# Questions tagged [exterior-algebra]

For questions on the exterior algebra, and related concepts such as the wedge product, the tensor algebra and differential forms.

1,258 questions
Filter by
Sorted by
Tagged with
33 views

### FEM vs DEC viewpoint [closed]

It occurred to me that the FEM (finite element method) and DEC (discrete exterior calculus) viewpoints are different, and I'm wondering if you agree with my comparison. Consider solving the Laplace ...
• 295
43 views

### Can we construct the exterior algebra just from simple multivectors?

$\newcommand\K{\mathbb K} \newcommand\Ext{\mathop{\textstyle\bigwedge}} \newcommand\Lip{\mathrm{Lip}} \newcommand\ev{\mathrm{ev}} \newcommand\Gr{\mathrm{Gr}}$Let $V$ be a finite-dimensional $\K$-...
• 6,990
35 views

### Show that the dual Lefschetz operator applied to a two-form $\alpha$ is explicitly given by $\Lambda \alpha =\sum_i \alpha(x_i, y_i)$.

Choose an orthonormal basis $x_1, y_1 = J(x_1), \dots , x_n, y_n = J(x_n)$ of an euclidian vector space $V$ endowed with a compatible almost complex structure $J$. Show that the dual Lefschetz ...
• 363
61 views

### Wedge product and isomorphism between $\bigwedge^{k}T_{p}^{*}M$ and $\left(\bigwedge^{k}T_{p}M\right)^{*}$

Wikipedia states that there is the following isomorphism, $$\bigwedge^{k}T_{p}^{*}M \cong \left(\bigwedge^{k}T_{p}M\right)^{*}$$ More concretely, we think of the k-form, $\omega$ as either a linear ...
• 873
39 views

### Is the 'square matrix' in a 2d linear transformation just a bivector? [closed]

$\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}=\begin{pmatrix} b_1 \\ b_2 \end{pmatrix}$ If so, how do we call the operation of ...
18 views

### Exterior powers of Weyl algebra

I'm working with first Weyl algebra $\frac{k[x,y]}{<xy-yx-1>}$and I want to compute its cohomology using Koszul complex, so I need to find its exterior powers as a module over ground field k, ...
157 views

### What operation on matrices corresponds to the curl of a vector field?

Given the total derivative $Df$ of a (sufficiently) smooth function $f:\mathbb{R}^n \to \mathbb{R}^n$, the trace of the total derivative matrix corresponds to the divergence of $f$ (considered as a ...
1 vote
15 views

### Question on homotopy for Koszul Complex

I am working on illustrating the relationship between the originally defined Koszul complex and its dual version by examining the definitions of the differentials we define for the Koszul complex. For ...
• 59
173 views

• 901
37 views

### What's the definition of dual number at perspect of exterior algebra?

In Dual Number it said that "It may also be defined as the exterior algebra of a one-dimensional vector space with $\varepsilon$ as its basis element." But I can't find the detailed rigorous ...
44 views

### Prove that the isotopy generated by a time-dependent symplectic vector field is a symplectomorphism

Let $M$ a compact and connected smooth manifold. Suppose $X_t$ is a time-dependent symplectic vector field and let $\phi_t$ be the isotopy generated by $X_t$. Prove that $\phi_t ∈ Symp(M, \omega)$ for ...
• 3,140
24 views

### How to determine that the quotient of the top power of two $2$-forms is $\leq 1$

For full context, I work on a complex manifold $M$ of dimension $n$ where I have a Hermitian $(1,1)$-form $\Omega$ and $(1,1)$-form $\tau$ which is negative semidefinite at a point. I am trying to see ...
• 1,665
80 views

### Kernel of the action of GL(V) on exterior square of V

I wonder whether anyone knows a reference for the following result? I can give a shortish proof, but would prefer to cite the literature if possible. Theorem Let $V=F^n$ be an $n$-dimensional $F$-...
• 474
73 views

• 2,231
39 views

• 3,936
1 vote
95 views

### What is the Grassmann algebra of $2\times 2$ complex matrices?

The traceless hermitian $2\times 2$ complex matrices form a real Euclidean space $\mathfrak E_3$ with dot product $a\cdot b:=\frac{1}{2}(ab+ba)/I$ where $I$ is the $2\times 2$ identity matrix. A ...
• 2,033
1 vote
57 views

### Trying to understand two differential forms

I'm struggling with two particular differential forms: the $\flat$ 1-form and the $\natural$ 2-form defined as follows: Let $X = (X_1,X_2,X_3)$ be a vector field in an open subset $O$ of $\Bbb R^3$. ...
• 433
Let $R$ and $S$ be two commutative rings and let $f\colon R\to S$ be a homomorphism of rings. For any $S$-module $N$ denote with $f^*N$ the module obtained from $N$ by restriction of ...