L eq K : AlgMatLie, AlgMatLie -> BoolElt
Returns true if, and only if, the Lie algebras L and K are equal.
L ne K : AlgMatLie, AlgMatLie -> BoolElt
Returns true if, and only if, the Lie algebras L and K are not equal.
L subset K : AlgMatLie, AlgMatLie -> BoolElt
Returns true if, and only if, the Lie algebra L is contained in the Lie
algebra K.
Returns true if, and only if, the Lie algebra L is not contained in the Lie
algebra K.
L meet M : AlgMatLie, AlgMatLie -> AlgMatLie
The intersection of the Lie algebras L and M is returned.
Note that L and M have a common superalgebra.
L * M : AlgMatLie, AlgMatLie -> AlgMatLie
The Lie algebra product [L, M] of the algebras L and M is
returned. Note that L and M must have a common superalgebra.
The (left-normed) n-th power of the (structure constant) Lie algebra
L, i.e., (( ... (L * L) * ... ) * L) is constructed.
The map giving the morphism from the (structure constant) Lie algebra
L to M is constructed. Either L is a subalgebra of M,
in which case the embedding of L into M is returned, or M is a quotient
algebra of L, in which case the natural epimorphism from L onto M is
returned.
IsIsomorphic(L, M) : AlgLie, AlgLie -> BoolElt, .
HL: AlgLie Default: false
HM: AlgLie Default: false
Returns true if the Lie algebras L and M are isomorphic. It is
currently implemented for trivial cases (such as when the dimensions differ),
reductive Lie algebras, solvable Lie algebras up to dimension 4,
nilpotent Lie algebras up to dimension 6 (some special cases excluded).
The solvable and nilpotent cases are handled using the databases for such
algebras described in Section Solvable and Nilpotent Lie Algebras Classification).
In the case of reductive Lie algebras, split maximal toral subalgebras for L
and M may be provided in the optional arguments HL and HM, respectively.
If these are not provided an attempt is made to compute them, a process which
may fail, particularly in characteristic 0.
This intrinsic has two return values: the first a boolean describing whether L
and M are isomorphic. If so, the second is an isomorphism from L to M,
otherwise the second is a string describing the reason for non-isomorphism.
An error is thrown if isomorphism cannot be determined.
HL: AlgLie Default: false
HM: AlgLie Default: false
Returns true if Magma can determine isomorphism between Lie algebras
L and M. If so, the second return value is whether L and M are
isomorphic, and the third is an isomorphism or a string (describing the
reason for non-isomorphism). Refer to IsIsomorphic for more details
on applicability and the meanings of the return values.
Returns true if the mapping m between two Lie algebras is an isomorphism
of Lie algebras.
We demonstrate that B 2 and C 2 are isomorphic over Q.
> k := Rationals();
> L := LieAlgebra("B2", k); M := LieAlgebra("C2", k);
> b, c := IsIsomorphic(L, M);
> b;
true
> IsIsomorphism(c);
true
> c(L.1);
(0 0 1 0 0 0 0 0 0 0)
We demonstrate that B 3 and C 3 are non-isomorphic over Q.
> L := LieAlgebra("B3", k); M := LieAlgebra("C3", k);
> b, c := IsIsomorphic(L, M);
> b;
false
> c;
21-dim component of L1 of type R1: Adjoint root datum of dimension 3 of type B3
didn't match R2:
Adjoint root datum of dimension 3 of type C3
We demonstrate that two distinct isogenies of B 2 are isomorphic over Q.
> L := LieAlgebra("B2", k : Isogeny := "Ad");
> M := LieAlgebra("B2", k : Isogeny := "SC");
> b, c := IsIsomorphic(L, M);
> b;
true
For larger nilpotent algebras Magma cannot decide on the isomorphism question.
> L := LieAlgebra("B4", k);
> pL, _, _ := StandardBasis(L);
> subL := sub<L | pL>;
> subL;
Lie Algebra of dimension 16 with base ring Rational Field
> M := LieAlgebra("C4", k);
> pM, _, _ := StandardBasis(M);
> subM := sub<M | pM>;
> subL;
Lie Algebra of dimension 16 with base ring Rational Field
> IsNilpotent(subL), IsNilpotent(subM);
true true
> a,b,c := IsKnownIsomorphic(subL, subM);
> a;
false
We demonstrate that in characteristic 3 the Lie algebras of type G 2 and
A 2 have isomorphic nontrivial ideals.
> k := GF(3);
> CSL := CompositionSeries(LieAlgebra("G2", k));
> CSL;
[
Lie Algebra of dimension 7 with base ring GF(3),
Lie Algebra of dimension 14 with base ring GF(3)
]
> L := CSL[1];
> CSM := CompositionSeries(LieAlgebra("A2", k));
> CSM;
[
Lie Algebra of dimension 7 with base ring GF(3),
Lie Algebra of dimension 8 with base ring GF(3)
]
> M := CSM[1];
> a,b,c := IsKnownIsomorphic(L, M);
> a;
true
> b, c;
true Mapping from: AlgLie: L to AlgLie: M given by a rule
> IsIsomorphism(c);
true
CoefficientRing(L) : AlgMatLie -> Rng
BaseRing(L) : AlgLie -> Rng
BaseRing(L) : AlgMatLie -> Rng
The coefficient ring (or base ring) over which the Lie algebra L is defined.
Dimension(L) : AlgMatLie -> RngIntElt
The dimension of the Lie algebra L.
# L : AlgMatLie -> RngIntElt
The cardinality of the Lie algebra L, if the coefficient ring is finite.
This returns a sequence of integers, of length equal to the dimension
of L. If the i-th element of this sequence is ai then ai is the
minimal non-negative integer such that aiei = 0. So if L is defined
over a field, then the sequence consists of zeros.
> T:= [ <1,2,2,2>, <2,1,2,2> ];
> t:= [0,4];
> L:= LieAlgebra< t | T : Rep:= "Dense" >;
> Moduli(L);
[ 0, 4 ]
ChangeRing(L, S) : AlgMatLie, Rng -> AlgMatLie, Map
Given a Lie algebra L with base ring R, together with a ring S,
this function constructs the Lie algebra M with base ring S obtained
by coercing the coefficients of elements of L into S. The homomorphism
from L to M is produced as second return value.
ChangeRing(L, S, f) : AlgMatLie, Rng, Map -> AlgMatLie, Map
Given a Lie algebra L with base ring R, together with a ring S and a
map f: R -> S, this function constructs the Lie algebra M
with base ring S obtained by mapping the coefficients of elements of
L into S via f. The homomorphism from L to M is produced as
the second return value.
BasisElement(A, i) : AlgMatLie, RngIntElt -> AlgMatLieElt
A . i : AlgLie, RngIntElt -> AlgLieElt
A . i : AlgMatLie, RngIntElt -> AlgMatLieElt
The i-th basis element of the algebra L.
Basis(A) : AlgMatLie -> [ AlgMatLieElt ]
The basis of the algebra L, as a sequence of elements of L.
IsIndependent(Q) : [ AlgMatLieElt ] -> BoolElt
IsIndependent(Q) : { AlgLieElt } -> BoolElt
IsIndependent(Q) : { AlgMatLieElt } -> BoolElt
Given a set or sequence Q of elements of the R-algebra L, this functions returns
true if these elements are linearly independent over R; otherwise false.
ExtendBasis(S, L) : AlgMatLie, AlgMatLie -> [ AlgElt ]
ExtendBasis(Q, L) : [ AlgLieElt ], AlgLie -> [ AlgElt ]
ExtendBasis(Q, L) : [ AlgMatLie ], AlgMatLie -> [ AlgElt ]
Given an algebra L and either a subalgebra S of dimension m of L or a
sequence Q of m linearly independent elements of L, this function returns
a sequence containing a basis of L such that the first m elements are the
basis of S resp. the elements in Q.
The WeylGroup functions are only available for structure constant Lie algebras.
SemisimpleType(L) : AlgMatLie -> MonStgElt
CartanName(L) : AlgLie -> MonStgElt
CartanName(L) : AlgMatLie -> MonStgElt
Let L be a Lie algebra.If L has a nondegenerate Killing form, then
(over some algebraic extension of the ground field) L is the direct
sum of absolutely simple Lie algebras. These Lie algebras have been
classified and the classes are named An, Bn, Cn, Dn,
E6, E7, E8, F4 and G2.
This function returns a single string
containing the types of the direct summands of L.
For a description of the algorithm used in the general case we refer
to [dG00], Para 5.17.1. For Lie algebras over fields of
characteristic 2 and 3 the algorithm used is described in
[Roo10], Chapter 5.
We compute the semisimple type of the Levi subalgebra of a subalgebra of
the simple Lie algebra of type D 7.
> L := LieAlgebra("D7", RationalField());
> L;
Lie Algebra of dimension 91 with base ring Rational Field
> K := Centralizer(L, sub<L | [L.1,L.2,L.3,L.4]>);
> K;
Lie Algebra of dimension 41 with base ring Rational Field
> _,S := HasLeviSubalgebra(K);
> S;
Lie Algebra of dimension 6 with base ring Rational Field
> SemisimpleType(S);
A1 A1
ReductiveType(L, H) : AlgLie, AlgLie -> RootDtm, MonStgElt, SeqEnum, SeqEnum
AssumeAlmostSimple: BoolElt Default: false
Let L be a Lie algebra of a reductive algebraic group, and H a split
maximal toral subalgebra of L. This function identifies the isomorphism
type of L.
This function has four return values. The first is the appropriate root
datum and the second return value a textual description of L.
The third return value is a sequence Q, containing a decomposition of
L into direct summands. Finally, the fourth return value is a
sequence P of records, such that P[i] contains additional information
(often a proof of correctness) of the identification of Q[i].
If a split maximal toral subalgebra H is not given, an attempt is made
to compute one by calling SplitMaximalToralSubalgebra if the
characteristic of the base field k is at least 5, or
SplitToralSubalgebra if char(k) is 2 or 3.
Note that, if k is infinite, such a subalgebra cannot in general be
computed so the second parameter H must be supplied for this function
to work.
If the optional parameter AssumeAlmostSimple is set
to true, the (possibly time consuming) step of computing a direct sum
decomposition of L is skipped.
Moreover, note that if L is the Lie algebra of a simple algebraic group
but itself non-simple (such as for example An of intermediate type in
characteristic n + 1),
the third return value Q may not be the direct sum decomposition of L
but simply [L].
We consider a particular Lie algebra of type A 3 over k = GF(2).
> RA3 := RootDatum("A3" : Isogeny := 2);
> L := LieAlgebra(RA3, GF(2));
> D := DirectSumDecomposition(L);
> D;
[
Lie Algebra of dimension 14 with base ring GF(2),
Lie Algebra of dimension 1 with base ring GF(2)
]
> R, str, Q, _ := ReductiveType(L);
> R;
RA3: Root datum of dimension 3 of type A3
> str;
Lie algebra of type A3[ 2]
> Q;
[
Lie Algebra of dimension 15 with base ring GF(2)
]
Note that this is an example where Q is not the direct sum decomposition
of L. Instead, L in its whole is recognised as the Lie algebra of a
simple algebraic group.
In the remainder of the example, we investigate the 14-dimensional ideal of L.
> M := D[1]; M;
Lie Algebra of dimension 14 with base ring GF(2)
> R, _, _, P := ReductiveType(M);
> R;
R: Adjoint root datum of dimension 2 of type G2
So this computation claims that L simeq M direct-sum k,
where M is of type G 2. Let us use the additional return values
to verify that fact.
> pos := P[1]`ChevBasData`BasisPos;
> neg := P[1]`ChevBasData`BasisNeg;
> cart := P[1]`ChevBasData`BasisCart;
> IsChevalleyBasis(M, RootDatum("G2"), pos, neg, cart);
true [ <1, 2, 0>, <1, 3, 0>, <1, 4, 0>, <2, 5, 0> ]
This demonstrates the fact that the Lie algebra of type G 2 is a
constituent of the Lie algebra of type A 3 over fields of
characteristic 2.
RootSystem(L) : AlgMatLie -> [ ModTupRngElt ], [ AlgMatLieElt ], [ ModTupRngElt ], AlgMatElt
Given a semisimple Lie algebra L with a split Cartan subalgebra, this
function computes the root system of L.
This function returns four values:
- (a)
- The roots of L with respect to the Cartan subalgebra which is output by
CartanSubalgebra(L). This is a sequence of vectors where the
positive roots come first, followed by the negative roots.
- (b)
- A sequence of elements of L which are the root vectors
corresponding to the roots of L (so the first element corresponds to
the first root and so on).
- (c)
- A sequence of simple roots.
- (d)
- The Cartan matrix of the root system with respect to the sequence of
simple roots.
We compute the root system of the simple Lie algebra of type G 2 over the
rational field.
> L := LieAlgebra("G2", RationalField());
> R, Rv, fund, C:=RootSystem(L);
> R;
[
(1 0),
(0 1),
(1 1),
(2 1),
(3 1),
(3 2),
(-1 0),
( 0 -1),
(-1 -1),
(-2 -1),
(-3 -1),
(-3 -2)
]
> Rv;
[ (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),
(0 0 0 0 0 1 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 1 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 1 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) ]
RootDatum(L) : AlgMatLie -> RootDtm
Here L is a semisimple Lie algebra. This function returns the
root datum D of L with respect to the Cartan subalgebra which
is output by CartanSubalgebra(L). We note that the order of
the positive roots in D is not necessarily the same as the order
in which they appear in the root system of L.
We set up the root datum of a Lie algebra, and extract the Cartan matrix.
> L:= LieAlgebra("F4", Rationals());
> rd := RootDatum(L);
> rd;
Root datum of type F4
> CartanMatrix(rd);
[ 2 0 -1 0]
[ 0 2 0 -1]
[-1 0 2 -1]
[ 0 -1 -2 2]
ChevalleyBasis(L) : AlgLie -> [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]
ChevalleyBasis(L, H) : AlgLie, AlgLie -> [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]
ChevalleyBasis(L) : AlgMatLie -> [ AlgMatLieElt ], [ AlgMatLieElt ], [ AlgMatLieElt ]
AssumeAlmostSimple: BoolElt Default: false
Given a semisimple Lie algebra L with a split maximal toral subalgebra H,
this function returns three sequences, x, y and h of elements of L.
They form a Chevalley basis of L. The first sequence gives basis
elements corresponding to positive roots, the second to the negative roots
and the third to basis elements in a Cartan subalgebra.
If a split maximal toral subalgebra H is not given, an attempt is made to
compute one.
For Lie algebras over fields of characteristic 2 and 3 the algorithm
used is described in [CR09]. In particular, this
involves computing a direct sum decomposition of L, which can be quite
time consuming. If there is reason to believe that L is (almost) simple,
the optional parameter AssumeAlmostSimple should be set to {true}.
We construct a Chevalley basis for two Lie algebras.
> L := LieAlgebra("A2", RationalField());
> x, y, h:= ChevalleyBasis(L);
> x; y; h;
[ (0 0 0 0 0 1 0 0), (0 0 0 0 0 0 1 0), (0 0 0 0 0 0 0 1) ]
[ (0 0 1 0 0 0 0 0), (0 1 0 0 0 0 0 0), (1 0 0 0 0 0 0 0) ]
[ (0 0 0 1 0 0 0 0), (0 0 0 0 1 0 0 0) ]
> L := LieAlgebra("A3", Rationals());
> print RootDatum(L) : Maximal;
Root datum of type A3 with simple roots
[ 1 0 1]
[ 1 -2 1]
[ 0 1 -2]
and simple coroots
[ 1 1 1]
[ 0 -1 0]
[ 0 0 -1]
Given a semisimple Lie algebra L with a split maximal toral subalgebra H,
and an irreducible root datum R, this function computes a Chevalley basis
of L with respect to H and R.
This basis is returned in the form of three sequences, x, y and h
of elements of L, where the first sequence gives basis elements corresponding
to positive roots, the second to the negative roots
and the third to basis elements in the toral subalgebra H.
Returns true if x, y and h form a Chevalley basis of the Lie algebra L
with respect to the root datum R. If so, return a sequence describing the extraspecial
signs as second return value.
We compute a Chevalley basis for a Lie algebra of type E 6 inside one of type E 7.
> R := RootDatum("E7");
> L1 := LieAlgebra(R, GF(2));
> p1,n1,c1 := StandardBasis(L1);
> L1;
Lie Algebra of dimension 133 with base ring GF(2)
> DynkinDiagram(R);
E7 1 - 3 - 4 - 5 - 6 - 7
|
2
> S, proj := sub<R | [1..6]>;
> S;
S: Root datum of dimension 7 of type E6
> #proj;
72
> projpos := [i : i in proj | i le NumPosRoots(R)];
> #projpos;
36
> L2 := sub<L1 | p1[projpos], n1[projpos]>;
> L2;
Lie Algebra of dimension 78 with base ring GF(2)
> H2 := L2 meet SplitMaximalToralSubalgebra(L1);
> H2;
Lie Algebra of dimension 6 with base ring GF(2)
> p2,n2,c2 := ChevalleyBasis(L2, H2, RootDatum("E6"));
> ok := IsChevalleyBasis(L2, RootDatum("E6"), p2, n2, c2);
> ok;
true
TwistedBasis(L, H, R) : AlgLie, AlgLie, RootDtm -> AlgLie, AlgLie, Rec, AlgMatElt
For a Lie algebra L, a split toral subalgebra H of L, and a twisted root
datum R, the function constructs a "twisted basis" of L.
Let k be the coefficient ring of L and K an extension field of k of
degree equal to the twisting degree of R. This function has 4 return
values. First, L' = L tensor K; second, a homomorphism φ from L to L',
third, a record containing a Chevalley basis of L' with respect to the untwisted
root datum of R; fourth, a matrix describing the action of the Frobenius
automorphism of K on the positive roots of the Chevalley basis of L'.
Such a basis constitutes a proof that L' is of type R. Consult
[Roo10], Chapter 5.3, for more details on such twisted bases.
We investigate a twisted basis of the Lie algebra of type () 2A 2
over the field with 5 elements. Let δ be the automorphism
of the root system of type A 2, let k = GF(5), and let K = GF(5 2).
> R := TwistedRootDatum(RootDatum("A2") : Twist := 2);
> L := TwistedLieAlgebra(R, GF(5));
> H := SplitToralSubalgebra(L);
> LK, phi, ChevBas, m := TwistedBasis(L, H, R);
> m;
[ 0 1]
[ 1 0]
This matrix m shows that δ acts as expected on the
Chevalley basis elements of LK = L tensor K.
We verify the correctness of m.
> K := CoefficientRing(LK);
> simp := ChevBas`BasisPos[[1..Rank(R)]];
> simp;
[ ( 0 0 0 0 0 1 ksi^8 0),
( 0 0 0 0 0 1 ksi^16 0) ]
> fr := FrobeniusMap(K);
> frv := func<x | Vector([ fr(i) : i in Eltseq(x)])>;
> [ Position(simp, frv(x)) : x in simp ];
[ 2, 1 ]
So indeed the Frobenius map (acting on the coordinates of LK)
acts as δ. This is equivalent [Roo10, Lemma 5.3] to
the basis elements of L being stable under the composition of the
Frobenius map (this time acting on the Chevalley basis of L tensor K)
and the root system automorphism δ.
We verify this assertion explicitly for this example.
> p := ChevBas`BasisPos;
> n := ChevBas`BasisNeg;
> c := ChevBas`BasisCart;
> pi := Sym(6)!(1, 2)(4, 5);
> ChevBasLK := VectorSpaceWithBasis([ Vector(x) : x in p cat n cat c]);
> piL := DiagramAutomorphism(LK, pi);
Now δ acts on L tensor K as T, and fr is still the Frobenius
automorphism of the field K. The images of the basis elements of L under
delta composed with fr are as follows:
> for i in [1..Dimension(L)] do
> b := phi(L.i);
> printf "i = %o, b = %o n", i, Coordinates(ChevBasLK, Vector(b));
> printf " pi(b)^fr = %o n", [ fr(i) : i in
> Coordinates(ChevBasLK, Vector(piL(b))) ];
> end for;
i = 1, b = [ 0, 0, 0, 0, 0, ksi^9, 0, 0 ]
(b*T)^fr = [ 0, 0, 0, 0, 0, ksi^9, 0, 0 ]
i = 2, b = [ 0, 0, 0, ksi^5, ksi, 0, 0, 0 ]
(b*T)^fr = [ 0, 0, 0, ksi^5, ksi, 0, 0, 0 ]
i = 3, b = [ 0, 0, 0, ksi^9, ksi^21, 0, 0, 0 ]
(b*T)^fr = [ 0, 0, 0, ksi^9, ksi^21, 0, 0, 0 ]
i = 4, b = [ 0, 0, 0, 0, 0, 0, ksi^5, ksi ]
(b*T)^fr = [ 0, 0, 0, 0, 0, 0, ksi^5, ksi ]
i = 5, b = [ 0, 0, 0, 0, 0, 0, ksi, ksi^5 ]
(b*T)^fr = [ 0, 0, 0, 0, 0, 0, ksi, ksi^5 ]
i = 6, b = [ ksi, ksi^5, 0, 0, 0, 0, 0, 0 ]
(b*T)^fr = [ ksi, ksi^5, 0, 0, 0, 0, 0, 0 ]
i = 7, b = [ ksi^21, ksi^9, 0, 0, 0, 0, 0, 0 ]
(b*T)^fr = [ ksi^21, ksi^9, 0, 0, 0, 0, 0, 0 ]
i = 8, b = [ 0, 0, ksi^9, 0, 0, 0, 0, 0 ]
(b*T)^fr = [ 0, 0, ksi^9, 0, 0, 0, 0, 0 ]
Thus, all the basis elements of L are stable under the composition of
the diagram automorphism δ and the Frobenius automorphism.
WeylGroup(GrpPermCox, L) : Cat, AlgLie -> GrpPermCox
The Weyl group of the reductive Lie algebra L, as a permutation Coxeter group
(see Chapter COXETER GROUPS).
The Weyl group of the reductive Lie algebra L, as a Coxeter group
(see Chapter COXETER GROUPS).
The Weyl group of the reductive Lie algebra L, as a reflection group
(see Chapter COXETER GROUPS).
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