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Dual RepresentationIf G is a group and ρ is a representation of it over the vector space V, then the dual representation is defined over the dual vector space as follows: - is the transpose of ρ(g-1) for all g in G.
is also a representation, as you may check explicitly. If is a Lie algebra and ρ is a representation of it over the vector space V, then the dual representation is defined over the dual vector space as follows: - is the transpose of -ρ(u) for all u in .
is also a representation, as you may check explicitly. Unfortunately, a general ring module does not admit a dual representation. See also complex conjugate representation For a unitary representation, the conjugate representation and the dual representation coincides.
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