Questions tagged [exterior-algebra]
The exterior-algebra tag has no usage guidance.
34
questions with no upvoted or accepted answers
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answer
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Analogy between the exterior power and the power set
The symmetric algebra of an object exists in every cocomplete $\otimes$-category. For the category of sets $\mathrm{Sym}(X)$ is the set of multi-subsets of $X$.
The usual definition of the exterior ...
- 61.5k
16
votes
0
answers
554
views
Are $0, 1, 4, 7, 8$ the only dimensions in which a bivector-valued cross product exists?
It is a well-known mathematical curiosity that ordinary (vector-valued) cross products over $\mathbb{R}$ exist only in dimensions $0, 1, 3$ and $7$ (this fact is related to Hurwitz's theorem that real ...
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7
votes
0
answers
138
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A generalization of matrix minors to non-integer values
I am interested to know if there exist a notion of $k$-minors of a real square matrix, for non-integer positive values of $k$
One approach I thought of was to use the fact that the $k$-minors are (...
- 6,621
6
votes
0
answers
254
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Is a Sobolev map with invertible smooth minors smooth?
$\newcommand{\Cof}{\text{cof}}$
Let $k,d$ be even integers, such that $d\ge3$ and $2 \le k \le d-1$. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for ...
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5
votes
0
answers
155
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The set of strongly positive forms is a closed cone
This question comes from the Complex Analytic and Differential Geometry by Demailly. Let $V$ be a $n$ dimensional complex space. Consider the exterior algebra $\Lambda V^* = \oplus \Lambda^{(p,q)}V^*$....
- 108
5
votes
0
answers
160
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Rational cohomology of p-adic general linear groups
I wanted to compute the cohomology ring $H^*(GL_n(\mathbb{Z}_p); \mathbb{Q}_p)$ (with $p$ fixed prime as usual). I found some incomplete notes stating that the computation should go as follows.
First ...
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5
votes
0
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281
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Federer's questions on the mass and comass norms
In Federer's book "Geometric measure theory", he says in section 1.8.3 (where $\|\cdot\|$ is the comass norm):
Very little appears to be known about the structure of the convex sets $\wedge^...
- 2,169
5
votes
0
answers
149
views
Counting square zero forms over finite fields
Let $p$ be an odd prime and let $R=\Lambda_{\mathbb{F}_p}[x_1,\dots,x_n]$ be the exterior algebra on $n$ generators over the finite field with $p$ elements. This is a graded-commutative ring.
Is ...
- 2,270
5
votes
0
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164
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Definition of modules over $C_\infty$-algebras ("commutative $A_\infty$-algebras")
Let $\Lambda$ be a finite-dimensional associative algebra. We can think of this as an $A_\infty$-algebra with vanishing $m_i$ for $i \ne 2$ and consider $A_\infty$-modules $M$ over it. A list of ...
- 10.2k
4
votes
0
answers
135
views
Structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ with $A$ a local integral domain
I am trying to see the structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ where $A$ is a local integral domain with small residue field.
Let $A$ be a local integral domain with maximal ideal $M$, residue ...
- 259
4
votes
0
answers
473
views
Hodge duality and the determinant of the product of two matrices
I stumbled onto the following identity, and I would like to know: Is it known by some name and are there some references I might cite (or is it actually too trivial to be mentioned anywhere)? Are ...
- 13.5k
4
votes
0
answers
159
views
Cannot multivectors be classified more easily than general tensors?
This is sort of a spinoff of Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? - seems to be almost hopeless, but maybe some partial ...
- 17.5k
4
votes
0
answers
221
views
Explicit description of graded (counital) cofree cocommutative coalgebras
Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$.
Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ ...
- 41
4
votes
0
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269
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Exterior powers and singular values on Hilbert spaces
I am currently writing an article relating to multiplicative ergodic theorems for cocycles of bounded operators acting on a Hilbert space, and in parts of the argument it is necessary for me to refer ...
- 6,186
3
votes
0
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38
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Expression for the lattice operations on subspaces in Plucker embedding
Suppose that $V$ is a finite dimensional $\mathbb Q$-vector space. To each subspace $S$ of dimension $k$, we can associate the line from the origin of $\Lambda^k(V)$ through the point $s_1\wedge \...
- 580
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0
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302
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Is the image of the map $A \to \bigwedge^{k}A $ a weakly embedded submanifold?
$\newcommand{\End}{\operatorname{End}}$
$\newcommand{\GL}{\operatorname{GL}}$
Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd $2 \le k \le d-2$. Define
$H_{>k}=\{ A \in \End(...
- 6,621
3
votes
0
answers
298
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Exterior power of a torsion-free sheaf on a DVR
Let $R$ be a discrete valuation ring and $X$ be a regular, integral. projective $R$-scheme, flat over $R$. Let $F$ be a torsion-free coherent sheaf on $X$ of rank $n$, flat over $\mathrm{Spec}(R)$. Is ...
- 2,126
3
votes
0
answers
302
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Differential ideals of Pfaffian forms on jet bundles (Integrability)
(I asked this question on math.stackexchange, but got no reaction in several weeks. So, my conclusion is, that it is harder to answer than I thought, and maybe admissible for the attribute 'research ...
- 193
3
votes
0
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238
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Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold?
Let $(M,g)$ be a geodesically complete Riemannian manifold that is not necessarily compact. Futhermore, assume that $M$ has at most exponential volume growth (ie., locally doubling property). Let $\mu$...
- 51
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votes
0
answers
100
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Is there a non-degenerate solution for this PDE on $\mathbb{R}^3$?
$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\R}{\mathbb{R}}$
Does there exist a smooth map $f:\mathbb{R}^3 \to \mathbb{R}^3$, which satisfies
$$\tr \big( df \otimes \delta(df \wedge df) \big)=0,...
- 6,621
2
votes
0
answers
251
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What comes next in the sequence "symmetric algebras, exterior algebras, divided power algebras, ..."?
This question was posed by A Rock and a Hard Place in this discussion, where they mentioned the isomorphisms
\begin{align*}
\mathrm{L}\,\mathrm{Sym}^n_R(M[1]) &\cong (\mathrm{L}\,{\...
- 10.7k
2
votes
0
answers
128
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What's known about the matroid induced by the Plücker coordinates of the representation of a matroid?
Let $M$ be a linear matroid with ground set $E$ and independent subsets $\mathcal I$, represented by $\rho: E \rightarrow V$.
This induces a map
$$
\hat\rho: \mathcal I \rightarrow \mathbf P(\Lambda V)...
- 193
2
votes
0
answers
62
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Ranks of cycles with base field coefficients as a generalization of ranks of multivectors?
This must be probably only reference request since I am inclined to believe that I am asking about something well known but just cannot pin down appropriate keywords for searching.
The starting point:...
- 17.5k
2
votes
0
answers
287
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Can the solution manifold for an exterior differential system be represented using alternating multivectors?
Differential equations can be written as an ideal of n-forms. Solutions are manifolds where the forms pull back to zero. Is it possible, or useful, to represent the solution by multivectors? For ...
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1
vote
0
answers
102
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Exterior algebra of free modules over Hopf algebras
Let $H$ be a commutative, cocommutative Hopf algebra over a field $\mathbb{K}$, and $M$ a free Hopf module over $H$. Is the exterior algebra $\Lambda^k_\mathbb{K} M$ with the diagonal $H$-action
$$h \...
1
vote
0
answers
64
views
Function for unique volume element
This is an issue that I'm am trying to solve for a fine-tuning measure in particle physics, but it is purely mathematical. Consider three vectors $\{v_1, v_2, v_3\}$ in $\mathbb{R}^3$. I would like a ...
- 11
1
vote
0
answers
96
views
Poincaré lemma for gradient times its transpose
Poincaré lemma states that a vector $v_i(x)$ defined on a ball in $R^n$ is the gradient of a function if and only if
\begin{equation}
\partial_i v_j = \partial_j v_i
\end{equation}
or equivalently ...
- 11
1
vote
0
answers
54
views
Maximum number of matrices satisfying given rank conditions
Assume that we have $2k$ matrices $S_1,\ldots,S_k$ and $\Phi_1,\ldots,\Phi_k$ over some finite field $F$ such that
(i) $S_i\in F^{l/2\times l}$ and $\dim S_i=l/2$ for any $i\in\{1,\ldots,k\}$;
(ii)...
- 147
1
vote
0
answers
158
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The dual space mapping of a $p$-form
I know that the dual of a tensor product is the tensor product of the duals. For example, the dual space of $V\otimes V^*$ is $V^* \otimes V$. I also know that the induced dual space mapping of a ...
1
vote
0
answers
192
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Non-negative Quadratic forms with Exterior Forms
Hello All,
I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you.
Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product space over $\mathbb{...
0
votes
0
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27
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Tableau and its first prolongation for linear Pfaffian systems
This question concerns characterization of tableau associated with an exterior differential system (EDS).
On the one hand, we have prop 4.2 in the EDS book by Bryant et al.:
Given an EDS on a manifold ...
- 685
0
votes
0
answers
45
views
Linear maps which exterior power preserves particular vector
Is there any convenient way to describe all linear operators $A \colon V \to V$ whose exterior power $\Lambda^k A$ preserves some vector $v \in \Lambda^k V$ (not necessary decomposable)? $V$ is ...
- 13
0
votes
0
answers
164
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Koszul exterior connections
Let $(E,M)$ be a vector bundle over a riemannian manifold $M$ which is a module for the exterior forms of $M$. I define a Koszul exterior connection as an operator $\nabla$ such that:
$$
\nabla : E \...
- 377
0
votes
0
answers
178
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A vector calculus formula
Let me answer my own question, hoping to be forgiven for that.
I asked unsuccessfully that question on Mathematics. Let $A, B$ be vector fields in $\mathbb R^3$.
We have
$$
\text{curl}\bigl((A\cdot \...
- 15.2k