Alexander–Spanier cohomology: Difference between revisions
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In [[mathematics|mathematical]] field of [[representation theory]], a '''quaternionic representation''' is a [[group representation|representation]] on a [[complex number|complex]] vector space ''V'' with an invariant [[quaternionic structure]], i.e., an [[antilinear]] [[equivariant map]] | |||
:<math>j\colon V\to V\,</math> | |||
which satisfies | |||
:<math>j^2=-1.\,</math> | |||
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 (mathematics)|module]] over the [[division algebra]] of [[quaternion]]s). From this point of view, quaternionic representation of a [[group (mathematics)|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 | |||
:<math>\rho(gh)=\rho(g)\rho(h)\text{ for all }g, h \in G.\,</math> | |||
Quaternionic representations of [[associative algebra|associative]] and [[Lie algebra]]s 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 representation]]s, can be picked out by the [[Frobenius-Schur indicator]]. | |||
Quaternionic representations are similar to [[real representation]]s 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]] | |||
:<math>j\colon V\to V\,</math> | |||
which satisfies | |||
:<math>j^2=+1.\,</math> | |||
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 [[rotation]]s 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 | |||
:<math>\rho(g)^\dagger \rho(g)=\mathbf{1}\,</math> | |||
for all ''g'' in Spin(3). | |||
Another unitary example is the [[spin representation]] of Spin(5). An example of a nonunitary 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 + 8''k'', 4 + 8''k'', and 5 + 8''k'' dimensions, where ''k'' is an integer. In physics, one often encounters the [[spinor]]s 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''<sub>4''k''+1</sub>, ''B''<sub>4''k''+1</sub>, ''B''<sub>4''k''+2</sub>, ''C''<sub>''k''</sub>, ''D''<sub>4''k''+2</sub>, and ''E''<sub>7</sub>. | |||
==References== | |||
*{{Fulton-Harris}}. | |||
*{{citation |first=Jean-Pierre|last= Serre | title=Linear Representations of Finite Groups | publisher=Springer-Verlag | year=1977 | isbn= 0-387-90190-6}}. | |||
==See also== | |||
* [[Symplectic vector space]] | |||
[[Category:Representation theory]] | |||
Revision as of 08:52, 12 January 2014
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.
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[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
for all g in Spin(3).
Another unitary example is the spin representation of Spin(5). An example of a nonunitary 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
- Template:Fulton-Harris.
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