Grassmann Structures |
Grassmann Algebra / Numbers > s.a. exterior
algebra; integral equations.
* Idea: An algebra of anticommuting objects;
They can be seen as the "classical analogues" of anticommuting operators, or formal
variables used to define path integrals for fermionic fields although they cannot be treated as
normal numbers; They can also be used as "anticommuting coordinates" for supermanifolds.
* Example: In particular, we can associate
with any manifold M the Grassmann algebra defined by the set Λ*T*(M)
or Ω(M) of all forms on M (a submodule of the algebra of all tensor
fields), together with the operation of exterior product.
@ General: in Bishop & Crittenden 64;
da Rocha & Vaz AACA(06)mp [generalized, over Peano spaces];
Sverdlov a0808,
a0908 [novel definition].
@ Grassmann-Banach algebras:
Ivashchuk mp/00 [infinite-dimensional].
@ Integrals:
Creutz PRL(98) [numerical evaluation];
> s.a. manifolds [supermanifolds].
@ Grassmann coordinates:
Bonora & Tonin PLB(81);
Dzhunushaliev GRG(02) [interpretation].
@ Applications: Carrozza et al a1604-conf [in proofs of combinatorial identities].
> Online resources:
see Wikipedia page.
Grassmannian / Grassmann Manifold of k-Dimensional Planes
* Idea: The manifold of k-dimensional
planes through the origin of \(\mathbb R\)n, the most
famous example of a flag manifold; It can be generalized to the manifold of k-planes satisfying
some condition, e.g., Lagrangian submanifolds of a symplectic vector space.
$ Def: The compact manifold
\[{\rm G}_k(\mathbb R^n) \equiv {\rm Gr}(n,k,\mathbb R):= {\rm O}(n)/{\rm O}(k) \times {\rm O}(n-k)\]
of k-dimensional planes through the origin of \(\mathbb R^n\), which can be obtained from the Stiefel Manifold of k-frames Vk(\(\mathbb R^n\)) by
Gk(\(\mathbb R^n\)) = Vk(\(\mathbb R^n\))/O(k) .
* Example: Gr(n+1,1,\(\mathbb R\))
= \(\mathbb R\)Pn.
* Infinite Grassmann manifold:
Gk(\(\mathbb R\)∞)
is the direct limit of the sequence Gk(\(\mathbb R^k\))
⊂ Gk(\(\mathbb R^{k+1}\)) ⊂ ...
(thus, it is paracompact); It is used as the base space for the universal bundle
γk(\(\mathbb R^\infty\))
for non-paracompact or infinite-dimensional base space bundles.
* Oriented Grassmann manifold:
Gror(n, k, \(\mathbb R\)):=
SO(n)/SO(k) × SO(n−k).
* Complex Grassmann manifold:
Gr(n, k, \(\mathbb C\)):= U(n)/U(k) ×
U(n−k), the universal bundle
ξC(\(2n-2k\), U(k)).
@ Invariant forms: Stoll 77.
> Online resources:
see Wikipedia page.
Properties and Related Concepts > s.a. differential equations;
Flag Manifold; Stiefel Manifold.
* Topology:
The manifold Gk(\(\mathbb R^n\))
is given the quotient topology by the Stiefel manifold; dim Gr(n,
k, \(\mathbb R\)) = k(n−k).
* Relationships: Gr(n, k,
\(\mathbb R\)) is canonically isomorphic to Gr(n, n−k,
\(\mathbb R\)) via the assignment to each k-plane of its orthogonal
(n−k)-plane.
* Schubert cell: A cell defined by
a Schubert Symbol in a Grassmann manifold;
The set of all Schubert cells makes \(G_n({\mathbb R}^m\)) into a CW-complex.
Applications > s.a. integrable
systems; quantum oscillators.
* Idea: It is used as
the base space for the universal bundle \(\gamma^k({\mathbb R}^n)\)
[= ξ(n−k−1, O(k))?],
with fiber the vectors in each k-plane; Most k-plane
bundles can be mapped into this universal bundle, provided n is
sufficiently large (∞ for paracompact base space); For example, if
M is k-dimensional, embeddable in \(\mathbb R^n\),
there is a natural map from T(M) to \(\gamma^k({\mathbb R}^n)\),
the covering of the generalized Gauss Map.
@ General references: Fujii JAM(02)qp/01 [and quantum computation].
@ In physics: Alpay et al JMP(19)-a1806 [distribution spaces and stochastic processes].
> In physics: see fermions;
hidden variables; phase space
[for fermion fields]; supergravity; supersymmetric
field theories; torsion in physics.
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send feedback and suggestions to bombelli at olemiss.edu – modified 18 jul 2020