nLab exceptional isomorphism

Idea

Given a sequence of structures following some pattern, one loosely calls isomorphisms that pertain to some special elements in the sequence, without themselves following that pattern, exceptional or sporadic.

Examples

Spin groups

The archetypical example is isomorphisms of spin groups. These appear in the infinite sequence $\big\{Spin(p) \vert p \in \mathbb{N}\big\}$, and for low values of $p$, but not generally, there are isomorphisms to other classical Lie groups.

See at spin group – Exceptional isomorphisms.

Finite linear groups

There is an exceptional isomorphisms

$PSL_2(\mathbb{F}_5) \;\simeq\; I$

between the projective special linear group over the prime field $\mathbb{F}_5$ with the icosahedral group.

See at Icosahedral group – Exceptional isomorphism to PSL(F5).

Related concepts

• exceptional Lie group

• exceptional Jordan algebra

• exceptional geometry, exceptional generalized geometry, exceptional field theory

• universal exceptionalism

References

• Paul Garrett, Sporadic isogenies to orthogonal groups (July 2013) [pdf, pdf ]