Nonassociative rings and algebras Definition

Evolution algebra

Builds on Nonassociative algebra Structure constants Field

Definition

Let $K$ be a field and let $A$ be a nonassociative algebra over $K$ . The algebra $A$ is an evolution algebra if it has a basis $(e_i)_{i \in I}$ such that:

$e_i e_j = 0$ for all $i, j \in I$ with $i \ne j$ .
For each $i \in I$ , $e_i^2 = \sum_{k \in I} a_{ik} e_k$ for scalars $a_{ik} \in K$ , only finitely many of them nonzero.

Such a basis is a natural basis, and the $a_{ik}$ are the structure constants of $A$ in it.

Remarks

Example: $A = K^2$ with basis $e_1, e_2$ , products $e_1 e_2 = e_2 e_1 = 0$ and $e_1^2 = e_2$ , $e_2^2 = e_1$ . Both clauses hold by construction. This algebra is not associative: $(e_1 e_1) e_2 = e_2^2 = e_1$ , while $e_1 (e_1 e_2) = 0$ .

Non-example: the algebra $M_2(K)$ of $2 \times 2$ matrices. In an evolution algebra the products of basis vectors satisfy $e_i e_j = e_j e_i$ , so by bilinearity the algebra is commutative. Matrix multiplication is not commutative, so no basis of $M_2(K)$ satisfies the clauses.

A real evolution algebra is called nonnegative when all its structure constants $a_{ik}$ are nonnegative.

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Tag	`2E4BS58WEV`
URI	`m2://2E4BS58WEV`
Kind	definition
Writing	W0 stub
MSC	17D92, 17D99
Defines	Evolution algebra
Uses	Nonassociative algebra, Structure constants, Field
Changed	3 November 2026

Evolution algebra

Definition

Remarks

Reading path

Further reading

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