Dual representation

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In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation

[\bar]

is defined over the dual vector space [\bar] as follows:

[\bar(g)] is the transpose of ρ(g⁻¹)

for all g in G. Then [\bar] is also a representation, as may be checked explicitly. The dual representation is also known as the contragredient representation.

If [\mathfrak] is a Lie algebra and ρ is a representation of it over the vector space V, then the dual representation [\bar] is defined over the dual vector space [\bar] as follows:

[\bar(u)] is the transpose of −ρ(u) for all u in [\mathfrak].

[\bar] is also a representation, as you may check explicitly.

Unfortunately, a general ring module does not admit a dual representation. Modules of Hopf algebras do, however.

For a unitary representation, the conjugate representation and the dual representation coincides.

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