Example AlgLie_TwistedLieAlgebra (H110E19)

> 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)
> Rt := TwistedRootDatum(RootDatum("D4") : Twist := 3);
> k := GF(7);
> L, phi, Lp, H, Hp := TwistedLieAlgebra(Rt, k);
> L;
Lie Algebra of dimension 28 with base ring GF(7)
> Lp;
Lie Algebra of dimension 28 with base ring GF(7^3)
> phi(L.3);
(0 0 ksi^49 ksi^7 ksi 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
> IsSplitToralSubalgebra(L, H);
true
> IsSplitToralSubalgebra(Lp, Hp);
true
> forall{b : b in Basis(H) | phi(b) in Hp};
true

Cartan-Type Lie Algebras

Simple Lie algebras over fields of characteristic 0 have been classified and are precisely the twisted forms of Lie algebras of types A_l, B_l, C_l, D_l, 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 ≥7, 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, which are discussed in the next section. 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.

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∈N^m }.

Let n be a sequence of positive integers of length m and set N := ∑_i=1^m n_i. Define

O(m, n) := < x^(a) | 0 ≤a_i < p^n_i >

For i=1, ..., m denote by ∂_i the derivation of O(m) defined by

∂_i(x_j^(r)) = δ_{i, j} x_j^{(r - 1)}.

Now define

W(m, n) := ∑_i=1^m O(m)∂_i.

The algebra W(m, n) is the Witt algebra and has dimension m p^N 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)]).

Further define

Ω⁰(m, n) := O(m, n),

Ω¹(m, n) := Hom()_{O(m, n)}(W(m, n), O(m, n)),

Ω^r(m, n) := bigwedge^r Ω¹(m, n),

Ω(m, n) := bigoplus Ω^r(m, n).

Let m≥2 and ω_S = dx₁ ^ ... ^ dx_m. Define the following subalgebras of W(m, n):

S(m, n) := { D∈W(m, n) | D(ω_S) = 0 },

CS(m, n) := { D∈W(m, n) | D(ω_S) ∈Fω_S }.

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)p^N + 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 ω_H = ∑limits_i=1^r dx_i ^ dx_{i + r}. Define the following subalgebras of W(m, n):

H(m, n) := { D∈W(m, n) | D(ω_H) = 0 },

CH(m, n) := { D∈W(m, n) | D(ω_H) ∈Fω_H }.

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 p^N - 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 ω_K = dx_m + ∑limits_i=1^r (x_idx_{i + r} - x_{i + r}dx_i). Define the following subalgebra of W(m, n):

K(m, n) := { D∈W(m, n) | D(ω_K) ∈O(m, n)ω_K },

The algebra K(m, n) is the contact Lie algebra. The dimension of K(m, n) over F is p^N.

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]). If m + 3 ≢ 0 mod p, then K(m, n)⁽¹⁾ = K(m, n).

WittLieAlgebra(F, m, n) : Fld, RngIntElt, SeqEnum[RngIntElt] -> AlgLie, Map

    Check: BoolElt                      Default: false

The Witt algebra W(m, n) is constructed 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.

An invertible map from the polynomial ring P over F of degree 2m to W(m, n) is returned as second value, to assist in identifying the elements of W(m, n). For 1 ≤i≤m the i-th generator of P maps to x_i in W(m, n), and for m + 1 ≤i≤2m the i-th generator of P maps to δ_{i - m} in W(m, n).

> W, phi := WittLieAlgebra(GF(9), 2, [2,1]);
> W;
Lie Algebra of dimension 54 with base ring GF(3^2)
> IsSimple(W);
true
> P<x1, x2, d1, d2> := Domain(phi);
> phi(x1*d1);
(0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
> (phi(x1*d1)*phi(x1^2*x2*d2)) @@ phi;
2*x1^2*x2*d2

> 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) ]

    Check: BoolElt                      Default: false

> 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 []

    Check: BoolElt                      Default: false

> 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

> 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

    Check: BoolElt                      Default: false

> M, phi := MelikianLieAlgebra(GF(5), 2, 1);     
> M;
Lie Algebra of dimension 625 with base ring GF(5)
> IsSimple(M);
true

> P<x1, x2, d1, d2, dp1, dp2> := Domain(phi);
> V := VectorSpace(GF(5), Dimension(M));
> W := sub<V | [ V | phi(x1^i*x2^j*d) : i in [0..24], j in [0..4], 
>                                                     d in [d1,d2] ]>;
> O := sub<V | [ V | phi(x1^i*x2^j) : i in [0..24], j in [0..4] ]>;
> Wp := sub<V | [ V | phi(x1^i*x2^j*d) : i in [0..24], j in [0..4], 
>                                                      d in [dp1,dp2] ]>;
> Dimension(W), Dimension(O), Dimension(Wp);
250 125 250
> Dimension(W meet O), Dimension(W meet Wp), Dimension(O meet Wp);
0 0 0

> m := func< A, B | sub<V | [ V | M!a*M!b : a in Basis(A), b in Basis(B) ]> >;
> WxWp := m(W, Wp); [ WxWp subset VV : VV in [W, O, Wp] ]; 
[ false, false, true ]

> VV := [W, O, Wp]; VVnm := ["W", "O", "W'" ];
> mm := function(A, B)
>   AB := m(A, B);
>   for i in [1..#VV] do 
>     if AB eq VV[i] then return VVnm[i]; end if;
>   end for;
>   return "??";
> end function;
> mm(W, Wp);  
W'
> for i,j in [1..#VV] do
>   printf "[ %2o, %2o ] = %2o%o", VVnm[i], VVnm[j], mm(VV[i], VV[j]), 
>     (j eq 3) select "\n" else ", ";
> end for;
[  W,  W ] =  W, [  W,  O ] =  O, [  W, W' ] = W'
[  O,  W ] =  O, [  O,  O ] = W', [  O, W' ] =  W
[ W',  W ] = W', [ W',  O ] =  W, [ W', W' ] =  O