Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free $\mathcal{O}_X$-modules of rank $n$ and vector bundles of rank $n$. So, equivalently, principal $GL(n,\mathbb{C})$-bundles are given by locally free sheaves of rank $n$.

So...what about other groups? I guess that $SL(n,\mathbb{C})$ bundles are then locally free sheaves of rank $n$ with top exterior power trivial, but can we phrase everything in terms of the properties of a sheaf and a group?

My guess is that in this context, if we can do it, we'll end up with something that's not quite locally free sheaves of rank n for $GL(n,\mathbb{C})$, but which will be equivalent.

Note: I'm aware that we could just say something like "the sheaf of local sections of a $G$-bundle" but I'm looking for something intrinsic, a set of properties of the sheaf without reference to the geometric bundle, which can be reconstructed from the sheaf description.

Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free $\mathcal{O}_X$-modules of rank $n$ and vector bundles of rank $n$. So, equivalently, principal $\mathrm{GL}(n,\mathbb{C})$-bundles are given by locally free sheaves of rank $n$.

So...what about other groups? I guess that $\mathrm{SL}(n,\mathbb{C})$ bundles are then locally free sheaves of rank $n$ with top exterior power trivial, but can we phrase everything in terms of the properties of a sheaf and a group?

My guess is that in this context, if we can do it, we'll end up with something that's not quite locally free sheaves of rank n for $\mathrm{GL}(n,\mathbb{C})$, but which will be equivalent.

Note: I'm aware that we could just say something like "the sheaf of local sections of a $G$-bundle" but I'm looking for something intrinsic, a set of properties of the sheaf without reference to the geometric bundle, which can be reconstructed from the sheaf description.

Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free O_X-modules of rank n and vector bundles of rank n. So, equivalently, principal GL(n,C)-bundles are given by locally free sheaves of rank n.

So...what about other groups? I guess that SL(n,C) bundles are then locally free sheaves of rank n with top exterior power trivial, but can we phrase everything in terms of the properties of a sheaf and a group?

My guess is that in this context, if we can do it, we'll end up with something that's not quite locally free sheaves of rank n for GL(n,C), but which will be equivalent.

Note: I'm aware that we could just say something like "the sheaf of local sections of a G-bundle" but I'm looking for something intrinsic, a set of properties of the sheaf without reference to the geometric bundle, which can be reconstructed from the sheaf description.

Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free $\mathcal{O}_X$-modules of rank $n$ and vector bundles of rank $n$. So, equivalently, principal $GL(n,\mathbb{C})$-bundles are given by locally free sheaves of rank $n$.

So...what about other groups? I guess that $SL(n,\mathbb{C})$ bundles are then locally free sheaves of rank $n$ with top exterior power trivial, but can we phrase everything in terms of the properties of a sheaf and a group?

My guess is that in this context, if we can do it, we'll end up with something that's not quite locally free sheaves of rank n for $GL(n,\mathbb{C})$, but which will be equivalent.

Note: I'm aware that we could just say something like "the sheaf of local sections of a $G$-bundle" but I'm looking for something intrinsic, a set of properties of the sheaf without reference to the geometric bundle, which can be reconstructed from the sheaf description.