 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].

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(nk).
* Complex Grassmann manifold: Gr(n, k, $$\mathbb C$$):= U(n)/U(k) × U(nk), the universal bundle ξC($$2n-2k$$, U(k)).
@ Invariant forms: Stoll 77.

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(nk).
* Relationships: Gr(n, k, $$\mathbb R$$) is canonically isomorphic to Gr(n, nk, $$\mathbb R$$) via the assignment to each k-plane of its orthogonal (nk)-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(nk−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.