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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 : which satisfies : 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 : 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 : which satisfies : 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(). 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quaternionic representation」の詳細全文を読む スポンサード リンク
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