If 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: The term group can refer to several concepts: Look up Group in Wiktionary, the free dictionary In music, a group is another term for band or other musical ensemble. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

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: In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

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. A module is a self-contained component of a system, which has a well-defined interface to the other components; something is modular if it is constructed so as to facilitate easy assembly, flexible arrangement, and/or repair of the components. ...

See also complex conjugate representation If G is a group and ρ is a representation of it over the complex vector space V, then the complex conjugate representation ρ* is defined over the conjugate vector space V* as follows: ρ*(g) is the conjugate of ρ(g) for all g in G. ρ* is also a representation, as you may...

For a unitary representation, the conjugate representation and the dual representation coincides. In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group...