Question

A linear representation of a classical Lie group G is defined by rho:g→GL(V)

where g∈G is a group element and V is the representation space. The dimension of the representation space(which is a linear vector space) is usually called the dimension of the representation.

The adjoint representation Ad:G→GL(g)
is defined via conjugation of an algebra element A∈g
i.e via gAg−1
where g is the Lie algebra. So the dimension of the representation is the dimension of the algebra which is n²−1
for example for SU(n).

Consider the defining/standard representation which is defined via a inclusion map to GL(n,C)
. Then the dimension of the representation should be given by dimension of Cn≅R2n
which is 2n
. However we know that the fundamental representation and defining representation happens to be the same for classical Lie groups and fundamental representations are n
dimensional for SU(n),SO(n) and Sp(n)
. Why is this mismatch arising?

Answer 1

The mismatch in dimension occurring in the defining representation and fundamental representation of classical Lie groups arises due to the fact that the dimension of the representation space is not directly related to the dimension of the algebra.

Step-by-step explanation:

The mismatch in dimension occurring in the defining representation and fundamental representation of classical Lie groups arises due to the fact that the dimension of the representation space is not directly related to the dimension of the algebra. While the defining representation is realized in the complex vector space of dimension 2n, the fundamental representation is a real vector space of dimension n. The reason for this mismatch is rooted in the underlying mathematics of Lie groups and their representations.