Fdtxjr

In many books on Yang-Mills theories, written from a physicist's point of view, the curvature tensor is written as:

$$F_mu nu^a = partial_mu A_nu^a-partial_nu A_mu^a+gf^abcA_mu^bA_nu^c$$

with the covariant derivative as:

$$D_mu=Ipartial_mu-igT^aA^a_mu$$

where $I$ is unit matrix and $T$ are generators of the group and

$$ mathcalL = -frac12operatornameTr(F^2)=- frac14F^amu nu F_mu nu^a$$

this has been basically taken from Wikipedia

But in the book I have been using, with a more mathematical perspective (Baez & Muniain's Gauge Fields, Knots and Gravity) the vector 1-form $A$ is an $textEnd(E)$ valued $1$-form described as

$$A = A_mu i^j e^i otimes e_j otimes dx^mu $$

and the covariant derivative and curvature are

$$D_mu = partial_mu + A_mu,quad F_jk = partial_j A_k - partial_k A_j + [A_j, A_k].$$

How can I go from one notation to the other? I can't do it, I think I'm confused by the superscipts $a$ in the physicist's notation.

And what is $F_munu$ for a non-abelian gauge theory? A matrix?

The superscript $a$ in these equations is the group index, that is, it runs over the a set of elements of the representation (I believe the adjoint representation) of the group. The commutator of two quantities $X^b$^ and $Y^c$ becomes $f^abcX^bY^c$ where $f^abc$ are the (fully antisymmetric structure constants for the group. For example, if the YM group is $SU(2)$ which is isomorphic to $SO(3)$ then $f^abc$ is the usual Levi-Civita symbol $varepsilon_abc$, which makes the expression into the familiar commutator.

In the B&M book the group index is suppressed; the implication is that $A_mu$ is formed from group members and is not a group invariant. I think in the expression for $F_jk$ in that book makes the further assumption that the normalization of $A$ is such that the "coupling constant" ($g$ in the physics text) is one.

This is an answer to your comment on Mark's answer, but a bit long for a comment so here it goes.

Presumably, in BM $A$ takes value in a Lie subalgebra of $mathfrakgl(mathbbR,n)$ and they are taking $e^iotimes e_j$ as a basis for it (think of $e^iotimes e_j$ as the matrix with 1 in the position $(i,j)$ and $0$ elsewhere). So $A$ and $F$ will both carry two "group" indices (components with respect to a basis of the Lie algebra really), and the appropriate number of Greek cotangent space indices.

In other books, especially when dealing with specific Lie algebras, you label a basis of the Lie algebra with a single index, e.g. for $mathfraksu(2)$ you could take $t_k = i sigma_k$ with $sigma_k$ the Pauli matrices. You then expand
$A=A^i_mu t_iotimes dx ^mu$ which is the notation used in Wikipedia.