Articles of exterior algebra

Computation of the hom-set of a comodule over a coalgebra: $Ext_{E(x)}(k, E(x)) = P(y)$.

First of all, since every other book somehow mentions that this is trivial, I apologize if it turns out that I am just misunderstanding something in the definitions. So here goes: The motivation for this is the proof that $Ext_{\Gamma}(k, k) = P(y_1, y_2, …)$ where $\Gamma$ is a commutative, graded connected Hopf algebra of […]

Economically computing $d\beta$

$\displaystyle \beta = z\frac{x dy \wedge dz + y dz \wedge dx + z dx \wedge dy}{(x^2+y^2+z^2)^{2}}$ Show that $d\beta=0$. So, let $r=x^2+y^2+z^2$, $\begin{align} \displaystyle d\beta &= d(z\frac{x dy \wedge dz + y dz \wedge dx + z dx \wedge dy}{(x^2+y^2+z^2)^{2}}) \\ &= d(\frac{zx dy \wedge dz + zy dz \wedge dx + z^2 dx […]

Why $\bigwedge^{d-1}A=\bigwedge^{d-1}B \Rightarrow A= \pm B$

Let $V,W$ be $d$-dimensional vector spaces, and let $A,B \in \text{Hom}(V,W)$. Consider the induced maps on the exterior algebras: $\bigwedge^{d-1}A,\bigwedge^{d-1}B :\Lambda_{d-1}(V) \to \Lambda_{d-1}(W)$. Suppose that $\bigwedge^{d-1}A=\bigwedge^{d-1}B$, and that $A,B$ are invertible. I want to prove that $A=\pm B$. (Note that this implies $A=B$ in the case $d$ is even). The assumption implies $$\bigwedge^{d-1}(AB^{-1})=\text{Id}_{\Lambda_{d-1}(V)}.$$ Hence, the […]

algebraic manipulation of differential form

Suppose $\phi_1, \phi_2, \dots, \phi_k \in (\mathbb{R}^n)^*$, and $\mathbf{v}_1, \dots, \mathbf{v}_k \in \mathbb{R}^n$ $(\mathbb{R}^n)^*$ stands for the space of all linear transformations that goes from $\mathbb{R}^n \to \mathbb{R}$ is it true that: $$\phi_1\wedge\dots\wedge\phi_k(\mathbf{v}_1, \dots, \mathbf{v}_k) = \mathrm{det}[\phi_i(\mathbf{v}_j)]$$ This is a homework question (from Multivariable Mathematics, Shifrin, Ex.18 on Page 347) and there is a hint […]

Inner Products on Exterior Powers

Let $H$ is a real, $n$-dimensional vector space. Define $\varphi \colon \operatorname{GL}(H) \rightarrow \operatorname{GL}(\wedge^{k}H)$ by $A \mapsto \wedge^{k}A$ and $\psi_{\langle \cdot, \cdot \rangle} \colon \operatorname{GL}(\wedge^{k}H) \rightarrow T$ given by $J \mapsto\langle J(\cdot),J(\cdot)⟩$. Here, $\langle \cdot, \cdot \rangle$ is some inner product on $H$, $$ T = \{g \colon \wedge^{k}H\times \wedge^{k}H \, | \, g \text{ […]

Determinant of the transpose via exterior products

Let $V$ be a finite-dimensional vector space over $F$ and let $\tau:V \to V$ be a linear operator. Here’s my definition of the determinant: If $t:U \to U$ is a linear operator and $\dim(U)=n$ then $\det(t)$ is the unique number satisfying $$tu_1 \wedge \cdots \wedge tu_n = \det(t) u_1 \wedge \cdots \wedge u_n$$ for all […]

Proving the Poincare Lemma for $1$ forms on $\mathbb{R}^2$

I am trying to prove the Poincare Lemma for $1$ forms on $\mathbb{R^2}$. So I said that if I doing this, I start of with $$\omega = f_1(x_1,x_2) dx_1 + f_2(x_1,x_2)dx_2.$$ First thing I want to prove is $d\omega = 0$. So, I get $$d \omega = df_1 \wedge dx_1 + df_2\wedge x_2$$ $$ = […]

Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ iff $i(X)\omega=0$ and $L_X\omega=0$

Let $M$ be connected and let $\pi:M\times N \rightarrow N$ be the natural projection. Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ for some $p-$form $\alpha$ on $N$ if and only if $i(X)\omega=0$ and $L_X\omega=0$ for every vector field $X$ on $M\times N$ for which $d\pi(X(m,n))=0$ at each point $(m,n)\in M\times N$ Here, […]

Is there an intuitve motivation for the wedge product in differential geometry?

I’ve recently started studying differential forms and have been looking at differential forms. I’m struggling to understand the motivation for introducing the notion of the wedge product. Does it simply arise when generalising the notion of a “signed area/volume” in higher dimensional spaces, or is there a deeper reasoning behind it? If it is just […]

Cayley-Hamilton Theorem – Trace of Exterior Power Form

Let $V$ be an $n$-dimensional vector space over a field $F$ (the characteristic of which, for the purpose of this post, may be taken as $0$). Let $T$ be a linear operator on $V$ and $\lambda\in F$. Sometime ago, somewhere (I can’t recall where) I read that Formula. $\det(T-\lambda I)= \sum_{k=0}^n (-1)^k \text{trace}(\Lambda^k T)\lambda^{n-k}$ I […]