split torus 예문

예문모바일

If X is the symmetric space associated to G and T \ subset G is a maximal split torus then there exists a unique orbit of T in X which is a totally geodesic flat subspace in X.
Isomorphism classes of twisted forms of a split torus are parametrized by nonabelian flat cohomology H ^ 1 ( S, GL _ n ( \ mathbb { Z } ) ), where the coefficient group forms a constant sheaf.
On the other hand, if { } _ F \ mathbf T \ subset \ mathbf T is a maximal F-split torus its action on the F-Lie algebra of \ mathbf G gives rise to another root system { } _ F \ Phi.
Another invariant associated to the split torus { } _ F \ mathbf T is the " anisotropic kernel " : this is the semisimple algebraic group obtained as the derived subgroup of the centraliser of { } _ F \ mathbf T in \ mathbf G ( the latter is only a reductive group ).
In particular, twisted forms of a split torus " T " over a field " K " are parametrized by elements of the Galois cohomology pointed set H ^ 1 ( G _ K, GL _ n ( \ mathbb { Z } ) ) with trivial Galois action on the coefficients.