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Take a simple Lie algebra over the complex numbers, and consider the
connected component of its automorphism group that contains the identity.
This group acts on the Lie algebra, and there is interest in understanding
the nature of its orbits. The nilpotent orbits for simple Lie algebras
have been classified. We refer to the book by Collingwood and McGovern
[CM93] for the details of this classification.
The main technical tools used for the classification are the weighted
Dynkin diagram and the sl2-triple. The weighted Dynkin diagram is
the Dynkin diagram of the root system of the Lie algebra, with labels
that can be 0, 1, 2. A nilpotent orbit is uniquely determined by its
weighted Dynkin diagram. By the Jacobson-Morozov theorem a nilpotent
element of a semisimple Lie algebra can be embedded (as nilpositive
element) in an sl2-triple. Now two nilpotent elements are conjugate
(under the group) if and only if the corresponding sl2-triples are
conjugate. This yields a bijection between nilpotent orbits and
conjugacy classes of simple subalgebras isomorphic to sl2.
This section describes functions for working with the classification
of nilpotent orbits in simple Lie algebras. One of the main invariants
of a nilpotent orbit is its weighted Dynkin diagram. We represent
such a diagram by a sequence of its labels; they are mapped to the
nodes of the Dynkin diagram in the order determined by the Cartan matrix
of the root datum. Also, an sl2-triple is represented by a sequence
[f, h, e] of three elements of a Lie algebra; these satisfy the
commutation relations [h, e]=2e, [h, f]= - 2f, [e, f]=h.
Throughout this section we consider orbits that are not the zero orbit.
Given a simple Lie algebra L, and a sequence wd consisting of integers
that are 0, 1, or 2, this function returns true if wd corresponds to
a nilpotent orbit (in other words, if it is the weighted Dynkin diagram
of a nilpotent orbit). If wd does corresponds to a nilpotent orbit, an
sl2-triple in L, such that the third element lies in the nilpotent
orbit corresponding to the weighted Dynkin diagram is returned. If wd
does not correspond to a nilpotent orbit. the second return value is a
sequence consisting of three zeros of L.
We can use this function to find the classification of the nilpotent orbits
of a given Lie algebra. First we construct all possible weighted Dynkin
diagrams, and then we remove those that do not correspond to an orbit.
> L:= LieAlgebra( RootDatum("D4"), Rationals() );
> [ w : i,j,k,l in [0,1,2] | IsGenuineWeightedDynkinDiagram(L, w)
> where w := [i,j,k,l] ];
[
[ 0, 0, 0, 2 ],
[ 0, 0, 2, 0 ],
[ 0, 1, 0, 0 ],
[ 0, 2, 0, 0 ],
[ 0, 2, 0, 2 ],
[ 0, 2, 2, 0 ],
[ 1, 0, 1, 1 ],
[ 2, 0, 0, 0 ],
[ 2, 0, 2, 2 ],
[ 2, 2, 0, 0 ],
[ 2, 2, 2, 2 ]
]
This returns the nilpotent orbit in the simple Lie algebra L with
weighted Dynkin diagram given by the sequence wd. It is not
checked whether the weighted Dynkin diagram really corresponds to a
nilpotent orbit.
This returns the nilpotent orbit in the simple Lie algebra L having
representative e. Here e has to be a nilpotent element of
the Lie algebra. This condition is not checked by the function.
> L:= LieAlgebra( RootDatum("A2"), Rationals() );
> NilpotentOrbit( L, [2,2] );
Nilpotent orbit in Lie algebra of type A2
> NilpotentOrbit( L, L.1 );
Nilpotent orbit in Lie algebra of type A2
Given a simple Lie algebra L, this function returns the sequence
of all nilpotent orbits in the simple Lie algebra L.
We compute the nilpotent orbits of the Lie algebra of type D 4, and
observe that they are the same as those found in Example H110E59.
> L:= LieAlgebra( RootDatum("D4"), Rationals() );
> o:= NilpotentOrbits(L);
> [ WeightedDynkinDiagram(orb) : orb in o ];
[
[ 2, 2, 2, 2 ],
[ 2, 0, 2, 2 ],
[ 2, 2, 0, 0 ],
[ 0, 2, 0, 2 ],
[ 0, 2, 2, 0 ],
[ 0, 2, 0, 0 ],
[ 1, 0, 1, 1 ],
[ 2, 0, 0, 0 ],
[ 0, 0, 0, 2 ],
[ 0, 0, 2, 0 ],
[ 0, 1, 0, 0 ]
]
Here o is a nilpotent orbit in a simple Lie algebra of classical type
(i.e., of type An, Bn, Cn or Dn). The nilpotent orbits
for the Lie algebras of these types have been classified in terms of
partitions. This function returns the partition corresponding to the orbit.
> L:= LieAlgebra( RootDatum("D4"), Rationals() );
> orbs:= NilpotentOrbits( L );
> Partition( orbs[5] );
[ 4, 4 ]
> Partition( orbs[6] );
[ 3, 3, 1, 1 ]
Given a nilpotent orbit o in a simple Lie algebra L, this function
returns an sl2-triple, [f, h, e] of elements of L, such that e
lies in the nilpotent orbit.
Given a semisimple Lie algebra L of characteristic 0, and a nilpotent
element e of L, this function returns an sl2-triple [f, h, e] of
elements of L. It may also work for other Lie algebras, and in other
characteristics, but this is not guaranteed.
Given a nilpotent orbit o for a simple Lie algebra L, this function
returns an e∈L lying in the orbit.
Given a nilpotent orbit o for a simple Lie algebra L, this function
returns its weighted Dynkin diagram.
We take some nilpotent element in the Lie algebra of type E 8 and we
find the weighted Dynkin diagram of the orbit it lies in.
> L:= LieAlgebra( RootDatum("E8"), Rationals() );
> x,_,_:= ChevalleyBasis(L);
> orb:= NilpotentOrbit( L, x[1]+x[10]-x[30]+3*x[50]-2*x[100] );
> WeightedDynkinDiagram( orb );
[ 1, 0, 0, 0, 0, 0, 0, 1 ]
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