• Source: Quaternionic representation

In 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
:
V
→
V


{\displaystyle j\colon V\to V}


which satisfies





j

2


=
−
1.


{\displaystyle 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(V, H), 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




ρ
(
g
h
)
=
ρ
(
g
)
ρ
(
h
)

for all

g
,
h
∈
G
.


{\displaystyle \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.




Properties and related concepts


If V is a unitary representation and the quaternionic structure j is a unitary operator, then V admits an invariant complex symplectic form ω, and hence is a symplectic representation. This always holds if V is a representation of a compact group (e.g. a finite group) and in this case quaternionic representations are also known as symplectic representations. Such representations, amongst irreducible representations, can be picked out by the Frobenius-Schur indicator.
Quaternionic representations are similar to real representations in that they are isomorphic to their complex conjugate representation. Here a real representation is taken to be a complex representation with an invariant real structure, i.e., an antilinear equivariant map




j
:
V
→
V


{\displaystyle j\colon V\to V}


which satisfies





j

2


=
+
1.


{\displaystyle j^{2}=+1.}


A representation which is isomorphic to its complex conjugate, but which is not a real representation, is sometimes called a pseudoreal representation.
Real and pseudoreal representations of a group G can be understood by viewing them as representations of the real group algebra R[G]. Such a representation will be a direct sum of central simple R-algebras, which, by the Artin-Wedderburn theorem, must be matrix algebras over the real numbers or the quaternions. Thus a real or pseudoreal representation is a direct sum of irreducible real representations and irreducible quaternionic representations. It is real if no quaternionic representations occur in the decomposition.




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




ρ
(
g

)

†


ρ
(
g
)
=

1



{\displaystyle \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, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103..
Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, Springer-Verlag, ISBN 978-0-387-90190-9.




See also


Symplectic vector space

Kata Kunci Pencarian:

• Kuaternion
• Quaternionic representation
• Real representation
• Frobenius–Schur indicator
• Quaternionic discrete series representation
• Complex representation
• Quaternionic matrix
• Classical group
• Symplectic representation
• Representation theory of finite groups
• List of representation theory topics