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| dbo:abstract
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- In mathematics, if G is a group and Π is a representation of it over the complex vector space V, then the complex conjugate representation Π is defined over the complex conjugate vector space V as follows: Π(g) is the conjugate of Π(g) for all g in G. Π is also a representation, as one may check explicitly. If g is a real Lie algebra and π is a representation of it over the vector space V, then the conjugate representation π is defined over the conjugate vector space V as follows: π(X) is the conjugate of π(X) for all X in g. π is also a representation, as one may check explicitly. If two real Lie algebras have the same complexification, and we have a complex representation of the complexified Lie algebra, their conjugate representations are still going to be different. See spinor for some examples associated with spinor representations of the spin groups Spin(p + q) and Spin(p, q). If is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), π(X) is the conjugate of −π(X*) for all X in g For a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations. (en)
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| dbo:wikiPageExtracted
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- 2020-01-16 03:54:22Z (xsd:date)
- 2021-01-26 10:00:49Z (xsd:date)
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| dbo:wikiPageLength
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- 2248 (xsd:integer)
- 2257 (xsd:integer)
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| dbo:wikiPageModified
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- 2020-01-16 03:54:18Z (xsd:date)
- 2021-01-26 10:00:42Z (xsd:date)
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- 936010400 (xsd:integer)
- 1002855309 (xsd:integer)
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| rdfs:comment
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- In mathematics, if G is a group and Π is a representation of it over the complex vector space V, then the complex conjugate representation Π is defined over the complex conjugate vector space V as follows: Π(g) is the conjugate of Π(g) for all g in G. Π is also a representation, as one may check explicitly. If g is a real Lie algebra and π is a representation of it over the vector space V, then the conjugate representation π is defined over the conjugate vector space V as follows: π(X) is the conjugate of π(X) for all X in g. π is also a representation, as one may check explicitly. (en)
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| rdfs:label
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- Complex conjugate representation (en)
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