{A}}^{\ast }$ allows us to define a coalgebra structure on ${\mathcal{A}}^{\ast }$ [9].

Let $\mathcal{C}$ be a n-dimensional $\mathbb{K}$-vector space with basis $\mathcal{B}=\{e_1,\dots ,e_n\}$. We denote by ${\mathcal{B}}^{\ast }=\{f_1,\dots ,f_n\}$ the dual basis of $\mathcal{B}$ in ${\mathcal{C}}^{\ast }$, i.e. f_i(e_j) = δ_ij, for all i, j = 1, …, n.

Theorem 3.1

Let $(\mathcal{C},\mathrm{\Delta })$ be a coalgebra satisfying LVC-1 and LVC-2 (with respect to $\mathcal{B}$). Then the induced multiplication in ${\mathcal{C}}^{\ast }$ is given by: $\{\begin{array}{ll}{f}_r{f}_r={\beta }_{rr}^r{f}_r;& \\ f_r{f}_s=f_s{f}_r={\beta }_{sr}^r{f}_r+{\beta }_{rs}^s{f}_s,& r\ne s.\end{array}$

Proof.

Use Δ*(f_r ⊗ f_s)(e_k) = (f_r ⊗ f_s)Δ(e_k) to obtain: $(f_r{f}_s)(e_k)=\{\begin{array}{ll}0;& k\notin \{r,s\};\\ {\beta }_{ks}^k={\beta }_{rs}^r;& k=r\ne s;\\ {\beta }_{rk}^k={\beta }_{rs}^s;& k=s\ne r;\\ {\beta }_{kk}^k;& k=r=s;\end{array}$for all r, s, k = 1, …, n.

Remark 3.1

Note the multiplication in the dual algebra ${\mathcal{C}}^{\ast }$ is indeed defined by $f_r{f}_s=\sum _{k=1}^n{\beta }_{rs}^k{f}_k$, for all r, s = 1, …, n.

Corollary 3.2

Let $(\mathcal{C},\mathrm{\Delta })$ be an LV-coalgebra with genetic realization. Then ${\mathcal{C}}^{\ast }$ is not an LV-algebra (with respect to ${\mathcal{B}}^{\ast }$).

Proof.

Assume ${\mathcal{C}}^{\ast }$ with the induced multiplication (see Theorem 3.1) is an LV-algebra. Then we have ${\beta }_{kk}^k=1$ for all k = 1, …, n, and LVC-3 and LVC-4 together imply that ${\beta }_{rs}^k={\delta }_{kr}{\delta }_{ks}$. Hence f_rf_s = f_sf_r = 0 if r ≠ s, contradicting then that ${\mathcal{C}}^{\ast }$ is LV-algebra.

We remark that, given an LV-coalgebra $\mathcal{C}$ with genetic realization whose dual algebra ${\mathcal{C}}^{\ast }$ is also an LV-algebra Corollary 3.2 shows that $\mathcal{C}$ is then a group coalgebra, i.e. a trivial LV-coalgebra, and ${\mathcal{C}}^{\ast }$ a (nonzero) trivial evolution algebra.

Next given a coalgebra $\mathcal{C}$ we provide necessary and sufficient conditions for its dual algebra ${\mathcal{C}}^{\ast }$ to be an LV-algebra.

Theorem 3.3

Let $(\mathcal{C},\mathrm{\Delta })$ be a coalgebra with basis $\mathcal{B}=\{e_1,\dots ,e_n\}$. Then, there exists a skew-symmetric matrix $\mathbf{A}=(a_{ij}{)}_{i,j=1}^n$ such that Δ(e_k) = e_k ⊗ T_k + T_k ⊗ e_k, where $T_k=\sum _{i=1}^n(\frac{1}{2}+a_{ki})e_i$ if and only if $({\mathcal{C}}^{\ast },{\mathrm{\Delta }}^{\ast })$ is an LV-algebra with respect to the dual basis ${\mathcal{B}}^{\ast }$.

Proof.

Let $(\mathcal{C},\mathrm{\Delta })$ be a coalgebra with comultiplication Δ(e_k) = e_k ⊗ T_k + T_k ⊗ e_k, with $T_k=\sum _{i=1}^n(\frac{1}{2}+a_{ki})e_i$ defined by a skew-symmetric matrix $\mathbf{A}=(a_{ij}{)}_{i,j=1}^n$. We note this results into: $\{\begin{array}{ll}{\beta }_{kk}^k=1;& \\ {\beta }_{ks}^k=\frac{1}{2}+a_{ks};& s=1,\dots ,n;s\ne k;\\ {\beta }_{rs}^k=0;& \text{otherwise.}\end{array}$Then, by Remark 3.1, $f_r{f}_s=\sum _{k=1}^n{\beta }_{rs}^k{f}_k={\beta }_{rs}^r{f}_r+{\beta }_{rs}^s{f}_s=(\frac{1}{2}+a_{rs})f_r+(\frac{1}{2}+a_{sr})f_s$, for all r, s = 1, …, n. Moreover, ${\mathcal{C}}^{\ast }$ is commutative and, since A is skew-symmetric, ${\mathcal{C}}^{\ast }$ is an LV-algebra.

Assume conversely $(\mathcal{C},\mathrm{\Delta })$ is a coalgebra, whose dual algebra ${\mathcal{C}}^{\ast }$ with the inherited multiplication m = Δ* is a (commutative) LV-algebra, with respect to ${\mathcal{B}}^{\ast }$, and write, for all r, s = 1, …, n: $m(f_r\otimes f_s)=f_r{f}_s=(\frac{1}{2}+a_{rs})f_r+(\frac{1}{2}+a_{sr})f_s$for a skew-symmetric matrix $\mathbf{A}=(a_{rs}{)}_{r,s=1}^n$. We consider the obvious identifications $({\mathcal{C}}^{\ast }{)}^{\ast }\cong \mathcal{C}$ and (Δ*)* ≅ Δ resulting from the finite-dimensionality of $\mathcal{C}$ [9, p.10]. Then Δ(e_k)(f_r ⊗ f_s) = <e_k, f_rf_s > implies $\mathrm{\Delta }(e_k)=\sum _{r,s=1}^n{\beta }_{rs}^k{e}_r\otimes e_s$, with ${\beta }_{rs}^k=(\frac{1}{2}+a_{rs}){\delta }_{kr}+(\frac{1}{2}+a_{sr}){\delta }_{ks}$, for all k, r, s = 1, …, n and, as a result:

1. If $k\notin \{r,s\}$, then ${\beta }_{rs}^k=0$,

2. If k = r ≠ s, then ${\beta }_{rs}^k={\beta }_{ks}^k=\frac{1}{2}+a_{ks}$,

3. If k = r = s, then ${\beta }_{rs}^k={\beta }_{kk}^k=1$.

Moreover ${\beta }_{rs}^k={\beta }_{sr}^k$ for all k, r, s = 1, …, n. Thus, taking into account that a_kk = 0, $\begin{array}{rl}\mathrm{\Delta }(e_k)& =e_k\otimes e_k+\sum _{r=1,r\ne k}^n{\beta }_{kr}^k{e}_k\otimes e_r+\sum _{r=1,r\ne k}^n{\beta }_{rk}^k{e}_r\otimes e_k\\ & =e_k\otimes (\frac{1}{2}+a_{kk})e_k+(\frac{1}{2}+a_{kk})e_k\otimes e_k\\ & +e_k\otimes (\sum _{r=1,r\ne k}^n(\frac{1}{2}+a_{kr})e_r)+(\sum _{r=1,r\ne k}^n(\frac{1}{2}+a_{kr})e_r)\otimes e_k\\ & =e_k\otimes (\sum _{r=1}^n(\frac{1}{2}+a_{kr})e_r)+(\sum _{r=1}^n(\frac{1}{2}+a_{kr})e_r)\otimes e_k\\ & =e_k\otimes T_k+T_k\otimes e_k.\end{array}$Hence the comultiplication in $\mathcal{C}$ is therefore given by the skew-symmetric matrix A defining the elements T_k's.

We note that, as defined in Theorem 3.3, $(\mathcal{C},\mathrm{\Delta })$ satisfies LVC-1 and LVC-2, but not LVC-3. Indeed we have $\sum _{i,j=1}^n{\beta }_{ij}^k=n$ for all k = 1, …, n, that is, comultiplication of the basis elements is not normalized.

Example 3.4

Let $\mathcal{A}$ be a 3-dimensional $\mathbb{K}$-vector space with basis $\mathcal{B}=\{e_1,e_2,e_3\}$. We define the following multiplication in $\mathcal{A}$: $\begin{array}{rl}& e_i{e}_i=e_i,i=1,2,3,\\ & e_1{e}_2=e_2{e}_1=e_2,\\ & e_1{e}_3=e_3{e}_1=\frac{1}{2}{e}_1+\frac{1}{2}{e}_3,\\ & e_2{e}_3=e_3{e}_2=e_2;\end{array}$and extend it by linearity to $m:\mathcal{A}\otimes \mathcal{A}\to \mathcal{A}$. Let us denote $\mathbf{M}\in M_{3\times 9}(\mathbb{K})$ the matrix associated to the multiplication m (as a linear map) in bases ${\mathcal{B}}^{\prime }=\{e_{ij}=e_i\otimes e_j{\}}_{i,j=1}^3$ of $\mathcal{A}\otimes \mathcal{A}$ and $\mathcal{B}$ of $\mathcal{A}$. Writing ${\mathcal{B}}^{\prime }=\{b_i{\}}_{i=1}^9=\{b_1=e_1\otimes e_1,b_2=e_1\otimes e_2,\dots ,b_9=e_3\otimes e_3\}$, we have $m(b_i)=\sum _{k=1}^3{m}_{ki}{e}_k$, for all i = 1, …, 9, and $\mathbf{M}=\left(\begin{array}{ccccccccc}1& 0& 1/2& 0& 0& 0& 1/2& 0& 0\\ 0& 1& 0& 1& 1& 1& 0& 1& 0\\ 0& 0& 1/2& 0& 0& 0& 1/2& 0& 1\end{array}\right).$Then $\mathrm{\Delta }:{\mathcal{A}}^{\ast }\to (\mathcal{A}\otimes \mathcal{A}{)}^{\ast }\cong {\mathcal{A}}^{\ast }\otimes {\mathcal{A}}^{\ast }$ defines a coalgebra structure on ${\mathcal{A}}^{\ast }$, with associated matrix (as a linear map) the transpose matrix M^T of M. Moreover $({\mathcal{A}}^{\ast },\mathrm{\Delta })$ satisfies LVC-1 and LVC-2 but not LVC-3, since columns of M^T are not normalized. Hence $({\mathcal{A}}^{\ast },\mathrm{\Delta })$ is not an LV-coalgebra. If, assuming then $char(\mathbb{K})\ne 5$, we column-normalize M^T, rows of the resulting matrix $\tilde{\mathbf{M}}=\left(\begin{array}{ccccccccc}1/2& 0& 1/4& 0& 0& 0& 1/4& 0& 0\\ 0& 1/5& 0& 1/5& 1/5& 1/5& 0& 1/5& 0\\ 0& 0& 1/4& 0& 0& 0& 1/4& 0& 1/2\end{array}\right)$i.e. columns of ${\tilde{\mathbf{M}}}^T$, define a new comultiplication $\overline{\mathrm{\Delta }}:{\mathcal{A}}^{\ast }\to {\mathcal{A}}^{\ast }\otimes {\mathcal{A}}^{\ast }$ given by: $\begin{array}{rl}& \overline{\mathrm{\Delta }}(f_1)=\frac{1}{2}{f}_1\otimes f_1+\frac{1}{4}(f_1\otimes f_3+f_3\otimes f_1),\\ & \overline{\mathrm{\Delta }}(f_2)=\frac{1}{5}{f}_2\otimes f_2+\frac{1}{5}(f_1\otimes f_2+f_2\otimes f_1+f_2\otimes f_3+f_3\otimes f_2),\\ & \overline{\mathrm{\Delta }}(f_3)=\frac{1}{2}{f}_3\otimes f_3+\frac{1}{4}(f_1\otimes f_3+f_3\otimes f_1),\end{array}$also satisfying LVC-3. Hence $({\mathcal{A}}^{\ast },\overline{\mathrm{\Delta }})$ is an LV-coalgebra.

4. Algebraic properties of Lotka–Volterra coalgebras

Cocommutativity of LV-coalgebras follows from LVC-2. Recall a coalgebra $\mathcal{C}$ is cocommutative if τΔ = Δ, where $\tau :\mathcal{C}\otimes \mathcal{C}\to \mathcal{C}\otimes \mathcal{C}$ is the twist map given by τ(a ⊗ b) = b ⊗ a for all $a,b\in \mathcal{C}$. LV-coalgebras are however not necessarily coassociative.

Example 4.1

Let $\mathcal{C}$ be a 2-dimensional coalgebra with comultiplication (with respect to a basis $\mathcal{B}=\{e_1,e_2\}$): $\begin{array}{rl}\mathrm{\Delta }(e_1)& =e_1\otimes e_1,\\ \mathrm{\Delta }(e_2)& =\frac{1}{2}{e}_2\otimes e_2+\frac{1}{4}(e_1\otimes e_2+e_2\otimes e_1).\end{array}$Clearly $(\mathcal{C},\mathrm{\Delta })$ is an LV-coalgebra and (id ⊗ Δ)Δ(e₂) ≠ (Δ ⊗ id)Δ(e₂). Hence $\mathcal{C}$ is not coassociative.

Definition 4.2

A coalgebra $\mathcal{C}$ is counital if it exists a linear map (counit) $\epsilon :\mathcal{C}\to \mathbb{K}$ such that $(\epsilon \otimes id)\mathrm{\Delta }=id=(id\otimes \epsilon )\mathrm{\Delta }$.

Proposition 4.3

Let $\mathcal{C}$ be an LV-coalgebra. Then $\mathcal{C}$ is counital if and only if there exists a nonzero $\mathbf{a}=({\gamma }_1,\dots ,{\gamma }_n{)}^T\in {\mathbb{K}}^n$ such that, for all k = 1, …, n, (1) $\begin{array}{rl}& \sum _{i=1}^n{\beta }_{ik}^k{\gamma }_i=1,\end{array}$(1) (2) $\begin{array}{rl}& {\gamma }_k(1-{\beta }_{kk}^k)=0.\end{array}$(2)

Proof.

Assume $\epsilon :\mathcal{C}\to \mathbb{K}$ is a (nonzero) counit for $\mathcal{C}$ and write $\epsilon (e_i)={\gamma }_i$, i = 1, …, n. Then, by LVC-2, we have: $\begin{array}{rl}{e}_k& =(id\otimes \epsilon )\mathrm{\Delta }(e_k)=(\epsilon \otimes id)\mathrm{\Delta }(e_k)\\ & =(\epsilon \otimes id)({\beta }_{kk}^k{e}_k\otimes e_k+\sum _{i=1,i\ne k}^n{\beta }_{ik}^k(e_i\otimes e_k+e_k\otimes e_i))\\ & ={\beta }_{kk}^k{\gamma }_k{e}_k+\sum _{i=1,i\ne k}^n{\beta }_{ik}^k({\gamma }_i{e}_k+{\gamma }_k{e}_i)\\ & =\left(\sum _{i=1}^n{\beta }_{ik}^k{\gamma }_i\right)e_k+\sum _{i=1,i\ne k}^n{\beta }_{ik}^k{\gamma }_k{e}_i.\end{array}$Thus, ε is a counit of $\mathcal{C}$ if and only if, for all k = 1, …, n it holds that $\sum _{i=1}^n{\beta }_{ik}^k{\gamma }_i=1,\text{and}{\gamma }_k\sum _{i=1,i\ne k}^n{\beta }_{ik}^k=0;$or equivalently, by LVC-3, $\sum _{i=1}^n{\beta }_{ik}^k{\gamma }_i=1,\text{and}{\gamma }_k(1-{\beta }_{kk}^k)=0.$Conversely, any such nonzero $\mathbf{a}=({\gamma }_1,\dots ,{\gamma }_n{)}^T\in {\mathbb{K}}^n$ defines a (nonzero) counit for $\mathcal{C}$.

Example 4.4

1. Let $\mathcal{C}=\mathbb{K}G$ be a trivial LV-coalgebra, i.e. a group coalgebra. Since then we have ${\beta }_{ij}^k={\delta }_{ki}{\delta }_{kj}$ for all i, j, k = 1, …, n, ${\gamma }_k(1-{\beta }_{kk}^k)=0$ holds straightforwardly for all k, and $\sum _{i=1}^n{\beta }_{ik}^k{\gamma }_i=1$ becomes ${\beta }_{kk}^k{\gamma }_k=1$. Thus, by Proposition 4.3, $\mathcal{C}=\mathbb{K}G$ has a unique counit defined by $\epsilon (e_k)=1$, for all k = 1, …, n.

2. The 2-dimensional LV-coalgebra defined in Example 4.1 admits no nonzero counit.

It follows from Example 4.4(ii) above that having natural basis elements that are group-like is not enough for ensuring LV-coalgebras to be counital. However, having a nonzero counit is a sufficient condition for the existence of group-like elements (i.e. elements $x\in \mathcal{C}$ such that Δ(x) = x ⊗ x) in the case of LV-coalgebras with genetic realization.

Theorem 4.5

Let $\mathcal{C}$ be an LV-coalgebra with genetic realization and natural basis $\mathcal{B}=\{e_1,\dots ,e_n\}$. If $\mathcal{C}$ has a nontrivial counit ε, then $D=spa{n}_{\mathbb{R}}(e_k\mid k\in \mathrm{\Lambda })$, where $\mathrm{\Lambda }=\{k\mid \epsilon (e_k)={\gamma }_k\ne 0\}$ is a (group) subcoalgebra of $\mathcal{C}$. Moreover γ_k = 1 for all k ∈ Λ.

Proof.

Let ε be a nonzero counit for $\mathcal{C}$. Then $\mathrm{\Lambda }=\{k\mid \epsilon (e_k)={\gamma }_k\ne 0\}\ne \mathrm{\varnothing }$. Take k₀ ∈ Λ. Then, by Proposition 4.3(2), we have ${\gamma }_{k_0}(1-{\beta }_{k_0{k}_0}^{k_0})=0$, which implies ${\beta }_{k_0{k}_0}^{k_0}=1$ and, therefore, by LVC-4, ${\beta }_{rs}^{k_0}={\delta }_{k_{0}r}{\delta }_{k_{0}s}$, for all r, s = 1, …, n. Thus $\mathrm{\Delta }(e_{k_0})=e_{k_0}\otimes e_{k_0}$ and $e_{k_0}$ is a group-like element, and, as a result, $D=spa{n}_{\mathbb{R}}(e_k\mid k\in \mathrm{\Lambda })$ is a subcoalgebra of $\mathcal{C}$. The last assertion, γ_k = 1 for all k ∈ Λ, follows from Proposition 4.3(1).

Definition 4.6

A coalgebra $(\mathcal{C},\mathrm{\Delta })$ is baric if there exists a nonzero linear map $\varphi :\mathcal{C}\to \mathbb{K}$ such that (ϕ ⊗ ϕ)Δ = ϕ. The map ϕ is called character or weight function.

Any character ϕ is an element of ${\mathcal{C}}^{\ast }=Ho{m}_{\mathbb{K}}(\mathcal{C},\mathbb{K})$ and therefore can be uniquely written $\varphi =\sum _{i=1}^n{\alpha }_i{e}_i^{\ast }$, for some $\mathbf{a}=({\alpha }_1,\dots ,{\alpha }_n{)}^T\in {\mathbb{K}}^n$.

Proposition 4.7

Let $\mathcal{C}$ be an LV-coalgebra. A linear map $\varphi =\sum _{i=1}^n{\alpha }_i{e}_i^{\ast }\in {\mathcal{C}}^{\ast }$ is a character of $\mathcal{C}$ if and only if (3) ${\alpha }_k={\alpha }_k({\beta }_{kk}^k{\alpha }_k+2\sum _{i=1,i\ne k}^n{\beta }_{ik}^k{\alpha }_i),\mathrm{for}\mathrm{all}k=1,\dots ,n.$(3)

Proof.

Let $\varphi =\sum _{i=1}^n{\alpha }_i{e}_i^{\ast }\in {\mathcal{C}}^{\ast }$. Then ϕ is a character of $\mathcal{C}$ if and only if, for all $e_k\in \mathcal{B}$ $\begin{array}{rl}{\alpha }_k& =\varphi (e_k)=(\varphi \otimes \varphi )\mathrm{\Delta }(e_k)\\ & =(\varphi \otimes \varphi )({\beta }_{kk}^k{e}_k\otimes e_k+\sum _{i=1,i\ne k}^n{\beta }_{ik}^k(e_i\otimes e_k+e_k\otimes e_i))\\ & ={\beta }_{kk}^k{\alpha }_k^2+2\sum _{i=1,i\ne k}^n{\beta }_{ik}^k{\alpha }_i{\alpha }_k.\end{array}$Hence, ϕ is a character if and only if for all k = 1, …, n, ${\alpha }_k={\alpha }_k({\beta }_{kk}^k{\alpha }_k+2\sum _{i=1,i\ne k}^n{\beta }_{ik}^k{\alpha }_i).$

Corollary 4.8

The linear map $\varphi =\sum _{i=1}^n{e}_i^{\ast }$ is a character of any LV-coalgebra $\mathcal{C}$.

Proof.

Let a = (α₁, …, α_n)^T = 1. By Proposition 4.7, ϕ is a character if and only if $1={\beta }_{kk}^k+2\sum _{i=1,i\ne k}^n{\beta }_{ik}^k\text{for all}k=1,\dots ,n$which follows straightforwardly for any LV-coalgebra $\mathcal{C}$. (See also [8, Theorem 4.3].)

Group-like elements of LV-coalgebras behave, with respect to weight maps (or characters), as do idempotent elements in genetic algebras [7, Lemma 4.1, p.65]. Indeed from the biological viewpoint, the existence of group-like elements can be understood as equilibrium states for the underlying biological system.

Proposition 4.9

Let ϕ be a character of an LV-coalgebra $\mathcal{C}$. For any group-like element $x\in \mathcal{C}$, either ϕ(x) = 0 or ϕ(x) = 1.

Proof.

Let $x\in \mathcal{C}$ be group-like. Then ϕ(x) = (ϕ ⊗ ϕ)Δ(x) = (ϕ ⊗ ϕ)(x ⊗ x) = ϕ(x)². Hence either ϕ(x) = 0 or ϕ(x) = 1.

Corollary 4.10

Characters of LV-coalgebras, if there exist, are not necessarily unique.

Proof.

Let $\mathcal{C}$ be the LV-coalgebra introduced in Example 3.4. The cubic matrix P associated to $\mathcal{C}$ unfolded by its frontal slices is $\mathbf{P}=\left(\begin{array}{ccccccccc}\frac{1}{2}& 0& \frac{1}{4}& 0& \frac{1}{5}& 0& 0& 0& \frac{1}{4}\\ 0& 0& 0& \frac{1}{5}& \frac{1}{5}& \frac{1}{5}& 0& 0& 0\\ \frac{1}{4}& 0& 0& 0& \frac{1}{5}& 0& \frac{1}{4}& 0& \frac{1}{2}\end{array}\right)$and Proposition 4.7(3) gives that $\varphi =\sum _{i=1}^n{\alpha }_i{e}_i^{\ast }$ is a (nonzero) character of $\mathcal{C}$ if and only if a = (α₁, α₂, α₃)^T is of the form: $\mathbf{a}=\{\begin{array}{ll}(\alpha ,0,2-\alpha ),& \alpha \in \mathbb{K};\\ (\alpha ,1,2-\alpha ),& \alpha \in \mathbb{K};\\ (0,5,0).& \end{array}$Thus the LV-coalgebra $\mathcal{C}$ defined in Example 3.4 has two 1-parameter families of characters together to one additional nontrivial character.

5. Constructions of LV-coalgebras

Achieving the construction of general objects enclosing simultaneously different (genetic) features has been a recurrent topic in the literature of genetic algebraic structures. See, for instance, [7, p.103] or [8, Section 5]. Here we consider different constructions of coalgebras preserving the property of being Lotka–Volterra.

Following [8, Subsection 5.1], given a n-dimensional $\mathbb{K}$-vector space V we denote by ${\mathcal{C}}_0(V)$ the set of all coalgebra structures defined on V or, equivalently, the set of all linear maps Δ: V → V ⊗ V. The set ${\mathcal{C}}_0(V)$ was shown to be a vector space in [8, Proposition 5.1].

Let $p\in \mathbb{N}$ and $\mathcal{B}=\{e_1,\dots ,e_n\}$ be an (ordered) basis of V. Given elements ${\mathrm{\Delta }}_1,\dots ,{\mathrm{\Delta }}_p\in {\mathcal{C}}_0(V)$, with ${\mathrm{\Delta }}_t(e_k)=\sum _{i,j=1}^n{\beta }_{ij}^{t,k}{e}_i\otimes e_{j}k=1,\dots ,n,t=1,\dots ,p;$and $({\alpha }_1,\dots ,{\alpha }_p{)}^T\in {\mathbb{K}}^p$, we write $\mathrm{\Delta }=\sum _{t=1}^p{\alpha }_t{\mathrm{\Delta }}_t$. Then $\mathrm{\Delta }(e_k)=\sum _{i,j=1}^n{\beta }_{ij}^k{e}_i\otimes e_{j}k=1,\dots ,n,$where for all i, j, k = 1, …, n, we have ${\beta }_{ij}^k=\sum _{t=1}^p{\alpha }_t{\beta }_{ij}^{t,k}$.

Theorem 5.1

Convex linear combinations of real LV-coalgebras are Lotka–Volterra.

Proof.

Let $\{{\mathcal{C}}_t=(V,{\mathrm{\Delta }}_t){\}}_{t=1}^p$ be real LV-coalgebras and $({\alpha }_1,\dots ,{\alpha }_p{)}^T\in {\mathbb{R}}^p$ be nonnegative and such that $\sum _{t=1}^p{\alpha }_t=1$. We claim that $\mathcal{C}=(V,\mathrm{\Delta })$, with $\mathrm{\Delta }=\sum _{t=1}^p{\alpha }_t{\mathrm{\Delta }}_t$, is also Lotka–Volterra. Indeed LVC-1 and LVC-2 follow straightforwardly from being each ${\mathcal{C}}_t$ LV-coalgebra, and LVC-3 follows from the Cauchy–Schwarz inequality.

Corollary 5.2

Any convex linear combination of real LV-coalgebras with genetic realization has genetic realization.

Proof.

LVC-4 follows from the nonnegativity of $({\alpha }_1,\dots ,{\alpha }_p{)}^T\in {\mathbb{R}}^p$.

Remark 5.1

The direct sum $\mathcal{C}=\underset{t=1}{\overset{p}{⨁}}{\mathcal{C}}_t$ of any finite family of coalgebras $\{{\mathcal{C}}_t=(V_t,{\mathrm{\Delta }}_t){\}}_{t=1}^p$ has a coalgebra structure resulting from $\mathcal{C}\otimes \mathcal{C}$ being isomorphic to $\underset{r,s=1}{\overset{p}{⨁}}({\mathcal{C}}_r\otimes {\mathcal{C}}_s)$ and, therefore, containing a copy of $\underset{t=1}{\overset{p}{⨁}}({\mathcal{C}}_t\otimes {\mathcal{C}}_t)$ [9, p.50].

Let $V=\underset{t=1}{\overset{p}{⨁}}{V}_t$, with V_t being a n_t-dimensional $\mathbb{K}$-vector space with basis ${\mathcal{B}}_t=\{e_{t,1},\dots e_{t,n_t}\}$. If ${\mathcal{C}}_t=(V_t,{\mathrm{\Delta }}_t)$ is a coalgebra, then Δ:V → V ⊗ V, given by Δ(x_t) = Δ_t(x_t), for all x_t ∈ V_t, defines a coalgebra structure on V. We write $\mathcal{B}=\{e_{t,k}{\}}_{\begin{array}{c}1\le k\le n_t\\ 1\le t\le p\end{array}}$

Theorem 5.3

Let $\{{\mathcal{C}}_t=(V_t,{\mathrm{\Delta }}_t){\}}_{t=1}^p$ be LV-coalgebras. Then, the direct sum $\mathcal{C}=(V=\underset{t=1}{\overset{p}{⨁}}{V}_t,\mathrm{\Delta })$ is an LV-coalgebra. If, moreover, each ${\mathcal{C}}_t$ is a real LV-coalgebra with genetic realization, so is $\mathcal{C}$.

Proof.

Write, for k = 1, …, n_t and t = 1, …, p, $\mathrm{\Delta }(e_{t,k})=\sum _{\begin{array}{c}r,s=1\\ 1\le i\le n_r\\ 1\le j\le n_s\end{array}}^p{\beta }_{r,i;s,j}^{t,k}{e}_{r,i}\otimes e_{s,t}.$On the other hand, $\mathrm{\Delta }(e_{t,k})={\mathrm{\Delta }}_t(e_{t,k})=\sum _{i,j=1}^{n_t}{\beta }_{ij}^{t,k}{e}_{t,i}\otimes e_{t,j}$. Thus we have ${\beta }_{r,i;s,j}^{t,k}=\{\begin{array}{ll}{\beta }_{ij}^{t,k},& r=s=t;\\ 0,& \text{otherwise.}\end{array}$Assume now ${\mathcal{C}}_t=(V_t,{\mathrm{\Delta }}_t)$ is an LV-coalgebra, for all t = 1, …, p. Then $(\mathcal{C},\mathrm{\Delta })$ clearly satisfies LVC-1 and LVC-2, whereas LVC-3 follows from the fact that Δ(x_t) = Δ_t(x_t), for all x_t ∈ V_t. The last statement about the genetic realization (i.e. $\mathcal{C}$ satisfying LVC-4) is clear.

The family of LV-coalgebras is therefore closed under convex linear combinations (when $\mathbb{K}=\mathbb{R}$) and direct sums, preserving also both constructions the genetic realization (i.e. LVC-4). LV-coalgebras are not however closed under tensor products or the cocommutative duplication.

Example 5.4

Let $(\mathcal{C},\mathrm{\Delta })$ be a coalgebra. It is a well-known result that the ($\mathbb{K}$-vector space) $\mathcal{C}\otimes \mathcal{C}$ admits a comultiplication $\overline{\mathrm{\Delta }}=(id\otimes \tau \otimes id)(\mathrm{\Delta }\otimes \mathrm{\Delta })$, where τ denotes the twist map, endowing $\mathcal{C}\otimes \mathcal{C}$ with a coalgebra structure [9, p.49].

Consider now the 2-dimensional coalgebra $(\mathcal{C},\mathrm{\Delta })$ with basis $\mathcal{B}=\{e_1,e_2\}$ and comultiplication: $\begin{array}{rl}\mathrm{\Delta }(e_1)& =\frac{1}{2}{e}_1\otimes e_1+\frac{1}{4}(e_1\otimes e_2+e_2\otimes e_1),\\ \mathrm{\Delta }(e_2)& =\frac{1}{2}{e}_2\otimes e_2+\frac{1}{4}(e_1\otimes e_2+e_2\otimes e_1).\end{array}$Clearly $\mathcal{C}$ is Lotka–Volterra with respect to $\mathcal{B}$, but it is not difficult to check that $(\mathcal{C}\otimes \mathcal{C},\overline{\mathrm{\Delta }})$ is not an LV-coalgebra (with respect to the basis $\overline{\mathcal{B}}=\{e_{ij}=e_i\otimes e_j{\}}_{i,j=1}^2$).

Finally let $(\mathcal{C}\vee \mathcal{C},\mathrm{\nabla })$ be the cocommutative duplication of the (cocommutative) coalgebra $\mathcal{C}$ [8, Definition 3.1], that is, $\mathcal{C}\vee \mathcal{C}=\mathcal{C}\otimes \mathcal{C}/\mathrm{\Sigma }$, where $\mathrm{\Sigma }=\{\sum _{i\in I}(x_i\otimes y_i-y_i\otimes x_i)\mid x_i,y_i\in \mathcal{C},i\in I,\mid I\mid <\mathrm{\infty }\},$is the symmetric tensor product of the (cocommutative) coalgebra $\mathcal{C}$. Recall $\mathcal{C}\vee \mathcal{C}$ is a cocommutative coalgebra with comultiplication (using Sweedler notation [9, p.32]) $\mathrm{\nabla }(x\vee y)=\sum _{(x),(y)}(x_{(1)}\vee y_{(1)})\otimes (x_{(2)}\vee y_{(2)}).$Then $\mathcal{C}\vee \mathcal{C}$ is a 3-dimensional coalgebra with basis ${\mathcal{B}}^{\prime }=\{e_1\vee e_1,e_1\vee e_2,e_2\vee e_2\}$ and, it holds that $\begin{array}{rl}\mathrm{\nabla }(e_1\vee e_1)& =\frac{1}{4}(e_1\vee e_1)\otimes (e_1\vee e_1)\\ & +\frac{1}{4}((e_1\vee e_1)\otimes (e_1\vee e_2)+(e_1\vee e_2)\otimes (e_1\vee e_1))\\ & +\frac{1}{16}((e_1\vee e_1)\otimes (e_2\vee e_2)+(e_2\vee e_2)\otimes (e_1\vee e_1))\\ & +\frac{1}{16}((e_1\vee e_2)\otimes (e_1\vee e_2)+(e_1\vee e_2)\otimes (e_1\vee e_2)).\end{array}$Hence $(\mathcal{C}\vee \mathcal{C},\mathrm{\nabla })$ is not Lotka–Volterra (with respect to ${\mathcal{B}}^{\prime }$).

6. Structure of LV-coalgebras

The different algebraic properties of the generators (i.e. natural basis elements) of algebraic objects with genetic significance usually reproduce their different behaviour within the underlying genetic system, when considered from the dynamical viewpoint. The classification of a natural basis elements into algebraically persistent and transient generators considered in [6, Subsection 3.4.2] led Tian to consider semi-direct sum decompositions of evolution algebras. Here we recover the notion of semi-direct sum decomposition, and apply it to Lotka–Volterra coalgebras.

Let $\mathcal{C}$ be an LV-coalgebra with natural basis $\mathcal{B}=\{e_1,\dots ,e_n\}$. We write [n] = {1, …, n} and $I_k=\mathbb{K}{e}_k$, for all k ∈ [n].

Proposition 6.1

Let k ∈ [n]. If I_k is a subcoalgebra of $\mathcal{C}$, then e_k is a group-like element. Otherwise I_k is a coideal.

Proof.

Take k ∈ [n] and assume I_k is a subcoalgebra of $\mathcal{C}$. This implies $\mathrm{\Delta }(e_k)={\beta }_{kk}^k{e}_k\otimes e_k$ and, therefore, by LVC-3, ${\beta }_{kk}^k=1$. Hence e_k is group-like. Assume otherwise, e_k is not group-like. Then ${\beta }_{ik}^k\ne 0$ for some i ≠ k, and $\begin{array}{rl}\mathrm{\Delta }(e_k)& ={\beta }_{kk}^k{e}_k\otimes e_k+\sum _{i=1,i\ne k}^k{\beta }_{ik}^k(e_i\otimes e_k+e_k\otimes e_i)\\ & =e_k\otimes \left(\sum _{i=1}^n{\beta }_{ik}^k{e}_i\right)+\left(\sum _{i=1,i\ne k}^n{\beta }_{ik}^k{e}_i\right)\otimes e_k.\end{array}$Therefore $\mathrm{\Delta }(e_k)\in I_k\otimes \mathcal{C}+\mathcal{C}\otimes I_k$. Hence I_k is a coideal.

Let us write $[n]=\mathrm{\Lambda }⨆{\mathrm{\Lambda }}_I$ (disjoint union), where $\mathrm{\Lambda }=\{k\in [n]\mid e_k\mathrm{is}\mathrm{group}-\mathrm{like}\}$ and, ${\mathrm{\Lambda }}_I=[n]\setminus \mathrm{\Lambda }$. Obviously $\mathrm{\Lambda }=\mathrm{\varnothing }$ or ${\mathrm{\Lambda }}_I=\mathrm{\varnothing }$ remains possible. Sufficient conditions for $\mathrm{\Lambda }\ne \mathrm{\varnothing }$ are given in Theorem 4.5.

Lemma 6.2

• D = ⊕_k∈Λ I_k is a subcoalgebra of $\mathcal{C}$.

• $I=\underset{k\in {\mathrm{\Lambda }}_I}{⨁}{I}_k$ is a subcoideal of $\mathcal{C}$.

Proof.

It follows from [9, p.18].

Theorem 6.3

Let $\mathcal{C}$ be an LV-coalgebra. Then $\mathcal{C}=D\dotplus I$, where $\dotplus$ denotes the direct sum of vector subspaces.

Proof.

Clear.

Remark 6.1

Following [6, Theorem 11] we will call such an LV-coalgebra decomposition to be a semidirect sum decomposition of $\mathcal{C}$. In such a decomposition (co)algebraic persistent elements can be identified to group-like elements, being (co)algebraic transient otherwise.

The dynamical behaviour of LV-coalgebras basis elements becomes apparent when considering their accompanying matrices. Write r = |Λ| and t = n−r. Then, by Theorem 6.3, the accompanying matrices of P have the following form (see Proposition 2.6):

Corollary 6.4

Reordering the natural basis elements, if necessary, so that, $\mathcal{B}=\{e_1,\dots ,e_r,e_{r+1},\dots ,e_n\}$, being e_k group-like, for k = 1, …, r: ${\mathbf{P}}_{(i)}={\mathbf{P}}_{(j)}=\left(\begin{array}{cc}{\mathbf{I}}_{r\times r}& {\mathbf{A}}_{r\times t}\\ {\mathbf{0}}_{t\times r}& {\mathbf{B}}_{t\times t}\end{array}\right)$

Proof.

It suffices to note that for group-like elements, i.e. for all k = 1, …, r, it holds that ${\beta }_{ij}^k={\delta }_{ik}{\delta }_{jk}$, i, j = 1, …, n and, therefore, $\sum _{i=1}^n{\beta }_{ik}^k={\beta }_{kk}^k=1$. On the other hand, by Theorem 6.3, for all j = 1, …, t, there exists some i = 1, …, r, such that $a_{ij}={\beta }_{i,r+j}^{r+j}\ne 0$.

6.1. Further comments

Under the additional assumption of being endowed with genetic realization (i.e. LVC-4), accompanying matrices of LV-coalgebras evidence the connection between these new family of coalgebras and dynamical systems. Indeed, then P_(i) turns out to be not only nonnegative, by LVC-4, but also column stochastic as a result of LVC-3. This settles a new connection between LV-coalgebras and Markov processes, where the topology of P_(i) classifying the matrix indices into essential and inessential classes translates into the (co)algebraic persistency and transiency of the natural basis elements. It is a forthcoming work to provide a detailed approach to the ergodic behaviour of LV-coalgebras from the viewpoint of Markov chains.

Acknowledgments

This paper was written while the third author was visiting the Department of Mathematics of the University of Chile, supported by FONDECYT 1170547, and wants to thank the members of the department there for their hospitality. The authors want to thank the referee for his/her comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors are supported by Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) grant 1170547. The third author is also supported by MTM2017-83506-C2-1-P (AEI/FEDER, UE).