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.

General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms We generalise the notion of separable equivalence, originally presented by Linckelmann in [13], to an equivalence relation on additive categories. We use this generalisation to show that from an initial equivalence between two algebras we may build equivalences between many related categories. We also show that separable equivalence preserves the representation type of an algebra. This generalises Linckelmann's result in [13], where he showed this in the case of symmetric algebras. We use these theorems to show that the group algebras of several small cyclic groups cannot be separably equivalent. This gives several examples of algebras that have the same complexity but are not separably equivalent.

Linckelmann showed in [Lin11] that a group algebra is separably equivalent to the group algebra of its Sylow p-subgroup. In this article we use this relationship, together with Mackey decomposition, to demonstrate that a group algebra of a group with an elementary abelian Sylow p-subgroup P, has representation dimension at most |P|.

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