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This section describes functions for working with the classification of
solvable Lie algebras of dimension 2, 3, and 4, and the classification of
nilpotent Lie algebras having dimensions 3,4,5, and 6.
The classification of solvable Lie algebras is
taken from [dG05], and applies to algebras over any base field.
The classification of nilpotent Lie algebras is taken from [dG07].
It lists the nilpotent Lie algebras over any base field, with the
exception of fields of characteristic 2, when the dimension is 6.
The functions
described here fall into two categories: functions for creating the
Lie algebras of the classification, and a function for
identifying a given solvable Lie algebra of dimension 2,3,4 or a given
nilpotent Lie algebra of dimension 3,4,5,6 as a member
of the list.
First we describe the classifications, in order to define names for the
Lie algebras that occur. We then describe the functions for working with
them in Magma.
We denote a solvable Lie algebra of dimension n by Lnk, where
k ranges between 1 and the number of classes of solvable Lie algebras
of dimension n. If the class depends on a parameter, say a,
then we denote the Lie algebra by Lnk(a). In such cases we also
state conditions under which Lnk(a) is isomorphic to Lnk(b)
(if there are any). We list the nonzero commutators only. The field
over which the Lie algebra is defined is denoted by F.
Here is the list of classes of solvable Lie algebras having dimension
not greater than 4:
- L21
- Abelian.
- L22
- [x2, x1]=x1.
- L31
- Abelian.
- L32
- [x3, x1]=x1, [x3, x2]=x2.
- L33(a)
- [x3, x1]=x2, [x3, x2]=ax1 + x2.
- L34(a)
- [x3, x1]=x2, [x3, x2]=ax1.
Condition of isomorphism: L34(a) isomorphic to L34(b) if and only if there is an α∈F * with a=α2 b.
- L41
- Abelian.
- L42
- [x4, x1]=x1, [x4, x2]=x2, [x4, x3]=x3.
- L43(a)
- [x4, x1]=x1, [x4, x2]=x3, [x4, x3]= - ax2 + (a + 1)x3.
- L44
- [x4, x2]=x3, [x4, x3]= x3.
- L45
- [x4, x2]=x3.
- L46(a, b)
- [x4, x1] = x2, [x4, x2]=x3, [x4, x3] = ax1 + bx2 + x3.
- L47(a, b)
- [x4, x1] = x2, [x4, x2]=x3, [x4, x3] = ax1 + bx2.
Isomorphism condition: L47(a, b) isomorphic to L47(c, d) if and only if there is an α∈F * with a=α3c and b=α2d.
- L48
- [x1, x2]=x2, [x3, x4]=x4.
- L49(a)
- [x4, x1] = x1 + ax2, [x4, x2]=x1, [x3, x1]=x1, [x3, x2]=x2.
Condition on the parameter a: T2 - T - a has no roots in F.
Isomorphism condition: L49(a) isomorphic to L49(b) if and only if the characteristic of F is not 2 and there is an α∈F * with a + (1/4) = α2(b + (1/4)), or the characteristic of F is 2 and X2 + X + a + b has roots in F.
- L410(a)
- [x4, x1] = x2, [x4, x2]=ax1, [x3, x1]=x1, [x3, x2]=x2.
Condition on F: the characteristic of F is 2.
Condition on the parameter a: a not∈F2.
Isomorphism condition: L410(a) isomorphic to L410(b) if and only if Y2 + X2b + a has a solution (X, Y)∈F x F with X != 0.
- L411(a, b)
- [x4, x1] = x1, [x4, x2] = bx2, [x4, x3]=(1 + b)x3, [x3, x1]=x2, [x3, x2]=ax1.
Condition on F: the characteristic of F is 2.
Condition on the parameters a, b: a != 0, b != 1.
Isomorphism condition: L411(a, b) isomorphic to L411(c, d) if and only if (a/c) and (δ2 + (b + 1)δ + b)/c are squares in F, where δ = (b + 1)/(d + 1).
- L412
- [x4, x1] = x1, [x4, x2]=2x2, [x4, x3] = x3, [x3, x1]=x2.
- L413(a)
- [x4, x1] = x1 + ax3, [x4, x2]=x2, [x4, x3] = x1, [x3, x1]=x2.
- L414(a)
- [x4, x1] = ax3, [x4, x3]=x1, [x3, x1]=x2.
Condition on parameter a: a != 0.
Isomorphism condition: L414(a) isomorphic to L414(b) if and only if there is an α∈F * with a=α2 b.
Over general fields the lists are not "precise" in the sense that some
classes that depend up on a parameter have an associated isomorphism
condition, but not a precise parametrization of the Lie algebras in that
class. However, for algebras over finite fields we are able to give a
precise list, by restricting the parameter values in some cases. In
this section we describe how this is done. Here F will be a finite
field of size q with primitive root γ.
- *
- If the characteristic of F is 2, then there are two algebras
of type L34(a), namely L34(0) and L34(1). If the characteristic
is not 2, then there are three algebras of this type, L34(0), L34(1),
L34(γ).
- *
- The class L47(a, b) splits into three classes:
L47(a, a) (a∈F), L47(a, 0) (a != 0), L47(0, b) (b != 0).
Among the algebras of the first class there are no isomorphisms. However,
for the other two classes we have the following:-
- (i)
- L47(a, 0) isomorphic to L47(b, 0) if and only if there is an
α∈F *
such that a=α3 b. If q ≡ 1 mod 3, then exactly a third of
the elements of F * are cubes, namely the γi with i divisible
by 3. So in this case we get three algebras, L47(1, 0),
L47(γ, 0), L47(γ2, 0). If q ≢ 1 mod 3
then F3=F, and hence there is only one algebra, namely L47(1, 0).
- (ii)
- L47(0, a) isomorphic to L47(0, b) if and only if there is an α
∈F *
such that a=α2 b. So if q is even then we get one algebra,
L47(0, 1). If q is odd we get two algebras, L47(0, 1),
L47(0, γ).
- *
- In [dG05] it is shown that there is only one Lie algebra
in the class L49(a). We let e be the smallest positive integer
such that T2 - T - γe has no roots in F. Then we take the Lie algebra
L49(γe) as representative of the class.
- *
- Over a finite field of characteristic 2 there are no Lie
algebras of type L410(a), as F2=F in that case.
- *
- There is only one Lie algebra of type L411(a, b) over a field
of characteristic 2, namely L411(1, 0).
- *
- If q is even then there is only one algebra of type L414(a),
namely L414(1). If q is odd, then there are two algebras,
L414(1) and L414(γ).
We denote a nilpotent Lie algebra of dimension r by Nrk, where
k ranges between 1 and the number of classes of nilpotent Lie algebras
of dimension r. If the class depends on a parameter, say a,
then we denote the Lie algebra by Nrk(a). The complete list of isomorphism
classes of nilpotent Lie algebras having dimensions 3, 4, 5 and 6, where
in dimension 6 we exclude base fields of characteristic 2 are as follows:
- N31
- Abelian.
- N32
- [x1, x2]=x3.
- N41
- Abelian.
- N42
- [x1, x2]=x3.
- N43
- [x1, x2]=x3, [x1, x3]=x4.
- N51
- Abelian.
- N52
- [x1, x2]=x3.
- N53
- [x1, x2]=x3, [x1, x3]=x4.
- N54
- [x1, x2] = x5, [x3, x4]=x5.
- N55
- [x1, x2]=x3, [x1, x3]= x5, [x2, x4] = x5.
- N56
- [x1, x2]=x3, [x1, x3]=x4, [x1, x4]=x5, [x2, x3]=x5.
- N57
- [x1, x2]=x3, [x1, x3]=x4, [x1, x4]=x5.
- N58
- [x1, x2]=x4, [x1, x3]=x5.
- N59
- [x1, x2]=x3, [x1, x3]=x4, [x2, x3]=x5.
There are nine 6-dimensional nilpotent Lie algebras denoted N6k for
k=1, ... , 9 which are the direct sum of N5k and a 1-dimensional
abelian ideal. Consequently, we get the following Lie algebras:-
- N610
- [x1, x2]=x3, [x1, x3]=x6, [x4, x5]=x6.
- N611
- [x1, x2]=x3, [x1, x3]=x4, [x1, x4]=x6, [x2, x3]=x6, [x2, x5]=x6.
- N612
- [x1, x2]=x3, [x1, x3]=x4, [x1, x4]=x6, [x2, x5]=x6.
- N613
- [x1, x2]=x3, [x1, x3]=x5, [x2, x4]=x5, [x1, x5]=x6, [x3, x4]=x6.
- N614
- [x1, x2]=x3, [x1, x3]=x4, [x1, x4]=x5, [x2, x3]=x5, [x2, x5]=x6, [x3, x4]= - x6.
- N615
- [x1, x2]=x3, [x1, x3]=x4, [x1, x4]=x5, [x2, x3]=x5, [x1, x5]=x6, [x2, x4]=x6.
- N616
- [x1, x2]=x3, [x1, x3]=x4, [x1, x4]=x5, [x2, x5]=x6, [x3, x4]= - x6.
- N617
- [x1, x2]=x3, [x1, x3]=x4, [x1, x4]=x5, [x1, x5]=x6, [x2, x3]= x6.
- N618
- [x1, x2]=x3, [x1, x3]=x4, [x1, x4]=x5, [x1, x5]=x6.
- N619(a)
- [x1, x2]=x4, [x1, x3]=x5, [x2, x4]=x6, [x3, x5]=a x6.
- N620
- [x1, x2]=x4, [x1, x3]=x5, [x1, x5]=x6, [x2, x4]=x6.
- N621(a)
- [x1, x2]=x3, [x1, x3]=x4, [x2, x3]=x5, [x1, x4]=x6, [x2, x5]= a x6 .
- N622(a)
- [x1, x2]=x5, [x1, x3]=x6, [x2, x4]= a x6, [x3, x4]=x5.
- N623
- [x1, x2]=x3, [x1, x3]=x5, [x1, x4]=x6, [x2, x4]= x5 .
- N624(a)
- [x1, x2]=x3, [x1, x3]=x5, [x1, x4]=a x6, [x2, x3]=x6, [x2, x4]= x5.
- N625
- [x1, x2]=x3, [x1, x3]=x5, [x1, x4]=x6.
- N626
- [x1, x2]=x4, [x1, x3]=x5, [x2, x3]=x6.
Note that for all classes that depend on a parameter a, the Lie algebra
with parameter a is isomorphic to the Lie algebra (from the same class) with
parameter b if and only if there is an α∈F * with a=α2 b.
This function returns the solvable Lie algebra Lnk over the field F.
The multiplication table is exactly the same as that given in the classification
of solvable algebras above, where the basis element xi corresponds to the
i-th basis element of the Lie algebra returned.
pars: SeqEnum Default: [ ]
If the Lie algebra Lnk depends on one or more parameters, then
the parameter pars specifies the parameter values corresponding
to the Lie algebra which is required.
> F<a>:= RationalFunctionField( Rationals() );
> K:= SolvableLieAlgebra( F, 3, 3 : pars:= [a] );
> K.3*K.1;
(0 1 0)
> K.3*K.2;
(a 1 0)
This function returns the nilpotent Lie algebra Nrk over the field F.
The multiplication table is exactly the same as that given in the classification
of nilpotent algebras above, where the basis element xi corresponds to the
i-th basis element of the Lie algebra returned.
pars: SeqEnum Default: [ ]
If the Lie algebra Nrk depends upon one or more parameters, then
the parameter pars specifies the parameter values corresponding
to the Lie algebra which is required.
> F<a>:= RationalFunctionField( Rationals() );
> K:= NilpotentLieAlgebra( F, 6, 19 : pars:= [a^3] );
> K.3*K.5;
( 0 0 0 0 0 a^3)
Given a finite field F and d an integer equal to 2, 3 or 4,
this function returns a sequence containing all solvable Lie algebras
of dimension d over the field F.
Given a finite field F and d an integer equal to 3, 4, 5 or 6,
this function returns a sequence containing all nilpotent Lie algebras of
dimension d over the field F. If the dimension is 6 then the
characteristic of F may not be 2.
Given a solvable Lie algebra L of dimension 2, 3, or 4, this function
returns data that identifies L with the isomorphic algebra in the
classification of solvable Lie algebras. (SLAC stands for Solvable Lie
Algebras Classification.) Three objects are returned: a string, a sequence
and a map.
The string gives the name of the Lie algebra as it occurs in the
classification, with information about the field and the parameters.
The sequence contains the parameters of the Lie algebra in the
classification to which L is isomorphic.
The map is an isomorphism from L to the corresponding Lie algebra
contained in the classification.
Given a nilpotent Lie algebra L of dimension 3, 4, 5 or 6 this function
returns data that identifies L with the isomorphic algebra in the
classification of nilpotent Lie algebras. (NLAC stands for Nilpotent
Lie Algebras Classification.) Three objects are returned: a string giving
the name of the algebra N in the classification, a sequence giving
the parameters for N, and the isomorphism mapping L to N.
Given an isomorphism f as returned by either IdDataSLAC or
IdDataNLAC, this function returns the matrix of that isomorphism.
The row convention is used, i.e., the i-th row contains the
coordinates of the image of the i-th basis element of the domain
of f.
We define a solvable Lie algebra of dimension 4 that depends on a
parameter a. We identify this Lie algebra as a member of the
classification.
> F<a>:= RationalFunctionField( Rationals() );
> T:= [ <1,2,2,1>, <1,2,3,a>, <1,4,4,a>, <2,1,2,-1>, <2,1,3,-a>, <4,1,4,-a> ];
> L:= LieAlgebra< F, 4 | T >;
> s,p,f:= IdDataSLAC( L );
> s;
L4_6( Univariate rational function field over Rational Field
Variables: a, 0, -a/(a^2 + 2*a + 1) )
> p;
[
0,
-a/(a^2 + 2*a + 1)
]
> MatrixOfIsomorphism( f );
[0 (-a - 1)/(a - 1) (-a - 1)/(a - 1) (a^2 + 2*a + 1)/(a^2 - a)]
[0 -1/(a - 1) -a/(a - 1) (a + 1)/(a - 1)]
[0 -1/(a^2 - 1) -a/(a^2 - 1) a/(a - 1)]
[1/(a + 1) 0 0 0]
So generically, the Lie algebra is isomorphic to L 46( 0,
- a/(a 2 + 2a + 1)).
We see that the parameters are not defined if a= - 1. Furthermore, the
isomorphism is not defined if a=∓ 1, or a=0. We investigate those cases.
> a:= 1;
> T:= [ <1,2,2,1>, <1,2,3,a>, <1,4,4,a>, <2,1,2,-1>, <2,1,3,-a>, <4,1,4,-a> ];
> L:= LieAlgebra< Rationals(), 4 | T >;
> s,p,f:= IdDataSLAC( L );
> s;
L4_3( Rational Field, 0 )
> MatrixOfIsomorphism( f );
[ 0 1 1 0]
[ 0 0 -1 1]
[ 0 0 0 1]
[ 1 0 0 0]
> a:= -1;
> T:= [ <1,2,2,1>, <1,2,3,a>, <1,4,4,a>, <2,1,2,-1>, <2,1,3,-a>, <4,1,4,-a> ];
> L:= LieAlgebra< Rationals(), 4 | T >;
> s,p,f:= IdDataSLAC( L );
> s;
L4_7( Rational Field, 0, 1 )
> MatrixOfIsomorphism( f );
[ 0 1/2 1/2 1/2]
[ 0 1/2 -1/2 -1/2]
[ 0 1/2 -1/2 1/2]
[ 1 0 0 0]
> a:= 0;
> T:= [ <1,2,2,1>, <1,2,3,a>, <1,4,4,a>, <2,1,2,-1>, <2,1,3,-a>, <4,1,4,-a> ];
> L:= LieAlgebra< Rationals(), 4 | T >;
> s,p,f:= IdDataSLAC( L );
> s;
L4_4( Rational Field )
> MatrixOfIsomorphism( f );
[ 0 0 1 0]
[ 0 1 0 -1]
[ 0 1 0 0]
[ 1 0 0 0]
We see that for a=1 the Lie algebra is isomorphic to L 43(0), and the
isomorphism is defined over any field. If a= - 1, then the Lie algebra
is isomorphic to L 47( 0, 1 ). However, the isomorphism is not defined
if the characteristic of the field is 2. But then we are back in the
case a=1. Finally, for a=0 the Lie algebra is isomorphic to
L 44.
The positive part of the simple Lie algebra of type G 2 is a nilpotent
Lie algebra of dimension six. We identify it in the classification of nilpotent
algebras, both in characteristic 0, and in characteristic 3.
> L:= LieAlgebra( "G2", Rationals() );
> x,y,h:= ChevalleyBasis( L );
> x;
[ (0 0 0 0 0 0 0 0 1 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 1 0 0 0 0), (0 0 0
0 0 0 0 0 0 0 1 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 1 0 0), (0 0 0 0 0 0 0
0 0 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 1) ]
So we see that the positive part is spanned by basis vectors L.i with
i=9, 10, 11, 12, 13 and 14. So
> K:= sub< L | [ L.i : i in [9,10,11,12,13,14] ] >;
> name,pp,f:= IdDataNLAC( K );
> name;
N6_16( Rational Field )
> MatrixOfIsomorphism( f );
[ 1 0 0 0 0 0]
[ 0 1 0 0 0 0]
[ 0 0 1 0 0 0]
[ 0 0 0 1/2 0 0]
[ 0 0 0 0 1/6 0]
[ 0 0 0 0 0 1/6]
> L:= LieAlgebra( "G2", GF(3) );
> K:= sub< L | [ L.i : i in [9,10,11,12,13,14] ] >;
> name,pp,f:= IdDataNLAC( K );
> name;
N6_19( Finite field of size 3, 0 )
We see that in characteristic 3 L is isomorphic to a Lie algebra from a
different class.
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