I am working with the Hodge star and a non-orthonormal basis, and I can't see where exactly something is going wrong.
Let $V$ be an oriented $n$-dimensional vector space, and let $\Lambda V$ be the exterior algebra, considered as a subspace of the tensor algebra $TV$, where my convention is: $a\wedge b=a\otimes b - b\otimes a$, and similarly for $a_1\wedge \dotsm \wedge a_k$ (using all permutations in $S_k$).
Also let $g$ be a metric on $V$, so that we have a metric on $TV$, and thus on $\Lambda V$. Explicitly: $$\langle a\otimes b,c\otimes d\rangle := \langle a,c \rangle\langle b,d\rangle$$ which makes $$\langle a\wedge b,c\wedge d\rangle := 2(\langle a,c \rangle\langle b,d\rangle - \langle a,d \rangle\langle b,c\rangle)$$ This definition of $g$ causes the unfortunate fact that if $(e^i)$ is an orthonormal basis of $V$, then $(e^i\wedge e^j)_{i<j}$ will be orthogonal, but each element will have norm $\sqrt 2$.
The volume element is defined as: if $(e^i)$ is an orthonormal basis, then let $\text{vol}_g = e^1\wedge\dotsm \wedge e^n$.
The Hodge star is defined by the equation $$a\wedge \star b = \langle a,b\rangle \text{vol}_g$$ and is supposed to be an isometry.
Problem
Restrict to $n=4$. I tried computing $\star e^1\wedge e^2$. We know that it should be some multiple of $e^3\wedge e^4$, say $$e^1\wedge e^2 = \kappa e^3\wedge e^4$$ So we have $$\kappa e^1\wedge e^2\wedge e^3\wedge e^4 = e^1\wedge e^2\wedge \star (e^1\wedge e^2) = 2\text{vol}_ge^1\wedge e^2\wedge e^3\wedge e^4$$ But this causes $\kappa =2$, which would mean the Hodge star is not an isometry! So something must have gone wrong.
I believe I am taking definitions from multiple incompatible setups, but I don't understand what should change. I suspect it is the relationship of:
- The choice to induce the metric on $TV$ rather than one directly on $\Lambda V$ which makes $e^i\wedge e^j$ orthonormal.
- The choice of embedding $\Lambda V \subseteq TV$ as I did.
- The definition of $\text{vol}_g$ (maybe it is just wrong?)
Please let me know if you can find the error in my understanding/computation.