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The radical of a Lie algebra is the maximal soluble ideal.
A Lie algebra is called reductive
if its radical is equal to its centre, and semisimple if its radical is trivial.
A Lie algebra is almost reductive (resp. simple, semisimple) if the corresponding
group of Lie type is reductive (resp. simple, semisimple).
Note that these concepts are equivalent if the field has characteristic zero.
The commands in the section construct almost reductive Lie algebras over an
arbitrary field.
Such Lie algebras have a corresponding root datum.
The matrix versions of these commands give the standard matrix representation,
which is the smallest degree representation (with a few exceptions for small
characteristic fields).
Subsections
Some of the functions in this sections have an optional flag Isogeny.
See Section Constructing Groups of Lie Type for the possible values of this flag.
SemisimpleLieAlgebra(N, k) : MonStgElt, Rng -> AlgLie
Isogeny: . Default: "Ad"
The (almost) semisimple Lie algebra over the ring k with Cartan type N
given as a string.
SemisimpleMatrixLieAlgebra(N, k) : MonStgElt, Rng -> AlgMatLie
Isogeny: . Default: "Ad"
The (almost) semisimple matrix Lie algebra over the ring k with Cartan type N
given as a string.
The twisted (almost) semisimple Lie algebra over the finite field k with Cartan type N
given as a string and twist given by the permutation p.
The twist should either be a permutation of the indices of the simple roots, or
of the indices of all roots.
This example demonstrates the use of the Isogeny option.
Over a field of characteristic zero, this option only effects the basis used.
In characteristic p, it sometimes effects the isomorphism type of the algebra.
For type An with
p|(n + 1), the default Isogeny is "Ad" (adjoint), which gives an algebra with
nontrivial derived subalgebra but no centre:
> L := LieAlgebra("A4", GF(5));
> Dimension(L);
24
> Dimension(L*L);
23
If you take Isogeny to be "SC" (simply connected), you get a perfect algebra with a nontrivial centre.
> L := LieAlgebra("A4", GF(5) : Isogeny:="SC");
> Dimension(L);
24
> Dimension(L*L);
24
> Dimension(Centre(L));
1
If p2|(n + 1) there is an intermediate isogeny type which has both
a centre and a nontrivial derived algebra:
> L := LieAlgebra("A24", GF(5) : Isogeny:=5);
> Dimension(L);
624
> Dimension(L*L);
623
> Dimension(Centre(L));
1
Similar results can be obtained by constructing the Lie algebra from a root datum.
This kind of phenomenon happens whenever the characteristic divides the
order of the fundamental group of your root datum.
See [Hog82] for more details.
> R := RootDatum("E6");
> #FundamentalGroup(R);
3
> L := LieAlgebra(R,GF(3));
> L;
Lie Algebra of dimension 78 with base ring GF(3)
> L*L;
Lie Algebra of dimension 77 with base ring GF(3)
ReductiveLieAlgebra(R, k) : RootDtm, Rng -> AlgLie
The (almost) reductive Lie algebra with root datum R over the ring k
(see resp. ROOT DATA).
ReductiveMatrixLieAlgebra(R, k) : RootDtm, Rng -> AlgMatLie
The (almost) reductive matrix Lie algebra with root datum R over the ring k
(see resp. ROOT DATA).
SemisimpleLieAlgebra(R, k) : RootSys, Rng -> AlgLie
The (almost) semisimple Lie algebra with crystallographic root system R over the ring
k (see Chapter ROOT SYSTEMS).
SemisimpleLieAlgebra(R, k) : RootSys, Rng -> AlgMatLie
The (almost) semisimple Lie algebra with crystallographic root system R over the ring
k (see Chapter ROOT SYSTEMS).
The twisted (almost) semisimple Lie algebra over the finite field k with root system
R and twist given by the permutation p.
The twist should either be a permutation of the indices of the simple roots, or
of the indices of all roots.
We construct some semisimple Lie algebras.
> SemisimpleLieAlgebra( "G2 B3", Rationals() );
Lie Algebra of dimension 35 with base ring Rational Field
> SemisimpleLieAlgebra( "E8", GF(2) );
Lie Algebra of dimension 248 with base ring GF(2)
> DynkinDiagram("E6");
E6 1 - 3 - 4 - 5 - 6
|
2
> LieAlgebra( "E6", GF(5), Sym(6)!(1,6)(3,5) );
Lie Algebra of dimension 78 with base ring GF(5)
LieAlgebra(C, k) : AlgMatElt, Rng -> AlgLie
The (almost) semisimple Lie algebra with crystallographic Cartan matrix C over the ring
k (see Section Cartan Matrices).
MatrixLieAlgebra(C, k) : AlgMatElt, Rng -> AlgMatLie
The (almost) semisimple matrix Lie algebra with crystallographic Cartan matrix C over the ring
k (see Section Cartan Matrices).
LieAlgebra(D, k) : GrphDir, Rng -> AlgLie
The (almost) semisimple Lie algebra with Dynkin digraph D over the ring k (see
Section Dynkin Digraphs).
MatrixLieAlgebra(D, k) : GrphDir, Rng -> AlgMatLie
The (almost) semisimple matrix Lie algebra with Dynkin digraph D over the ring k (see
Section Dynkin Digraphs).
This function constructs the (almost) simple Lie algebra of Cartan type Xn over the
ring k. The result is a Lie algebra defined by a multiplication table. Here
X is a string which can be one of "A", "B", "C", "D",
"E", "F" or "G" and n is a positive integer.
In a few cases the Lie algebra returned by this function is not simple;
examples are the Lie algebras of type An over a field of characteristic
p>0 where p divides n + 1, and the Lie algebras of type D1 and D2.
This function constructs the (almost) simple matrix Lie algebra of Cartan type Xn over the
ring k. The result is a Lie algebra defined by a multiplication table. Here
X is a string which can be one of "A", "B", "C", "D",
"E", "F" or "G" and n is a positive integer.
In a few cases the Lie algebra returned by this function is not simple;
examples are the Lie algebras of type An over a field of characteristic
p>0 where p divides n + 1, and the Lie algebras of type D1 and D2.
We construct some standard simple Lie algebras.
> SimpleLieAlgebra("D", 7, RationalField());
Lie Algebra of dimension 91 with base ring Rational Field
> SimpleLieAlgebra("G", 2, GF(5));
Lie Algebra of dimension 14 with base ring GF(5)
Simple Lie algebras over fields of characteristic 0 have been
classified and are only all twisted forms of Lie algebras of types
Aell, Bell, Cell, Dell, E6, E7, E8, F4 and G2
(see previous Subsection).
Over fields of finite characteristic p, the analogues of
these algebras are called classical-type (including the exceptional
algebras).
Over such fields there are other simple Lie algebras, the first of them found
by Witt sometimes before 1937. For p geq7, the only non-classical
simple Lie algebras are the Lie algebras of
Cartan-type,
which we discuss in this section.
For p=5, one further class of simple Lie algebras occurs:
Melikian algebras.
In characteristic 2 and 3, the classification of simple Lie algebras
is not yet complete.
Cartan-type Lie algebras are non-classical Lie algebras which arise
from infinite dimensional algebras of differential operators over C:
 - (generalised) Witt algebras,
- special and conformal special Lie algebras,
- Hamiltonian and conformal Hamiltonian Lie algebras,
- and contact Lie algebras.
The notation and the description of these Lie algebras
closely follow Strade and Farnsteiner [Str04] and [SF88].
Where the notation of the two books differs, we follow [Str04].
Let F be a finite field of characteristic p>0 and m a positive integer.
We refer for the definition of O(m) and x(a) to [Str04, 2.1].
The basis of O(m) is { x(a) | 0≤a, a∈Nm }.
Let n be a sequence of positive integers of length m and set
N := ∑i=1m ni. Define
O(m, n) := < x(a) | 0 ≤ai < pni >
For i=1, ..., m denote by ∂i the derivation of O(m) defined by
∂i(xj(r)) = δi, j xj(r - 1).
Now define
W(m, n) := ∑i=1m O(m)∂i.
The algebra W(m, n) is the Witt algebra
and has dimension m pN over F.
In particular, W(1, [1]) is the standard p-dimensional Witt algebra.
The Witt algebra W(m, n) is simple unless p=2 and m=1 ([SF88, 4.2.4(1)])
and is restrictable if and only if n=[1, ..., 1] ([SF88, 4.2.4(2)]).
Now define
Ω0(m, n) := O(m, n),
Ω1(m, n) := Hom()O(m, n)(W(m, n), O(m, n)),
Ωr(m, n) := bigwedger Ω1(m, n),
Ω(m, n) := bigoplus Ωr(m, n).
Let m≥2 and ωS = dx1 ^ ... ^ dxm. Define the following
subalgebras of W(m, n):
S(m, n) := { D∈W(m, n) | D(ωS) = 0 },
CS(m, n) := { D∈W(m, n) | D(ωS) ∈FωS }.
The algebra S(m, n) is the special and CS(m, n) is the
conformal special
Lie algebra. The dimension of S(m, n) over F is (m - 1)pN + 1 and
the dimension of CS(m, n) is dim S(m, n) + 1.
Suppose m≥3. Then the algebra S(m, n)(1) is simple ([SF88, 4.3.5(1)])
and is restrictable if and only if n=[1, ..., 1] ([SF88, 4.3.5(2)]).
Let p>2, m = 2r ≥2 and let ωH = ∑limitsi=1r dxi ^ dxi + r. Define the following
subalgebras of W(m, n):
H(m, n) := { D∈W(m, n) | D(ωH) = 0 },
CH(m, n) := { D∈W(m, n) | D(ωH) ∈FωH }.
The algebra H(m, n) is the Hamiltonian and CH(m, n) is the
conformal Hamiltonian
Lie algebra. The dimension of H(m, n) over F is pN - 1 and
the dimension of CH(m, n) is dim H(m, n) + 1.
The algebra H(m, n)(2) is simple ([SF88, 4.4.5(1)])
and is restrictable if and only if n=[1, ..., 1] ([SF88, 4.4.5(2)]).
And, if m>2, then H(m, n)(2) = H(m, n)(1).
Let p>2, m = 2r + 1 ≥3 and let ωK = dxm + ∑limitsi=1r (xidxi + r - xi + rdxi). Define the following
subalgebra of W(m, n):
K(m, n) := { D∈W(m, n) | D(ωK) ∈O(m, n)ωK },
The algebra K(m, n) is the contact
Lie algebra. The dimension of K(m, n) over F is pN.
The algebra K(m, n)(1) is simple ([SF88, 4.5.5(1)])
and is restrictable if and only if n=[1, ..., 1] ([SF88, 4.5.6]).
And, if m + 3 not = 0 mod p, then K(m, n)(1) = K(m, n).
Check: BoolElt Default: false
Construct the Witt algebra W(m, n) over the finite field F, where
m must be a positive integer and n a sequence of positive integers of length m.
If the optional argument Check is true, the algebra is checked to be Lie
upon construction.
We compute the Witt algebra W(2, [2, 3]) over GF(9):
> W := WittLieAlgebra( GF(9), 2, [2,3] );
> W;
Lie Algebra of dimension 486 with base ring GF(3^2)
> IsSimple(W);
true
and the standard Witt algebra W(1, [1]) over GF(2):
> W := WittLieAlgebra( GF(2), 1, [1] );
> W;
Lie Algebra of dimension 2 with base ring GF(2)
> IsSimple(W);
false
> IsRestrictedLieAlgebra(W);
true [ (0 0), (0 1) ]
ConformalSpecialLieAlgebra(F, m, n) : Fld, RngIntElt, SeqEnum[RngIntElt] -> AlgLie, AlgLie, AlgLie
Check: BoolElt Default: false
Construct the (conformal) special Lie algebra (C)S(m, n) over the finite field F, where
m≥2 must be an integer and n a sequence of positive integers of length m.
If the optional argument Check is true, the algebra is checked to be Lie
upon construction. SpecialLieAlgebra returns W(m, n), in which S(m, n) is contained,
as the second return value. Similarly, ConformalSpecialLieAlgebra returns S(m, n)
it contains and W(m, n) it is contained in, as second and third return values.
We compute both S(3, [1, 2, 1]) and CS(3, [1, 2, 1]) over GF(9):
> CS,S,W := ConformalSpecialLieAlgebra( GF(9), 3, [1,2,1] );
> CS;S;W;
Lie Algebra of dimension 164 with base ring GF(3^2)
Lie Algebra of dimension 163 with base ring GF(3^2)
Lie Algebra of dimension 243 with base ring GF(3^2)
> IsSimple(S);
false
> IsSimple(S*S);
true
> IsRestrictedLieAlgebra(S*S);
false []
ConformalHamiltonianLieAlgebra(F, m, n) : Fld, RngIntElt, SeqEnum[RngIntElt] -> AlgLie, AlgLie, AlgLie
Check: BoolElt Default: false
Construct the (conformal) Hamiltonian Lie algebra (C)H(m, n) over the finite field F
of characteristic at least 3, where
m≥2 must be even and n a sequence of positive integers of length m.
If the optional argument Check is true, the algebra is checked to be Lie
upon construction. HamiltonianlLieAlgebra returns W(m, n), in which H(m, n) is contained,
as the second return value. Similarly, ConformalHamiltonianLieAlgebra returns H(m, n)
it contains and W(m, n) it is contained in, as second and third return values.
We compute both H(2, [2, 2]) and CH(2, [2, 2]) over GF(9):
> CH,H,W := ConformalHamiltonianLieAlgebra( GF(9), 2, [2,2] );
> CH;H;W;
Lie Algebra of dimension 81 with base ring GF(3^2)
Lie Algebra of dimension 80 with base ring GF(3^2)
Lie Algebra of dimension 162 with base ring GF(3^2)
> IsSimple(H);
false
> IsSimple(H*H);
true
> IsSimple(H*H*H);
true
> IsRestrictedLieAlgebra(H*H*H);
false []
Check: BoolElt Default: false
Construct the contact Lie algebra K(m, n) over the finite field F
of characteristic at least 3, where
m≥3 must be odd and n a sequence of positive integers of length m.
If the optional argument Check is true, the algebra is checked to be Lie
upon construction. ContactLieAlgebra returns W(m, n), in which K(m, n) is contained,
as the second return value.
We compute the contact Lie algebra K(3, [1, 1, 1]) over GF(5):
> K,W := ContactLieAlgebra( GF(5), 3, [1,1,1] );
> K;W;
Lie Algebra of dimension 125 with base ring GF(5)
Lie Algebra of dimension 375 with base ring GF(5)
> K*K eq K;
true
> IsSimple(K);
true
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