Quaternionic representation

{{Short description|Representation of a group or algebra in terms of an algebra with quaternionic structure}}

In the mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map

:j\colon V\to V

which satisfies

:j^2=-1.

Together with the imaginary unit i and the antilinear map k := ij, j equips V with the structure of a quaternionic vector space (i.e., V becomes a module over the division algebra of quaternions). From this point of view, quaternionic representation of a group G is a group homomorphism φ: G → GL(VH), the group of invertible quaternion-linear transformations of V. In particular, a quaternionic matrix representation of g assigns a square matrix of quaternions ρ(g) to each element g of G such that ρ(e) is the identity matrix and

:\rho(gh)=\rho(g)\rho(h)\text{ for all }g, h \in G.

Quaternionic representations of associative and Lie algebras can be defined in a similar way.

Examples

A common example involves the quaternionic representation of rotations in three dimensions. Each (proper) rotation is represented by a quaternion with unit norm. There is an obvious one-dimensional quaternionic vector space, namely the space H of quaternions themselves under left multiplication. By restricting this to the unit quaternions, we obtain a quaternionic representation of the spinor group Spin(3).

This representation ρ: Spin(3) → GL(1,H) also happens to be a unitary quaternionic representation because

:\rho(g)^\dagger \rho(g)=\mathbf{1}

for all g in Spin(3).

Another unitary example is the spin representation of Spin(5). An example of a non-unitary quaternionic representation would be the two dimensional irreducible representation of Spin(5,1).

More generally, the spin representations of Spin(d) are quaternionic when d equals 3 + 8k, 4 + 8k, and 5 + 8k dimensions, where k is an integer. In physics, one often encounters the spinors of Spin(d, 1). These representations have the same type of real or quaternionic structure as the spinors of Spin(d − 1).

Among the compact real forms of the simple Lie groups, irreducible quaternionic representations only exist for the Lie groups of type A4k+1, B4k+1, B4k+2, Ck, D4k+2, and E7.

References

  • {{Fulton-Harris}}.
  • {{citation | first=Jean-Pierre | last=Serre | title=Linear Representations of Finite Groups | publisher=Springer-Verlag | year=1977 | isbn=978-0-387-90190-9 | url-access=registration | url=https://archive.org/details/linearrepresenta1977serr }}.

See also