In
mathematics, if is a
group and is a
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
of it over the
complex vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
, then the complex conjugate representation is defined over the
complex conjugate vector space
In mathematics, the complex conjugate of a complex vector space V\, is a complex vector space \overline V, which has the same elements and additive group structure as V, but whose scalar multiplication involves conjugation of the scalars. In oth ...
as follows:
: is the
conjugate of for all in .
is also a representation, as one may check explicitly.
If is a
real Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
and is a representation of it over the vector space , then the conjugate representation is defined over the conjugate vector space as follows:
: is the conjugate of for all in .
[This is the mathematicians' convention. Physicists use a different convention where the ]Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of two real vectors is an imaginary vector. In the physicist's convention, insert a minus in the definition.
is also a representation, as one may check explicitly.
If two real Lie algebras have the same
complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include t ...
, and we have a complex representation of the complexified Lie algebra, their conjugate representations are still going to be different. See
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
for some examples associated with spinor representations of the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As ...
s and .
If
is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),
: is the conjugate of for all in {{math, g
For a finite-dimensional
unitary representation 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 ' ...
, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.
See also
*
Dual representation
Notes
Representation theory of groups