Vacuum solution (general relativity)

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. In mathematics, if G is a group and ρ is a linear representation of it on the real vector space V, then the dual representation Template:Overline is defined over the dual vector space Template:Overline as follows:

Template:Overline(g) is the transpose of ρ(g⁻¹)

for all g in G. Then Template:Overline is also a representation, as may be checked explicitly. The dual representation is also known as the contragredient representation. For a complex vector space, take ${\overline {\rho }}(g):=\rho (g)^{\dagger }$

If ${\mathfrak {g}}$ is a Lie algebra and ρ is a representation of it over the vector space V, then the dual representation Template:Overline is defined over the dual vector space Template:Overline as follows:

Template:Overline(u) is the transpose of −ρ(u) for all u in ${\mathfrak {g}}$.

Template:Overline is also a representation, as can be explicitly checked.

For a unitary representation, the conjugate representation and the dual representation coincide, up to equivalence of representations.

Generalization

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

See also

• Complex conjugate representation
• Kirillov Character Formula