# Operations on Lie Algebras ## `L eq K: AlgLie, AlgLie -> BoolElt` ## `L eq K: AlgMatLie, AlgMatLie -> BoolElt` Returns `true` if, and only if, the Lie algebras $L$ and $K$ are equal. ## `L ne K: AlgLie, AlgLie -> BoolElt` ## `L ne K: AlgMatLie, AlgMatLie -> BoolElt` Returns `true` if, and only if, the Lie algebras $L$ and $K$ are not equal. ## `L subset K: AlgLie, AlgLie -> BoolElt` ## `L subset K: AlgMatLie, AlgMatLie -> BoolElt` Returns `true` if, and only if, the Lie algebra $L$ is contained in the Lie algebra $K$. ## `L notsubset K: AlgLie, AlgLie -> BoolElt` Returns `true` if, and only if, the Lie algebra $L$ is not contained in the Lie algebra $K$. ## `L meet M: AlgLie, AlgLie -> AlgLie` ## `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: AlgLie, AlgLie -> AlgLie` ## `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. ## `L ^ n: AlgLie, RngIntElt -> AlgLie` The (left-normed) $n$-th power of the (structure constant) Lie algebra $L$, i.e., $((\ldots (L*L)* \ldots)*L)$ is constructed. ## `Morphism(L, M): AlgLie, AlgLie -> Map` 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, .` ```magma 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](solvable-nilpotent.md#sectslac)). 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. ## `IsKnownIsomorphic(L, M): AlgLie, AlgLie -> BoolElt, BoolElt, .` ```magma 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`](#function-isisomorphiclie) for more details on applicability and the meanings of the return values. ## `IsIsomorphism(m): Map[AlgLie, AlgLie] -> BoolElt` Returns `true` if the mapping $m$ between two Lie algebras is an isomorphism of Lie algebras. ## `Example: Isomorphic (ex-4fbecb)` We demonstrate that B$_2$ and C$_2$ are isomorphic over ${\mathbb{Q}}$. ```magma > 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 ${\mathbb{Q}}$. ```magma > 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 ${\mathbb{Q}}$. ```magma > 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. ```magma > L := LieAlgebra("B4", k); > pL, _, _ := StandardBasis(L); > subL := sub; > subL; Lie Algebra of dimension 16 with base ring Rational Field > M := LieAlgebra("C4", k); > pM, _, _ := StandardBasis(M); > subM := sub; > 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 ``` ## `Example: Isomorphic G2 (ex-3b3b3e)` We demonstrate that in characteristic $3$ the Lie algebras of type G$_2$ and A$_2$ have isomorphic nontrivial ideals. ```magma > 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 ``` ## Basic Invariants ### `CoefficientRing(L): AlgLie -> Rng` ### `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): AlgLie -> RngIntElt` ### `Dimension(L): AlgMatLie -> RngIntElt` The dimension of the Lie algebra $L$. ### `# L: AlgLie -> RngIntElt` ### `# L: AlgMatLie -> RngIntElt` The cardinality of the Lie algebra $L$, if the coefficient ring is finite. ### `Moduli(L): AlgLie -> SeqEnum` This returns a sequence of integers, of length equal to the dimension of $L$. If the $i$-th element of this sequence is $a_i$ then $a_i$ is the minimal non-negative integer such that $a_ie_i = 0$. So if $L$ is defined over a field, then the sequence consists of zeros. ### `Example: Moduli Of Lie Algebra (ex-29430b)` ```magma > T:= [ <1,2,2,2>, <2,1,2,2> ]; > t:= [0,4]; > L:= LieAlgebra< t | T : Rep:= "Dense" >; > Moduli(L); [ 0, 4 ] ``` ## Changing Base Rings ### `ChangeRing(L, S): AlgLie, Rng -> AlgLie, Map` ### `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): AlgLie, Rng, Map -> AlgLie, Map` ### `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 $f: R \rightarrow S$, 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. ## Bases ### `BasisElement(A, i): AlgLie, RngIntElt -> AlgLieElt` ### `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): AlgLie -> [ AlgLieElt ]` ### `Basis(A): AlgMatLie -> [ AlgMatLieElt ]` The basis of the algebra $L$, as a sequence of elements of $L$. ### `IsIndependent(Q): [ AlgLieElt ] -> BoolElt` ### `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): AlgLie, AlgLie -> [ AlgElt ]` ### `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 $Q$. ## Operations for Semisimple and Reductive Lie Algebras ### `SemisimpleType(L): AlgLie -> MonStgElt` ### `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 $A_n$, $B_n$, $C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$. 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 [[de Graaf, 2000](../../references.md#cite-wdg-lie-alg)], §5.17.1. For Lie algebras over fields of characteristic $2$ and $3$ the algorithm used is described in [[Roozemond, 2010](../../references.md#cite-roozemond2010)], Chapter 5. ### `Example: Semisimple Type (ex-6372db)` We compute the semisimple type of the Levi subalgebra of a subalgebra of the simple Lie algebra of type $D_7$. ```magma > L := LieAlgebra("D7", RationalField()); > L; Lie Algebra of dimension 91 with base ring Rational Field > K := Centralizer(L, sub); > 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): AlgLie -> RootDtm, MonStgElt, SeqEnum, SeqEnum` ### `ReductiveType(L, H): AlgLie, AlgLie -> RootDtm, MonStgElt, SeqEnum, SeqEnum` ```magma 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`](sub.md#function-splitmaximaltoralsubalgebra) if the characteristic of the base field $k$ is at least $5$, or [`SplitToralSubalgebra`](sub.md#function-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 A$_n$ of intermediate type in characteristic $n+1$), the third return value $Q$ may not be the direct sum decomposition of $L$ but simply [$L$]. ### `Example: Reductive Type (ex-c3afbb)` We consider a particular Lie algebra of type A$_3$ over $k =$GF$(2)$. ```magma > 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 $L$. ```magma > 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 $L \simeq M \oplus k$, where $M$ is of type G$_2$. Let us use the additional return values to verify that fact. ```magma > 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): AlgLie -> [ ModTupRngElt ], [ AlgLieElt ], [ ModTupRngElt ], AlgMatElt` ### `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. ### `Example: Root System (ex-cfabbc)` We compute the root system of the simple Lie algebra of type $G_2$ over the rational field. ```magma > 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): AlgLie -> RootDtm` ### `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 $L$. ### `Example: Root Datum (ex-afa536)` We set up the root datum of a Lie algebra, and extract the Cartan matrix. ```magma > 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 ]` ```magma 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 [[Cohen and Roozemond, August 2009](../../references.md#cite-cohenroozemond2009)]. 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`. ### `Example: Chevalley Basis (ex-fc1498)` We construct a Chevalley basis for two Lie algebras. ```magma > 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] ``` ### `ChevalleyBasis(L, H, R): AlgLie, AlgLie, RootDtm -> [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]` 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 $H$. ### `IsChevalleyBasis(L, R, x, y, h): AlgLie, RootDtm, [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ] -> BoolElt, [ Tup ]` 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. ### `Example: Chevalley Basis Small Char (ex-f1d947)` We compute a Chevalley basis for a Lie algebra of type E$_6$ inside one of type E$_7$. ```magma > 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; > 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; > 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, $L' = L \otimes K$; second, a homomorphism $\phi$ 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 [[Roozemond, 2010](../../references.md#cite-roozemond2010)], Chapter 5.3, for more details on such twisted bases. ### `Example: Twisted Basis (ex-cf55bc)` We investigate a twisted basis of the Lie algebra of type ${}^2$A$_2$ over the field with $5$ elements. Let $\delta$ be the automorphism of the root system of type A$_2$, let $k =$GF($5$), and let $K =$GF($5^2$). ```magma > 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 $\delta$ acts as expected on the Chevalley basis elements of `LK` $= L \otimes K$. We verify the correctness of `m`. ```magma > 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; > [ Position(simp, frv(x)) : x in simp ]; [ 2, 1 ] ``` So indeed the Frobenius map (acting on the coordinates of LK) acts as $\delta$. This is equivalent [[Roozemond, 2010](../../references.md#cite-roozemond2010), 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 $L \otimes K$) and the root system automorphism $\delta$. We verify this assertion explicitly for this example. ```magma > 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 $\delta$ acts on $L \otimes K$ 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: ```magma > 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 $\delta$ and the Frobenius automorphism. The `WeylGroup` functions are only available for structure constant Lie algebras. ### `WeylGroup(L): AlgLie -> GrpPermCox` ### `WeylGroup(GrpPermCox, L): Cat, AlgLie -> GrpPermCox` The Weyl group of the reductive Lie algebra $L$, as a permutation Coxeter group (see Chapter ChapGrpPermCox). ### `WeylGroup(GrpFPCox, L): Cat, AlgLie -> GrpPermCox` The Weyl group of the reductive Lie algebra $L$, as a Coxeter group (see Chapter ChapGrpFPCox). ### `WeylGroup(GrpMat, L): Cat, AlgLie -> GrpPermCox` The Weyl group of the reductive Lie algebra $L$, as a reflection group (see Chapter ChapGrpFPCox).