Questions tagged [exterior-algebra]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
0 answers
169 views

Is the exterior algebra intrinsically formal?

Following 4.6 and 4.7 of this paper by Seidel and Thomas, a graded algebra $A$ is called intrinsically formal if any two dgas with cohomology $A$ are quasi-isomorphic. There is a sufficient condition ...
Faniel's user avatar
  • 603
0 votes
0 answers
25 views

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 ...
Josh Burby's user avatar
0 votes
0 answers
43 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 ...
ptashek's user avatar
  • 13
4 votes
1 answer
120 views

DG algebra structure on minimal free resolution of modules over regular local ring

Let $(Q, \mathfrak n, k)$ be a regular local ring. Let $I\subseteq \mathfrak n^2$ be an ideal, and fix a minimal generating set $\mathbb f= f_1,\cdots, f_n$ of $I$. The Koszul complex $E:= Q[e_1,...,...
uno's user avatar
  • 280
2 votes
1 answer
180 views

Orthogonal complements in exterior powers

I previously asked this on Mathematics Stack Exchange, to no result: Consider the standard induced inner product structure on $\wedge^k\mathbb{R}^d$ given by defining $$\langle u_1\wedge \cdots \wedge ...
Ian Morris's user avatar
  • 6,176
19 votes
3 answers
939 views

How big can a wedge of $n$ 2-forms in $\mathbb{R}^{2n}$ be?

$\def\RR{\mathbb{R}}$Let $\omega$ be a $2$-form on $\RR^{2n}$, where $\RR^{2n}$ has the usual Euclidean norm. The comass of $\omega$ is defined to be $\max_{|u|, |v| \leq 1} \omega(u,v)$. Here is the ...
Mikhail Katz's user avatar
  • 15.1k
7 votes
1 answer
412 views

How big can a wedge of 2-forms be?

The comass of a 2-form $\alpha$ is the maximal value of $\alpha(u,v)$ for a pair of unit vectors $u,v$. The symplectic form $\alpha$ on $\mathbb R^{2n}$ has the property that $|\alpha^{\wedge n}| = n!...
Mikhail Katz's user avatar
  • 15.1k
3 votes
1 answer
376 views

What is the name for algebras generated by elements, all of whose cubes vanish?

Given a ring $R$ with identity $1$, we can define the exterior algebra of order $k$ over $R$ to be the algebra over $R$, generated by elements $x_1, \dots, x_k$ satisfying $x_i^2 = 0$ for each index $...
Naysh's user avatar
  • 455
5 votes
0 answers
146 views

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 $\Lambda V^* = \oplus \Lambda^{(p,q)}V^*$....
Junyu Cao's user avatar
  • 108
5 votes
1 answer
139 views

When is a symmetric tensor equal to gradient times gradient?

On a ball in $\mathbb R^n$, a vector field $v_j$ is a gradient of a function when its exterior derivative vanishes. In other words, if $$\partial_i v_j-\partial_j v_i=0$$ then there exists a function $...
Davide Leroy's user avatar
0 votes
0 answers
160 views

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 \...
Antoine Balan's user avatar
1 vote
1 answer
101 views

nth-power of the dual Lefshetz operator

Let $(X,\omega)$ be a Kahler manifold, denote by $\Lambda$ the dual of the Lefshetz operator $\omega\wedge$ (see e.g. Dual Lefschetz Operator and Contraction with the Fundamental Form). Let $\zeta\in\...
BinAcker's user avatar
  • 767
1 vote
0 answers
102 views

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 \...
Lukas Miaskiwskyi's user avatar
2 votes
0 answers
100 views

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,...
Asaf Shachar's user avatar
  • 6,611
2 votes
0 answers
242 views

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}\,{\...
Emily's user avatar
  • 10k
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 ...
Liddo's user avatar
  • 259
2 votes
1 answer
210 views

Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a polynomial ring

Question 1. Let $R$ be a polynomial ring over a field $k$. Assume that $R$ is graded in the usual way (i.e., each variable has degree $1$). Let $I$ and $J$ be two ideals of $R$ such that $I$ is ...
darij grinberg's user avatar
1 vote
1 answer
812 views

Simple properties of the codifferential

The exterior derivative $d$ has many very nice algebraic relations. For example $d(\alpha\wedge\beta) = (d\alpha)\wedge \beta + (-1)^k\alpha\wedge(d\beta)$ $f^*(d \alpha)=d f^*(\alpha)$. $d\circ ...
RaphaelB4's user avatar
  • 4,296
2 votes
0 answers
128 views

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)...
Cornelius Brand's user avatar
5 votes
0 answers
156 views

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 ...
N.B.'s user avatar
  • 757
5 votes
0 answers
276 views

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^...
Quarto Bendir's user avatar
4 votes
1 answer
435 views

Exterior algebra of normed spaces

This question is related to my prior question, but this one is aimed, even though it's more general. If $V$ is a vector space, we define the exterior algebra of $V$ do be: $$\bigwedge V := \bigoplus_{...
MathMath's user avatar
  • 1,255
6 votes
1 answer
1k views

What is the role of topology on infinite dimensional exterior algebras?

Wedge products and exterior powers are discussed in W. Greub's book Multilinear algebra as follows. Definition: Let $E$ be an arbitrary vector space and $p \ge 2$. Then a vector space $\bigwedge^{p}E$ ...
MathMath's user avatar
  • 1,255
3 votes
1 answer
194 views

action of symmetric group on the second exterior power

Let $e_i \wedge e_j \ (i < j)$ be a basis for the $\mathbb Z$-module $\wedge^2 \Gamma$, where $\Gamma = \mathbb Z^n$. Clearly $S_n$ acts on the module $\wedge^2 \Gamma$ via $$\pi(e_i \wedge e_j) ...
A. Gupta's user avatar
  • 336
16 votes
0 answers
552 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 ...
pregunton's user avatar
  • 976
9 votes
2 answers
404 views

Can we recover all $k$-minors of a square matrix from some of them?

This is a cross-post. Let $k,n$ be natural numbers, $1<k<n$. Suppose we have an "unknown" invertible $n \times n$ matrix $A$ over a field of characteristic zero. (we do not know the entries of ...
Asaf Shachar's user avatar
  • 6,611
5 votes
1 answer
791 views

Extension of a bilinear form to the exterior algebra

In Serre's Local Fields, at the beginning of the chapter III section 2, he has wrote "it is known that $T$ extends to a non-degenerate bilinear form on the exterior algebra of $V$", where $T$ is a ...
tanjia's user avatar
  • 337
3 votes
1 answer
734 views

Trace and exterior product

Let $V$ be a $2n$-dimensional complex vector space with base $\{e_1,\dotsc,e_n,f_1,\dotsc,f_n]\}$ Let $W \subset \wedge^n V$ be the subspace in the exterior product, with basis vectors $$ e_{i_1} \...
Per Alexandersson's user avatar
2 votes
1 answer
847 views

Exterior derivative independence from coordinate systems

In the book Mathematical Methods of Classical Mechanics by V.I. Arnold, the author introduces (p.189) the concept of exterior derivative as "the principal linear part of the increment" of the function ...
Lo Scrondo's user avatar
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 ...
Daniel's user avatar
  • 11
0 votes
0 answers
178 views

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 \...
Bazin's user avatar
  • 15k
1 vote
0 answers
95 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 ...
Damiano's user avatar
  • 11
7 votes
1 answer
289 views

Is every basis for $\bigwedge^kV$ satisfying a "complementary" property a rescaling of a "standard" basis?

This is a cross-post. Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$ be a basis for $\bigwedge^2V$, where each $\omega_{i_1,i_2}$ is decomposable. Suppose that for every $\...
Asaf Shachar's user avatar
  • 6,611
6 votes
1 answer
377 views

Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?

$\newcommand{\R}{\mathbb R}$ $\newcommand{\N}{\mathbb N}$ $\newcommand{\de}{\delta}$ $\newcommand{\sig}{\sigma}$ $\newcommand{\Average}[1]{\left\langle#1\right\rangle} $ $\newcommand{\IP}[2]{\Average{...
Asaf Shachar's user avatar
  • 6,611
5 votes
2 answers
1k views

What are the necessary and sufficient conditions for a two-form to be an exterior product of two one-forms?

Consider a two-form $\gamma \in \Lambda^2(V)$ where $V$ is a real vector space. Now I would like to know the necessary and sufficient conditions for $\gamma$ to be expressible as an exterior product ...
Void's user avatar
  • 161
4 votes
1 answer
396 views

Taylor spectrum of commuting operators

Taylor spectrum of commuting operators Fom the following paper (M. Ch—o, H. Motoyoshi, B. Na¡cevska Nastovska: On the joint spectra of commuting tuples of operators and a conjugation) we have Let ...
Student's user avatar
  • 1,152
12 votes
1 answer
506 views

Representations of degenerate Clifford algebras

Given a real finite-dimensional vector space $V$ with a symmetric bilinear form $b$, we define the Clifford algebra $Cl(V,b)$ as the quotient of the tensor algebra $\bigotimes V$ by the two sided ...
José Figueroa-O'Farrill's user avatar
7 votes
0 answers
138 views

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 (...
Asaf Shachar's user avatar
  • 6,611
3 votes
3 answers
485 views

Can we specify the value of harmonic forms at a point?

Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed. Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$. Does there exist an open ...
Asaf Shachar's user avatar
  • 6,611
2 votes
0 answers
62 views

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:...
მამუკა ჯიბლაძე's user avatar
7 votes
1 answer
282 views

A commutative variant of the exterior algebra

Consider $A = \mathbb{R}[t_1,\ldots,t_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t_1,\ldots,t_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through $$ p(y) = ...
Cornelius Brand's user avatar
17 votes
1 answer
634 views

An explicit reconstruction of a matrix from its minors

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ $\newcommand{\Cof}{\operatorname{cof}}$ Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd integer $...
Asaf Shachar's user avatar
  • 6,611
3 votes
0 answers
37 views

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 \...
Milo Brandt's user avatar
3 votes
0 answers
301 views

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(...
Asaf Shachar's user avatar
  • 6,611
4 votes
1 answer
211 views

Is the map $A \to \bigwedge^{k}A $ from matrices above rank $k$ proper?

$\newcommand{\End}{\operatorname{End}}$ Let $V$ be a $d$-dimensional real vector space. ($d \ge 3$). Fix an odd $2 \le k \le d-1$. Define $H_{>k}=\{ A \in \End(V) \mid \operatorname{rank}(A) > ...
Asaf Shachar's user avatar
  • 6,611
5 votes
1 answer
141 views

If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ equals $\bigwedge^k B$ for some complex matrix $B$, does it have a real source?

Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some complex $B \in \text{End}(\mathbb{C}^d)$. Does there exist $M \in \...
Asaf Shachar's user avatar
  • 6,611
12 votes
1 answer
401 views

Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$?

Let $V$ a real vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor: $$ \psi:\text{End}(V) \to \text{End}(\bigwedge^kV) \, \, \, \, , \, \, ...
Asaf Shachar's user avatar
  • 6,611
7 votes
1 answer
206 views

Is a Sobolev map with smooth minors smooth on the whole domain?

Let $d\ge 3$ and $2 \le k \le d-1$ be integers, where at least one of $k,d$ is odd. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for some $p \ge 1$. ...
Asaf Shachar's user avatar
  • 6,611
8 votes
2 answers
444 views

Obstructions for the wedge of coordinate differentials to be harmonic

Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property: For every $p \in M$ there exist a ...
Asaf Shachar's user avatar
  • 6,611
6 votes
0 answers
254 views

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 ...
Asaf Shachar's user avatar
  • 6,611