Quaternionic representation

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(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 : \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.

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(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 \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\colon V\to V which satisfies 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. \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 A 4k+1 , B 4k+1 , B 4k+2 , C k , D 4k+2 , and E 7 . ==References== . . ==See also== Symplectic vector space

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