This is a special case of non-Abelian gauge groups, which is especially relevant to characterize the fundamental forces for at least the standard model, and must set the motivation for alterations to it. SU(N) in its fundamental representation consists of a product of unitary N ⨉ N matrices with positive determinants, which are themselves associated with linear, special unitary transformations of a complex vector space. From previous entries, we should know both that SU(N) is a continuous Lie group, and the definitions of Lie groups. For the intuition of the following, Lie groups should be thought of as differentiable manifolds. Elements of a Lie group may be expanded about the unit element as
with traceless and Hermitian generators t, structure constants f, and some distance parameter ξ from unity. This distance may be represented by the exponential of iξt, meaning an infinite series of infinitesimal transformations. The generators span a vector space, with dimension N² - 1 for SU(N). In the following, we note the Lie algebra of a Lie group G as g.
SU(N) is semisimple insofar that its Lie algebra su(N) is a direct sum of simple Lie algebras. See the course on fundamentals of algebra for a more rigorous definition of simple algebras. Within physics, it's convenient to map the group action of SU(N) onto a space determined by the spin properties of a particle, using a representation of dimension 2J + 1 where J is the total spin (half integer, starting with 0). The Weyl theorem implies that this representation is completely reducible. For irreducible representations r of SU(N), their generators can be normalized down to a coefficient that is consistent with the structure constants.
In Yang-Mills theory, the standard representation of a Lie group is spanned by the generators, i.e. its own Lie algebra, so that the adjoint representation are given directly by the structure constants with maximal dimension.
with a representation-dependent constant C. For SU(2), the generators can be normalized into the 2D fundamental representations built through the 2-Pauli-matrices.
For later use, define the center of SU(N)
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