Grassmann manifolds
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== Introduction == | == Introduction == | ||
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− | Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let $K$ be the real, complex or quaternion field and $V$ a vector space over $K$. A Grassmann manifolds of $k$-dimensional subspaces is a set $G_k(V)$ of $k$-dimensional subspaces. The set $G_k(V)$ is a quotient of a subset of $V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$ as the quotient topology. | + | Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let $K$ be the real, complex or quaternion field and $V$ a vector space over $K$ of dimension $n$ and let $k\leq n$. A Grassmann manifolds of $k$-dimensional subspaces is a set $G_k(V)$ of $k$-dimensional subspaces. The set $G_k(V)$ is a quotient of a subset of $V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$ as the quotient topology. |
+ | {{beginthm|Theorem|{{cite|{Milnor&Stasheff1974}}}}} $G_k(V)$ is a Hausdorff, compact space. {{endthm}} | ||
− | {{beginthm|Theorem|{{cite|}}}} $G_k(V)$ is a | + | {{beginthm|Theorem|{{cite|{Milnor&Stasheff1974}}}}} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$. {{endthm}} |
− | + | Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$ {{endthm}} | |
− | + | Grassmann manifold is a homogeneous space of the general linear group. General linear group $GL(V)$ acts transitively on $G_k(V)$ with an isotropy group consisting of automorphisms preserving a given subspace. If the space $V$ is equipped with a scalar product (hermitian metric resp.) then the group of isometries $O(V)$ acts transitively and the isotropy group of $W$ is $\O(W^\bot)\times O(W)$. | |
− | + | {{beginthm|Proposition|{{cite|{Milnor&Stasheff1974}}}}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$ {{endthm}} | |
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+ | The Grassmannians $G_1(V)$ are the well-known projective spaces, denoted $P(V)$. Note that $G_1(F^2)=S^d$ and if we identify $S^d$ with the one-point compactification of $\F$ the projection $p$ corresponds to the map $p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$. Note, that the same formula works for $\F=\OO$, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$ for $d=1,2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fiber of $p_d$ is a sphere $S^{d-1}$. Check directly that the Hopf maps are locally trivial, thus fibrations. | ||
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− | \ | + | There is an embedding of the Grassmannian $G_k(\F^n)$ in the cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on $G_k(\F^n)$. |
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+ | Prove that there is a free action of the group $O(k,\F)$ on $V_k(\F^n)$ sucht that the orbit space is homeomorphic to $G_k(\F^n)$. Similarly for the noncompact Stiefel manifold. | ||
\end{zad} | \end{zad} | ||
− | + | Prove that the map $p:V_k(\F^n)\to G_k(\F^n)$ is locally trivial (even a principal $O(k,\F)$-bundle), thus a fibration. | |
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They are examples of coadjoint orbits \cite{Kirillov2004} | They are examples of coadjoint orbits \cite{Kirillov2004} |
Revision as of 12:26, 26 November 2010
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Contents |
1 Introduction
Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let be the real, complex or quaternion field and a vector space over of dimension and let . A Grassmann manifolds of -dimensional subspaces is a set of -dimensional subspaces. The set is a quotient of a subset of consisting of linearly independent -tuples of vectors with the subspace topology. We define topology on as the quotient topology.
Note that the Grassmann manifold around is locally modelled on the vector space </div>
Grassmann manifold is a homogeneous space of the general linear group. General linear groupTex syntax erroracts transitively on with an isotropy group consisting of automorphisms preserving a given subspace. If the space is equipped with a scalar product (hermitian metric resp.) then the group of isometries acts transitively and the isotropy group of is .
Tex syntax error, however the higher dimensional projective spaces over octonions do not exist. The maps for
Tex syntax errorare called the Hopf maps and they play a very important role in homotopy theory; a fiber of is a sphere . Check directly that the Hopf maps are locally trivial, thus fibrations.
Tex syntax errorin the cartesian space which assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on
Tex syntax error.
Tex syntax erroron
Tex syntax errorsucht that the orbit space is homeomorphic to
Tex syntax error. Similarly for the noncompact Stiefel manifold.
\end{zad}
Prove that the mapTex syntax erroris locally trivial (even a principal
Tex syntax error-bundle), thus a fibration.
They are examples of coadjoint orbits [Kirillov2004]
Theorem 1.4.
2 Construction and examples
...
3 Invariants
...
4 Classification/Characterization
...
5 Further discussion
...
6 References
- [Kirillov2004] A. A. Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics 64, American Mathematical Society, Providence, RI, 2004. MR2069175 (2005c:22001) Zbl 02121486
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [[Template:{Milnor&Stasheff1974|[{Milnor&Stasheff1974]]] {{{Milnor&Stasheff1974}}