An axial algebra over the field $\mathbb F$ is a commutative algebra generated by idempotents whose adjoint action has multiplicity-free minimal polynomial. For semisimple associative algebras this leads to sums of copies of $\mathbb F$. Here we consider the first nonassociative case, where adjoint minimal polynomials divide $(x-1)x(x-\eta)$ for fixed $0\neq\eta\neq 1$. Jordan algebras arise when $\eta=\frac{1}{2}$, but our motivating examples are certain Griess algebras of vertex operator algebras and the related Majorana algebras. We study a class of algebras, including these, for which axial automorphisms like those defined by Miyamoto exist, and there classify the $2$-generated examples. For $\eta \neq \frac{1}{2}$ this implies that the Miyamoto involutions are $3$-transpositions, leading to a classification.Comment: 41 pages; comments welcom

We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category.We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions.We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples.We also take the opportunity to fix some terminology in this rapidly expanding subject.

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We apply diagram geometry and amalgam techniques to give a new proof of a theorem of K.-W. Phan, characterizing the special unitary group as a group generated by certain systems of subgroups SUð2; q 2 Þ.