Revision #652 → #1547 · back to history
addedThe A2 root system (six vectors in R²)c1ad4d26bed8
addedRoot system12689bd76f4d
addedEquivalent forms of conditions 3 and 4115114dee186
addedCrystallographic root systemd0e28f2f2188
addedReduced root system7c7a2bd156db
addedIntegrality equivalent to reflection differing by integer multiple617a13532c42
addedThe pairing operator is not an inner product02c099d78c35
addedRank of a root system8d5cc96039bc
addedIrreducible root systemeb95eebde85a
addedIsomorphic root systemsbbcbcfe93722
addedRoot latticee551920a1b94
addedWeyl group of a root systemcd90cc67eb0f
addedThe Weyl group is finitea9965a7802ce
addedThe unique rank-one root system A13bfbdadbb916
addedThe four rank-two root systems836b171d0100
addedA root system is not determined by its lattice1f27b102a066
addedIntersection with a subspace is a root system59cba3e2231c
addedPossible angles between two roots29a8cddf9372
addedRoot of a semisimple Lie algebra relative to a Cartan subalgebra182b1f0063d8
addedRoots of a semisimple Lie algebra form a root system7d3aeebd0fd3
addedCosine of angle between roots is constrained5afaa5c3db1d
addedPossible angle and length-ratio pairs0e40919bd725
addedSet of positive rootsd322f1a1d0eb
addedNegative rootsed5629f1605c
addedEvery set of positive roots arises from a hyperplane0517c82cef91
addedSimple root (base)f09f49972681
addedSimple roots form a basis with special properties2bbc0795f40a
addedSets of positive roots differ by the Weyl group0131073f90ca
addedDual root system2707056bb908
addedDouble coroot equals the root3dfceeef3d6d
addedDual of simple roots are simple roots of the dualdeb887477d51
addedSelf-dual root systems6197f45db0f5
addedIntegral element83f3d9c280d7
addedWeight latticec4c874965e4f
addedRoots are integral elements83d9ac97e136
addedIrreducible root system (via partition)23441908a358
addedIrreducible root systems correspond to Dynkin diagrams949885665c53
addedConstructing the Dynkin diagram from simple roots00732b82689f
addedWeyl group acts transitively on choices of simple roots042ff17a4b19
addedDynkin diagram is independent of choice of simple rootsf8290a3f9d03
addedIrreducible iff connected Dynkin diagramdb62305a4ab9
addedDynkin diagram of the dual reverses arrows81594e1fc3f3
addedWeyl chamber7de39941ac4f
addedFundamental Weyl chamber2caaae0be403
addedWeyl group permutes the Weyl chambersfd5e6930fc4d
addedWeyl group acts freely and transitively on Weyl chambers65fed2e1a547
addedEach Weyl orbit meets a chamber closure in one point2b670ad51100
addedIrreducible root systems classify Lie-theoretic objects377020654733
addedRoots are weights of the adjoint representationeb5235de2de2
addedSemisimple Lie algebra simple iff root system irreducibled17ae131d9c6
addedCartan subalgebras give isomorphic root systems6db1098b0404
addedRoot system determines the Lie algebra up to isomorphism3f2ef42ac496
addedEach irreducible root system has an associated simple Lie algebrae49b68541d8e
addedFour classical families and five exceptional cases0ffa4bd6903d
addedAt most two root lengths167515479272
addedSimply laced root system97fec42b7b2d
addedEqual-length roots lie in same Weyl orbit1f21d7faf31b
addedThe Aₙ root system6a947bf5c471
addedAₙ root lattice7510bf3dd810
addedThe Bₙ root systemf78f118ebcee
addedBₙ root lattice is all integer vectorse2dfeaf62f5f
addedB1 is isomorphic to A1cd14f8cfb802
addedThe Cₙ root system0ab179dd01f1
addedCₙ root lattice86f1c0935e81
addedC2 is isomorphic to B2b02c31cbc680
addedThe Dₙ root system89bd1ce4f880
addedDₙ root lattice equals Cₙ root latticeee6686a9bf94
addedD3 coincides with A347e619e48dfa
addedD4 has triality symmetry9de6ed8c22ef
addedThe E8 root system5327b7291de2
addedE8 has 240 rootsf2060ab8a264
addedThe E7 root system37dd0eb81885
addedThe E6 root systemdb71e77901f6
addedThe lattices Γ8 and Γ'8 are isomorphic7349aa8dfad1
addedThe F4 root system48a18ea936fd
addedF4 root lattice is the Hurwitz quaternionse6fac3f73210
addedThe G2 root systemf24e5c2db2b5
addedG2 root lattice equals A2 root latticee937c2f189fa
addedThe root poset orderingd7c15dc4f6e5
addedDegrees of fundamental invariants from the root poset9c4f4bd1f06e