Returns true if, and only if, the Lie algebras L and K are equal.
Returns true if, and only if, the Lie algebras L and K are not equal.
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.
The intersection of the Lie algebras L and M is returned. Note that L and M have a common superalgebra.
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.
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.
> 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 B3 and C3 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 B2 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
> 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
The coefficient ring (or base ring) over which the Lie algebra L is defined.
The dimension of the Lie algebra L.
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 ]
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.
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.
The i-th basis element of the algebra L.
The basis of the algebra L, as a sequence of elements of L.
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.
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.
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.
> 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
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].
> 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 G2So this computation claims that L simeq M direct-sum k, where M is of type G2. 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 G2 is a
constituent of the Lie algebra of type A3 over fields of
characteristic 2.
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.
> 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) ]
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.
> 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]
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}.
> 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.
> 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
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.
> 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.
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).