In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

is defined over the dual vector space as follows: 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. Then is also a representation, as may be checked explicitly. The dual representation is also known as the contragredient representation.

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 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. Modules of Hopf algebras do, however. 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. ... In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ...

See also complex conjugate representation, Kirillov Character Formula. 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...