Confusion about the Moyal-Weyl twist

[gasgas]

I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime

Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra by construction respects the Lie bracket of the tangent space such that v⊗w-w⊗v=[v,w] for all v,w in the tangent space and [.,.] the Lie bracket. As is always done in differential geometry, we may take the basis vector fields to be commutative, which implies that in the enveloping algebra ∂_μ⊗∂_ν=∂_ν⊗∂_μ, i.e. the tensor product that appears in the definition of F is symmetric and contracted with an antisymmetric tensor which should give zero.

Am I misunderstanding something?

 

[martinbn]

Can you give some of the definitions?

 

[gasgas]

Can you give some of the definitions?

I will just link the source I'm using if that's fine. All the definitions are in the first chapter of this PhD thesis https://arxiv.org/pdf/1210.1115

 

[martinbn]

You have a Lie algebra $\mathfrak g$, and its universal enveloping algebra $U(\mathfrak g)$. The vectors $\partial_\mu$ are viewed as elements in $U(\mathfrak g)$ and $\partial\mu\otimes\partial\nu$ are elements in $U(\mathfrak g)\otimes U(\mathfrak g)$. So they don't commute.

 

[gasgas]

You have a Lie algebra $\mathfrak g$, and its universal enveloping algebra $U(\mathfrak g)$. The vectors $\partial_\mu$ are viewed as elements in $U(\mathfrak g)$ and $\partial\mu\otimes\partial\nu$ are elements in $U(\mathfrak g)\otimes U(\mathfrak g)$. So they don't commute.

I see, so a more precise way to say this is that m($\partial\mu, \partial\nu$)=m($\partial\nu, \partial\mu$) where m is multiplication in U(g), while this does not apply to an element of U(g)$\otimes$U(g). Thank you!