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].**
**@

@

>

**Grassmannian / Grassmann Manifold of k-Dimensional Planes**

*

$

G* _{k}*(\(\mathbb R\)

of *k*-dimensional planes through the origin of \(\mathbb R\)^{n},
which can be obtained from the Stiefel Manifold of *k*-frames V* _{k}*(\(\mathbb R\)

G* _{k}*(\(\mathbb R\)

* __Example__: Gr(*n*+1,1,\(\mathbb R\)) = \(\mathbb R\)P^{n}.

* __Infinite Grassmann manifold__:
G* _{k}*(\(\mathbb R\)

*

*

@

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**Properties and Related Concepts** > s.a. Flag
Manifold; Stiefel
Manifold.

* __Topology__: G* _{k}*(\(\mathbb R\)

*

*

**Applications** > s.a. integrable
systems; quantum oscillators.

* __Idea__: It is used as
the base space for the universal
bundle *γ*^{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 *γ*^{k}(\(\mathbb R\)^{n}),
the covering of the generalized Gauss Map.

@ __References__: Fujii JAM(02)qp/01
[and quantum computation].

> __Related topics__: see hidden variables; phase space [for fermion fields]; supergravity; supersymmetric field theories; torsion in physics.

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