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 V_k(\(\mathbb R^n\)) by

G_k(\(\mathbb R^n\)) = V_k(\(\mathbb R^n\))/O(k) .

* Example: Gr(n+1,1,\(\mathbb R\)) = \(\mathbb R\)Pⁿ.
* Infinite Grassmann manifold: G_k(\(\mathbb R\)^∞) is the direct limit of the sequence G_k(\(\mathbb R^k\)) ⊂ G_k(\(\mathbb R^{k+1}\)) ⊂ ... (thus, it is paracompact); It is used as the base space for the universal bundle γ^k(\(\mathbb R^\infty\)) for non-paracompact or infinite-dimensional base space bundles.
* Oriented Grassmann manifold: Gr_or(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 ξ_C(\(2n-2k\), 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 G_k(\(\mathbb R^n\)) 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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