Dual representation

In mathematics, if $G$ is a group and $ρ$ is a linear representation of it on the vector space $V$ , then the dual representation $ρ*$ is defined over the dual vector space $V *$ as follows:

ρ*(g)

is the transpose of

ρ(g -1)

, that is,

ρ*(g)

=

ρ(g -1) T

for all

g \in G

.

The dual representation is also known as the contragradient representation.

If $g$ is a Lie algebra and $π$ is a representation of it on the vector space $V$ , then the dual representation $π*$ is defined over the dual vector space $V *$ as follows:

π*(X)

=

-π(X) T

for all

X \in g

.

The motivation for this definition is that Lie algebra representation associated to the dual of a Lie group representation is computed by the above formula. But the definition of the dual of a Lie algebra representation makes sense even if it does not come from a Lie group representation.

In both cases, the dual representation is a representation in the usual sense.

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