It's not really funny to calculate the exponentials of ##(3\times 3)-##matrices and easy to make mistakes. I guess, that's why I haven't really seen it anywhere. I would start with ##(2\times 2)-##matrices and the Lie group of matrices of the form ##\begin{bmatrix}a&b\\0&0\end{bmatrix}## with ##a \neq 0## and the Lie algebra ##[X,Y]=Y## which is easier than with simple Lie groups which have entries on both sides of the diagonal. It's also the reason why I used the differentiation of curves in the Lie group to show the connection to its Lie algebra. Exponentiation is basically the way back, an integration.nigelscott said:When I try to calculate the matrix exponential I get strange results. Any pointers you could give would be of great help.
The adjoint representation of SU(2) is a mathematical concept used in the study of group theory and quantum mechanics. It refers to the linear transformation that maps the group elements of SU(2) onto themselves.
The adjoint representation of SU(2) is closely related to the Lie algebra of the group. It is a faithful representation, meaning that it preserves the structure and properties of the group. In other words, the Lie algebra can be derived from the adjoint representation.
The adjoint representation of SU(2) has many applications in physics, particularly in the study of quantum mechanics and particle physics. It is used to describe the symmetries and transformations of physical systems, such as the spin of particles.
The fundamental representation of SU(2) is a 2-dimensional representation that describes the spin-1/2 particles. In contrast, the adjoint representation is a 3-dimensional representation that describes the interactions between particles and their symmetries.
Yes, the concept of the adjoint representation can be extended to other groups, such as SU(N) and SO(N). However, the specific properties and equations may differ for each group, as they have different Lie algebras and representations.