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**

*

$

\[{\rm G}_k(\mathbb R^n) \equiv {\rm Gr}(n,k,\mathbb R):= {\rm O}(n)/{\rm O}(k) \times {\rm O}(n-k)\]

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

G* _{k}*(\(\mathbb R^n\))
= V

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

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

*

*

@

>

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

* __Topology__:
The manifold G* _{k}*(\(\mathbb R^n\))
is given the quotient topology by the Stiefel manifold; dim Gr(

*

*

**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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