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# 61 Lie Algebras

### Sections

1. Lie objects
2. Constructing Lie algebras
3. Distinguished Subalgebras
4. Series of Ideals
5. Properties of a Lie Algebra
6. Direct Sum Decompositions
7. Semisimple Lie Algebras and Root Systems
8. Restricted Lie algebras
9. The Adjoint Representation
10. Universal Enveloping Algebras
11. Finitely Presented Lie Algebras
12. Modules over Lie Algebras and Their Cohomology
13. Modules over Semisimple Lie Algebras
14. Tensor Products and Exterior and Symmetric Powers

A Lie algebra L is an algebra such that xx=0 and x(yz)+y(zx)+z(xy)=0 for all x,y,z Î L. A common way of creating a Lie algebra is by taking an associative algebra together with the commutator as product. Therefore the product of two elements x,y of a Lie algebra is usually denoted by [x,y], but in GAP this denotes the list of the elements x and y; hence the product of elements is made by the usual `*`. This gives no problems when dealing with Lie algebras given by a table of structure constants. However, for matrix Lie algebras the situation is not so easy as `*` denotes the ordinary (associative) matrix multiplication. In GAPthis problem is solved by wrapping elements of a matrix Lie algebra up as LieObjects, and then define the `*` for LieObjects to be the commutator (see lie objects);

## 61.1 Lie objects

Let x be a ring element, then `LieObject(x)` wraps x up into an object that contains the same data (namely x). The multiplication `*` for Lie objects is formed by taking the commutator. More exactly, if l1 and l2 are the Lie objects corresponding to the ring elements r1 and r2, then l1 * l2 is equal to the Lie object corresponding to r1 * r2 - r2 * r2. Two rules for Lie objects are worth noting:

- An element is not equal to its Lie element.

- If we take the Lie object of an ordinary (associative) matrix then this is again a matrix; it is therefore a collection (of its rows) and a list. But it is not a collection of collections of its entries, and its family is not a collections family.

- `LieObject(` obj`) A`

  Let obj be a ring element. Then `LieObject(` obj`)` is the corresponding Lie object. If obj lies in the family F, then `LieObject(` obj`)` lies in the family LieFamily( F ) (see LieFamily).

      gap> m:= [ [ 1, 0 ], [ 0, 1 ] ];;
      gap> lo:= LieObject( m );
      LieObject( [ [ 1, 0 ], [ 0, 1 ] ] )
      gap> m*m;
      [ [ 1, 0 ], [ 0, 1 ] ]
      gap> lo*lo;
      LieObject( [ [ 0, 0 ], [ 0, 0 ] ] )

- `IsLieObject(` obj`) C`
- `IsLieObjectCollection(` obj`) C`

  An object lies in `IsLieObject` if and only if it lies in a family constructed by `LieFamily`.

      gap> m:= [ [ 1, 0 ], [ 0, 1 ] ];;
      gap> lo:= LieObject( m );
      LieObject( [ [ 1, 0 ], [ 0, 1 ] ] )
      gap> IsLieObject( m );
      false
      gap> IsLieObject( lo );
      true

- `LieFamily(` Fam`) A`

  is a family F in bijection with the family Fam, but with the Lie bracket as infix multiplication. That is, for x, y in Fam, the product of the images in F will be the image of x * y - y * x.

  The standard type of objects in a Lie family F is ``F`!.packedType`.

  The bijection from Fam to F is given by `Embedding(` Fam`, F )`; this bijection respects addition and additive inverses.

- `UnderlyingFamily(` Fam`) A`

  If Fam is a Lie family then `UnderlyingFamily(` Fam`)` is a family F such that ``Fam`= LieFamily( F )`.

  ## 61.2 Constructing Lie algebras

  In this section we describe functions that create Lie algebras. Creating and working with subalgebras goes exactly in the same way as for general algebras; so for that we refer to Chapter algebras.

- `LieAlgebraByStructureConstants(` R`,` sctable`) F`
- `LieAlgebraByStructureConstants(` R`,` sctable`,` name`) F`
- `LieAlgebraByStructureConstants(` R`,` sctable`,` name1`,` name2`, ... ) F`

  `LieAlgebraByStructureConstants` does the same as `AlgebraByStructureConstants`, except that the result is assumed to be a Lie algebra. Note that the function does not check whether sctable satisfies the Jacobi identity. (So if one creates a Lie algebra this way with a table that does not satisfy the Jacobi identity, errors may occur later on.)

      gap> T:= EmptySCTable( 2, 0, "antisymmetric" );;
      gap> SetEntrySCTable( T, 1, 2, [ 1/2, 1 ] );
      gap> L:= LieAlgebraByStructureConstants( Rationals, T );
      <Lie algebra of dimension 2 over Rationals>

- `LieAlgebra(` L`) F`
- `LieAlgebra(` F`,` gens`) F`
- `LieAlgebra(` F`,` gens`,` zero`) F`
- `LieAlgebra(` F`,` gens`, "basis" ) F`
- `LieAlgebra(` F`,` gens`,` zero`, "basis" ) F`

  For an associative algebra L, `LieAlgebra(` L`)` is the Lie algebra isomorphic to L as a vector space but with the Lie bracket as product.

  `LieAlgebra(` F`,` gens`)` is the Lie algebra over the division ring F, generated as Lie algebra by the Lie objects corresponding to the vectors in the list gens.

  Note that the algebra returned by `LieAlgebra` does not contain the vectors in gens. The elements in gens are wrapped up as Lie objects (see lie objects). This allows one to create Lie algebras from ring elements with respect to the Lie bracket as product. But of course the product in the Lie algebra is the usual `*`.

  If there are three arguments, a division ring F and a list gens and an element zero, then `LieAlgebra(` F`,` gens`,` zero`)` is the corresponding F-Lie algebra with zero element the Lie object corresponding to zero.

  If the last argument is the string `"basis"` then the vectors in gens are known to form a basis of the algebra (as an F-vector space).

  Note that even if each element in gens is already a Lie element, i.e., is of the form `LieElement(` elm`)` for an object elm, the elements of the result lie in the Lie family of the family that contains gens as a subset.

      gap> A:= FullMatrixAlgebra( GF( 7 ), 4 );;
      gap> L:= LieAlgebra( A );
      <Lie algebra of dimension 16 over GF(7)>
      gap> mats:= [  [[ 1, 0 ], [ 0, -1 ]], [[ 0, 1 ], [ 0, 0 ]], [[ 0, 0 ], [ 1, 0]] ];;
      gap> L:= LieAlgebra( Rationals, mats );
      <Lie algebra over Rationals, with 3 generators>

- `FreeLieAlgebra(` R`,` rank`) F`
- `FreeLieAlgebra(` R`,` rank`,` name`) F`
- `FreeLieAlgebra(` R`,` name1`,` name2`, ... ) F`

  Returns a free Lie algebra of rank rank over the ring R. `FreeLieAlgebra(` R`,` name1`,` name2`,...)` returns a free Lie algebra over R with generators named name1, name2, and so on. The elements of a free Lie algebra are written on the Hall-Lyndon basis.

      gap> L:= FreeLieAlgebra( Rationals, "x", "y", "z" );
      <Lie algebra over Rationals, with 3 generators>
      gap> g:= GeneratorsOfAlgebra( L );; x:= g[1];; y:=g[2];; z:= g[3];;
      gap> z*(y*(x*(z*y)));
      (-1)*((x*(y*z))*(y*z))+(-1)*((x*((y*z)*z))*y)+(-1)*(((x*z)*(y*z))*y)

- `FullMatrixLieAlgebra(` R`,` n`) F`
- `MatrixLieAlgebra(` R`,` n`) F`
- `MatLieAlgebra(` R`,` n`) F`

  is the full matrix Lie algebra R n ×n , for a ring R and a nonnegative integer n.

      gap> FullMatrixLieAlgebra( GF(9), 10 );
      <Lie algebra over GF(3^2), with 19 generators>

- `RightDerivations(` B`) A`
- `LeftDerivations(` B`) A`
- `Derivations(` B`) A`

  These functions all return the matrix Lie algebra of derivations of the algebra A with basis B.

  `RightDerivations(` B`)` returns the algebra of derivations represented by their right action on the algebra A. This means that with respect to the basis B of A, the derivation D is described by the matrix [ di,j ] which means that D maps the i-th basis element bi to åj=1n dij bj.

  `LeftDerivations(` B`)` returns the Lie algebra of derivations represented by theor left action on the algebra A. So the matrices contained in the algebra output by `LeftDerivations(` B`)` are the transposes of the matrices contained in the output of `RightDerivations(` B`)`.

  `Derivations` is just a synonym for `RightDerivations`.

      gap> A:= OctaveAlgebra( Rationals );
      <algebra of dimension 8 over Rationals>
      gap> L:= Derivations( Basis( A ) );
      <Lie algebra of dimension 14 over Rationals>

- `SimpleLieAlgebra(` type`,` n`,` F`) F`

  This function constructs the simple Lie algebra of type type and of rank n over the field F.

  type must be one of A, B, C, D, E, F, G, H, K, S, W. For the types A to G, n must be a positive integer. The last four types only exist over fields of characteristic p > 0. If the type is H, then n must be a list of positive integers of even length. If the type is K, then n must be a list of positive integers of odd length. For the other types, S and W, n must be a list of positive integers of any length. In some cases the Lie algebra returned by this function is not simple. Examples are the Lie algebras of type An over a field of characteristic p > 0 where p divides n+1, and the Lie algebras of type Kn where n is a list of length 1.

      gap> SimpleLieAlgebra( "E", 6, Rationals );
      <Lie algebra of dimension 78 over Rationals>
      gap> SimpleLieAlgebra( "A", 6, GF(5) );
      <Lie algebra of dimension 48 over GF(5)>
      gap> SimpleLieAlgebra( "W", [1,2], GF(5) );
      <Lie algebra of dimension 250 over GF(5)>
      gap> SimpleLieAlgebra( "H", [1,2], GF(5) );
      <Lie algebra of dimension 123 over GF(5)>

  ## 61.3 Distinguished Subalgebras

  Here we describe functions that calculate well-known subalgebras and ideals of a Lie algebra (such as the centre, the centralizer of a subalgebra, etc.).

- `LieCentre(` L`) A`
- `LieCenter(` L`) A`

  The Lie centre of the Lie algebra L is the kernel of the adjoint mapping, that is, the set { a Î L; "x Î L:a x = 0 }.

  In characteristic 2 this may differ from the usual centre (that is the set of all a Î L such that ax=xa for all x Î L). Therefore, this operation is named `LieCentre` and not `Centre`.

      gap> L:= FullMatrixLieAlgebra( GF(3), 3 );
      <Lie algebra over GF(3), with 5 generators>
      gap> LieCentre( L );
      <two-sided ideal in <Lie algebra of dimension 9 over GF(3)>, (dimension 1)>

- `LieCentralizer(` L`,` S`) O`

  is the annihilator of S in the Lie algebra L, that is, the set { a Î L; "s Î S:a*s = 0}. Here S may be a subspace or a subalgebra of L.

      gap> L:= SimpleLieAlgebra( "G", 2, Rationals );
      <Lie algebra of dimension 14 over Rationals>
      gap> b:= BasisVectors( Basis( L ) );;
      gap> LieCentralizer( L, Subalgebra( L, [ b[1], b[2] ] ) );
      <Lie algebra of dimension 1 over Rationals>

- `LieNormalizer(` L`,` U`) O`

  is the normalizer of the subspace U in the Lie algebra L, that is, the set NL(U) = { x Î L; [x,U] Ì U }.

      gap> L:= SimpleLieAlgebra( "G", 2, Rationals );
      <Lie algebra of dimension 14 over Rationals>
      gap> b:= BasisVectors( Basis( L ) );;
      gap> LieNormalizer( L, Subalgebra( L, [ b[1], b[2] ] ) );
      <Lie algebra of dimension 8 over Rationals>

- `LieDerivedSubalgebra(` L`) A`

  is the (Lie) derived subalgebra of the Lie algebra L.

      gap>  L:= FullMatrixLieAlgebra( GF( 3 ), 3 );
      <Lie algebra over GF(3), with 5 generators>
      gap> LieDerivedSubalgebra( L );
      <Lie algebra of dimension 8 over GF(3)>

- `LieNilRadical(` L`) A`

  This function calculates the (Lie) nil radical of the Lie algebra L.

  In the following two examples we temporarily increase the line length limit from its default value 80 to 81 in order to make the long output expressions fit each into one line.

      gap> mats:= [ [[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[0,1]] ];;
      gap> L:= LieAlgebra( Rationals, mats );;
      gap> SizeScreen([ 81, ]);;
      gap> LieNilRadical( L );
      <two-sided ideal in <Lie algebra of dimension 3 over Rationals>, (dimension 2)>
      gap> SizeScreen([ 80, ]);;

- `LieSolvableRadical(` L`) A`

  Returns the (Lie) solvable radical of the Lie algebra L.

      gap> L:= FullMatrixLieAlgebra( Rationals, 3 );;
      gap> SizeScreen([ 81, ]);;
      gap> LieSolvableRadical( L );
      <two-sided ideal in <Lie algebra of dimension 9 over Rationals>, (dimension 1)>
      gap> SizeScreen([ 80, ]);;

- `CartanSubalgebra(` L`) A`

  A Cartan subalgebra of a Lie algebra L is defined as a nilpotent subalgebra of L equal to its own Lie normalizer in L.

      gap> L:= SimpleLieAlgebra( "G", 2, Rationals );;
      gap> CartanSubalgebra( L );
      <Lie algebra of dimension 2 over Rationals>

  ## 61.4 Series of Ideals

- `LieDerivedSeries(` L`) A`

  is the (Lie) derived series of the Lie algebra L.

      gap> mats:= [ [[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[0,1]] ];;
      gap> L:= LieAlgebra( Rationals, mats );;
      gap> LieDerivedSeries( L );
      [ <Lie algebra of dimension 3 over Rationals>,
        <Lie algebra of dimension 1 over Rationals>,
        <Lie algebra of dimension 0 over Rationals> ]

- `LieLowerCentralSeries(` L`) A`

  is the (Lie) lower central series of the Lie algebra L.

      gap> mats:= [  [[ 1, 0 ], [ 0, 0 ]], [[0,1],[0,0]], [[0,0],[0,1]] ];;
      gap> L:=LieAlgebra( Rationals, mats );;
      gap> LieLowerCentralSeries( L );
      [ <Lie algebra of dimension 3 over Rationals>,
        <Lie algebra of dimension 1 over Rationals> ]

- `LieUpperCentralSeries(` L`) A`

  is the (Lie) upper central series of the Lie algebra L.

      gap> mats:= [  [[ 1, 0 ], [ 0, 0 ]], [[0,1],[0,0]], [[0,0],[0,1]] ];;
      gap> L:=LieAlgebra( Rationals, mats );;
      gap> LieUpperCentralSeries( L );
      [ <two-sided ideal in <Lie algebra of dimension 3 over Rationals>,
            (dimension 1)>, <Lie algebra over Rationals, with 0 generators> ]

  ## 61.5 Properties of a Lie Algebra

- `IsLieAbelian(` L`) P`

  is `true` if L is a Lie algebra such that each product of elements in L is zero, and `false` otherwise.

      gap>  T:= EmptySCTable( 5, 0, "antisymmetric" );;
      gap>  L:= LieAlgebraByStructureConstants( Rationals, T );
      <Lie algebra of dimension 5 over Rationals>
      gap> IsLieAbelian( L );
      true

- `IsLieNilpotent(` L`) P`

  A Lie algebra L is defined to be (Lie) nilpotentwhen its (Lie) lower central series reaches the trivial subalgebra.

      gap> T:= EmptySCTable( 5, 0, "antisymmetric" );;
      gap> L:= LieAlgebraByStructureConstants( Rationals, T );
      <Lie algebra of dimension 5 over Rationals>
      gap> IsLieNilpotent( L );
      true

- `IsLieSolvable(` L`) P`

  A Lie algebra L is defined to be (Lie) solvablewhen its (Lie) derived series reaches the trivial subalgebra.

      gap> T:= EmptySCTable( 5, 0, "antisymmetric" );;
      gap> L:= LieAlgebraByStructureConstants( Rationals, T );
      <Lie algebra of dimension 5 over Rationals>
      gap> IsLieSolvable( L );
      true

  ## 61.6 Direct Sum Decompositions

  In this section we describe two functions that calculate a direct sum decomposition of a Lie algebra; the so-called Levi decomposition and the decomposition into a direct sum of ideals.

- `LeviMalcevDecomposition(` L`) A`

  A Levi-Malcev subalgebra of the algebra L is a semisimple subalgebra complementary to the radical of L. This function returns a list with two components. The first component is a Levi-Malcev subalgebra, the second the radical. This function is implemented for associative and Lie algebras.

      gap> L:= FullMatrixLieAlgebra( Rationals, 5 );;
      gap> LeviMalcevDecomposition( L );
      [ <Lie algebra of dimension 24 over Rationals>,
        <two-sided ideal in <Lie algebra of dimension 25 over Rationals>,
            (dimension 1)> ]

- `DirectSumDecomposition(` L`) A`

  This function calculates a list of ideals of the algebra L such that L is equal to their direct sum. Currently this is only implemented for semisimple associative algebras, and Lie algebras (semisimple or not).

      gap> L:= FullMatrixLieAlgebra( Rationals, 5 );;
      gap> DirectSumDecomposition( L );
      [ <two-sided ideal in
            <two-sided ideal in <Lie algebra of dimension 25 over Rationals>,
                (dimension 1)>, (dimension 1)>,
        <two-sided ideal in <two-sided ideal in
                <Lie algebra of dimension 25 over Rationals>, (dimension 24)>,
            (dimension 24)> ]

  ## 61.7 Semisimple Lie Algebras and Root Systems

  This section contains some functions for dealing with semisimple Lie algebras and their root systems.

- `SemiSimpleType(` L`) A`

  Let L be a semisimple Lie algebra, i.e., a direct sum of simple Lie algebras. Then `SemiSimpleType` returns the type of L, i.e., a string containing the types of the simple summands of L.

      gap> L:= SimpleLieAlgebra( "E", 8, Rationals );;
      gap> b:= BasisVectors( Basis( L ) );;
      gap> K:= LieCentralizer( L, Subalgebra( L, [ b[61]+b[79]+b[101]+b[102] ] ) );
      <Lie algebra of dimension 102 over Rationals>
      gap> lev:= LeviMalcevDecomposition(K);;
      gap> SemiSimpleType( lev[1] );
      "B3 A1"

- `ChevalleyBasis(` L`) A`

  Here L must be a semisimple Lie algebra with a split Cartan subalgebra. Then `ChevalleyBasis(` L`)` returns a list consisting of three sublists. Together these sublists form a Chevalley basis of L. The first list contains the positive root vectors, the second list contains the negative root vectors, and the third list the Cartan elements of the Chevalley basis.

      gap> L:= SimpleLieAlgebra( "G", 2, Rationals );
      <Lie algebra of dimension 14 over Rationals>
      gap> ChevalleyBasis( L );
      [ [ v.1, v.2, v.3, v.4, v.5, v.6 ], [ v.7, v.8, v.9, v.10, v.11, v.12 ],
        [ v.13, v.14 ] ]

- `IsRootSystem(` obj`) C`

  Category of root systems.

- `IsRootSystemFromLieAlgebra(` obj`) C`

  Category of root systems that come from (semisimple) Lie algebras. They often have special attributes such as `UnderlyingLieAlgebra`, `PositiveRootVectors`, `NegativeRootVectors`, `CanonicalGenerators`.

- `RootSystem(` L`) A`

  `RootSystem` calculates the root system of the semisimple Lie algebra L with a split Cartan subalgebra.

      gap> L:= SimpleLieAlgebra( "G", 2, Rationals );
      <Lie algebra of dimension 14 over Rationals>
      gap> R:= RootSystem( L );
      <root system of rank 2>
      gap> IsRootSystem( R );
      true
      gap> IsRootSystemFromLieAlgebra( R );
      true

- `UnderlyingLieAlgebra(` R`) A`

  Here R is a root system coming from a semisimple Lie algebra L. This function returns L.

- `PositiveRoots(` R`) A`

  The list of positive roots of the root system R.

- `NegativeRoots(` R`) A`

  The list of negative roots of the root system R.

- `PositiveRootVectors(` R`) A`

  A list of positive root vectors of the root system R that comes from a Lie algebra L. This is a list in bijection with the list `PositiveRoots(` L`)`. The root vector is a non-zero element of the root space (in L) of the corresponding root.

- `NegativeRootVectors(` R`) A`

  A list of negative root vectors of the root system R that comes from a Lie algebra L. This is a list in bijection with the list `NegativeRoots(` L`)`. The root vector is a non-zero element of the root space (in L) of the corresponding root.

- `SimpleSystem(` R`) A`

  A list of simple roots of the root system R.

- `CartanMatrix(` R`) A`

  The Cartan matrix of the root system R, relative to the simple roots in `SimpleSystem(` R`)`.

- `BilinearFormMat(` R`) A`

  The matrix of the bilinear form of the root system R. If we denote this matrix by B, then we have B(i,j) = (ai,aj), where the ai are the simple roots of R.

- `CanonicalGenerators(` R`) A`

  Here R must be a root system coming from a semisimple Lie algebra L. This function returns 3l generators of L, x1, ¼, xl,y1, ¼, yl,h1,¼, hl, where xi lies in the root space corresponding to the i-th simple root of the root system of L, yi lies in the root space corresponding to - the i-th simple root, and the hi are elements of the Cartan subalgebra. These elements satisfy the relations hi*hj=0, xi*yj=dijhi, hj*xi = cij xi, hj*yi = -cij yi, where cij is the entry of the Cartan matrix on position ij.

  Also if a is a root of the root system R (so a is a list of numbers), then we have the relation hi*x = a[i] x, where x is a root vector corresponding to a.

      gap> L:= SimpleLieAlgebra( "G", 2, Rationals );;
      gap> R:= RootSystem( L );;
      gap> UnderlyingLieAlgebra( R );
      <Lie algebra of dimension 14 over Rationals>
      gap> PositiveRoots( R );
      [ [ 2, -1 ], [ -3, 2 ], [ -1, 1 ], [ 1, 0 ], [ 3, -1 ], [ 0, 1 ] ]
      gap> x:= PositiveRootVectors( R );
      [ v.1, v.2, v.3, v.4, v.5, v.6 ]
      gap> g:=CanonicalGenerators( R );
      [ [ v.1, v.2 ], [ v.7, v.8 ], [ v.13, v.14 ] ]
      gap> g[3][1]*x[1];
      (2)*v.1
      gap> g[3][2]*x[1];
      (-1)*v.1   # i.e., x[1] is the root vector belonging to the root [ 2, -1 ]
      gap> BilinearFormMat( R );
      [ [ 1/12, -1/8 ], [ -1/8, 1/4 ] ]

  The next few sections deal with the Weyl group of a root system. A Weyl group is represented by its action on the weight lattice. A weight is by definition a linear function l: H® F (where F is the ground field), such that the values l(hi) are all integers (where the hi are the Cartan elements of the `CanonicalGenerators`). On the other hand each weight is determined by these values. Therefore we represent a weight by a vector of integers; the i-th entry of this vector is the value l(hi). Now the elements of the Weyl group are represented by matrices, and if `g` is an element of a Weyl group and `w` a weight, then `w*g` gives the result of applying `g` to `w`. Another way of applying the i-th simple reflection to a weight is by using the function `ApplySimpleReflection` (see below).

  A Weyl group is generated by the simple reflections. So `GeneratorsOfGroup( W )` for a Weyl group `W` gives a list of matrices and the i-th entry of this list is the simple reflection corresponding to the i-th simple root of the corresponding root system.

- `IsWeylGroup(` G`) P`

  A Weyl group is a group generated by reflections, with the attribute `SparseCartanMatrix` set.

- `SparseCartanMatrix(` W`) A`

  This is a sparse form of the Cartan matrix of the corresponding root system. If we denote the Cartan matrix by `C`, then the sparse Cartan matrix of W is a list (of length equal to the length of the Cartan matrix), where the `i`-th entry is a list consisting of elements `[ j, C[i][j] ]`, where `j` is such that `C[i][j]` is non-zero.

- `WeylGroup(` R`) A`

  The Weyl group of the root system R. It is generated by the simple reflections. A simple reflection is represented by a matrix, and the result of letting a simple reflection `m` act on a weight `w` is obtained by `w*m`.

      gap> L:= SimpleLieAlgebra( "F", 4, Rationals );;
      gap> R:= RootSystem( L );;
      gap> W:= WeylGroup( R );
      <matrix group with 4 generators>
      gap> IsWeylGroup( W );
      true
      gap> SparseCartanMatrix( W );
      [ [ [ 1, 2 ], [ 3, -1 ] ], [ [ 2, 2 ], [ 4, -1 ] ],
        [ [ 1, -1 ], [ 3, 2 ], [ 4, -1 ] ], [ [ 2, -1 ], [ 3, -2 ], [ 4, 2 ] ] ]
      gap> g:= GeneratorsOfGroup( W );;
      gap> [ 1, 1, 1, 1 ]*g[2];
      [ 1, -1, 1, 2 ]

- `ApplySimpleReflection(` SC`,` i`,` wt`) O`

  Here SC is the sparse Cartan matrix of a Weyl group. This function applies the i-th simple reflection to the weight wt, thus changing wt.

      gap> L:= SimpleLieAlgebra( "F", 4, Rationals );;
      gap> W:= WeylGroup( RootSystem( L ) );;
      gap> C:= SparseCartanMatrix( W );;
      gap> w:= [ 1, 1, 1, 1 ];;
      gap> ApplySimpleReflection( C, 2, w );
      gap> w;
      [ 1, -1, 1, 2 ]

- `LongestWeylWordPerm(` W`) A`

  Let g0 be the longest element in the Weyl group W, and let {a1,¼, al} be a simple system of the corresponding root system. Then g0 maps ai to -as(i), where s is a permutation of (1,¼,l). This function returns that permutation.

      gap> L:= SimpleLieAlgebra( "E", 6, Rationals );;
      gap> W:= WeylGroup( RootSystem( L ) );;
      gap> LongestWeylWordPerm( W );
      (1,6)(3,5)

- `ConjugateDominantWeight(` W`,` wt`) O`
- `ConjugateDominantWeightWithWord(` W`,` wt`) O`

  Here W is a Weyl group and wt a weight (i.e., a list of integers). This function returns the unique dominant weight conjugate to wt under W.

  `ConjugateDominantWegihtWithWord(` W`,` wt`)` returns a list of two elements. The first of these is the dominant weight conjugate do wt. The second element is a list of indices of simple reflections that have to be applied to wt in order to get the dominant weight conjugate to it.

      gap> L:= SimpleLieAlgebra( "E", 6, Rationals );;
      gap> W:= WeylGroup( RootSystem( L ) );;
      gap> C:= SparseCartanMatrix( W );;
      gap> w:= [ 1, -1, 2, -2, 3, -3 ];;
      gap> ConjugateDominantWeight( W, w );
      [ 2, 1, 0, 0, 0, 0 ]
      gap> c:= ConjugateDominantWeightWithWord( W, w );
      [ [ 2, 1, 0, 0, 0, 0 ], [ 2, 4, 2, 3, 6, 5, 4, 2, 3, 1 ] ]
      gap> for i in [1..Length(c[2])] do
      > ApplySimpleReflection( C, c[2][i], w );
      > od;
      gap> w;
      [ 2, 1, 0, 0, 0, 0 ]

- `WeylOrbitIterator(` W`,` wt`) O`

  Returns an iterator for the orbit of the weight wt under the action of the Weyl group W.

      gap> L:= SimpleLieAlgebra( "E", 6, Rationals );;
      gap> W:= WeylGroup( RootSystem( L ) );;
      gap> orb:= WeylOrbitIterator( W, [ 1, 1, 1, 1, 1, 1 ] );
      <iterator>
      gap> NextIterator( orb );
      [ 1, 1, 1, 1, 1, 1 ]
      gap> NextIterator( orb );
      [ -1, -1, -1, -1, -1, -1 ]
      gap> orb:= WeylOrbitIterator( W, [ 1, 1, 1, 1, 1, 1 ] );
      <iterator>
      gap> k:= 0;
      0
      gap> while not IsDoneIterator( orb ) do
      > w:= NextIterator( orb ); k:= k+1;
      > od;
      gap> k;
      51840    # i.e., the size of the Weyl group of E6.

  ## 61.8 Restricted Lie algebras

  A Lie algebra L over a field of characteristic p > 0 is called restricted if there is a map x® xp from L into L (called a p-map) such that ad xp=(ad x)p, (a x)p=ap xp and (x+y)p=xp+yp+åi=1p-1 si(x,y), where si: L×L® L are certain Lie polynomials in two variables. Using these relations we can calculate yp for all y Î L, once we know xp for x in a basis of L. Therefore a p-map is represented in GAP by a list containing the images of the basis vectors of a basis B of L. For this reason this list is an attribute of the basis B.

- `IsRestrictedLieAlgebra(` L`) P`

  Test whether L is restricted.

      gap> L:= SimpleLieAlgebra( "W", [2], GF(5));
      <Lie algebra of dimension 25 over GF(5)>
      gap> IsRestrictedLieAlgebra( L );
      false
      gap> L:= SimpleLieAlgebra( "W", [1], GF(5));
      <Lie algebra of dimension 5 over GF(5)>
      gap> IsRestrictedLieAlgebra( L );
      true

- `PthPowerImages(` B`) A`

  Here `B` is a basis of a restricted Lie algebra. This function returns the list of the images of the basis vectors of `B` under the p-map.

      gap> L:= SimpleLieAlgebra( "W", [1], GF(11) );
      <Lie algebra of dimension 11 over GF(11)>
      gap> B:= Basis( L );
      CanonicalBasis( <Lie algebra of dimension 11 over GF(11)> )
      gap> PthPowerImages( B );
      [ 0*v.1, v.2, 0*v.1, 0*v.1, 0*v.1, 0*v.1, 0*v.1, 0*v.1, 0*v.1, 0*v.1, 0*v.1 ]

- `PthPowerImage(` B`,` x`) O`

  B is a basis of a restricted Lie algebra L. This function calculates for an element x of L the image of x under the p-map.

      gap> L:= SimpleLieAlgebra( "W", [1], GF(11) );;
      gap> B:= Basis( L );;
      gap> x:= B[1]+B[11];
      v.1+v.11
      gap> PthPowerImage( B, x );
      v.1+v.11

- `JenningsLieAlgebra(` G`) A`

  Let G be a nontrivial p-group, and let G = G1 É G2 É ¼ É Gm=1 be its Jennings series (see JenningsSeries). Then the quotients Gi/Gi+1 are elementary abelian p-groups, i.e., they can be viewed as vector spaces over `GF`(p). Now the Jennings-Lie algebra L of G is the direct sum of those vector spaces. The Lie bracket on L is induced by the commutator in G. Furthermore, the map g® gp in G induces a p-map in L making L into a restricted Lie algebra. In the canonical basis of L this p-map is added as an attribute. A Lie algebra created by `JenningsLieAlgebra` is naturally graded. The attribute `Grading` is set.

- `PCentralLieAlgebra(` G`) A`

  Here G is a nontrivial p-group. `PCentralLieAlgebra(` G`)` does the same as `JenningsLieAlgebra(` G`)` except that the p-central series is used instead of the Jennings series (see PCentralSeries). This function also returns a graded Lie algebra. However, it is not necessarily restricted.

      gap> G:= SmallGroup( 3^6, 123 );
      <pc group of size 729 with 6 generators>
      gap> L:= JenningsLieAlgebra( G );
      <Lie algebra of dimension 6 over GF(3)>
      gap> HasPthPowerImages( Basis( L ) );
      true
      gap> PthPowerImages( Basis( L ) );
      [ v.6, 0*v.1, 0*v.1, 0*v.1, 0*v.1, 0*v.1 ]
      gap> g:= Grading( L );
      rec( min_degree := 1, max_degree := 3, source := Integers,
        hom_components := function( d ) ... end )
      gap> List( [1,2,3], g.hom_components );
      [ <vector space over GF(3), with 3 generators>,
        <vector space over GF(3), with 2 generators>,
        <vector space over GF(3), with 1 generators> ]

  ## 61.9 The Adjoint Representation

  In this section we show functions for calculating with the adjoint representation of a Lie algebra (and the corresponding trace form, called the Killing form) (see also adjointbasis and indicesofadjointbasis).

- `AdjointMatrix(` B`,` x`) O`

  is the matrix of the adjoint representation of the element x w.r.t. the basis B. The adjoint map is the left multiplication by x. The i-th column of the resulting matrix represents the image of the the i-th basis vector of B under left multiplication by x.

      gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
      gap> AdjointMatrix( Basis( L ), Basis( L )[1] );
      [ [ 0, 0, -2 ], [ 0, 0, 0 ], [ 0, 1, 0 ] ]

- `AdjointAssociativeAlgebra(` L`,` K`) A`

  is the associative matrix algebra (with 1) generated by the matrices of the adjoint representation of the subalgebra K on the Lie algebra L.

      gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
      gap> AdjointAssociativeAlgebra( L, L );
      <algebra of dimension 9 over Rationals>
      gap> AdjointAssociativeAlgebra( L, CartanSubalgebra( L ) );
      <algebra of dimension 3 over Rationals>

- `KillingMatrix(` B`) A`

  is the matrix of the Killing form k with respect to the basis B, i.e., the matrix (k(bi,bj)) where b1,b2¼ are the basis vectors of B.

      gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
      gap> KillingMatrix( Basis( L ) );
      [ [ 0, 4, 0 ], [ 4, 0, 0 ], [ 0, 0, 8 ] ]

- `KappaPerp(` L`,` U`) O`

  is the orthogonal complement of the subspace U of the Lie algebra L with respect to the Killing form k, that is, the set U^ = { x Î L; k(x,y)=0 for all y Î L }.

  U^ is a subspace of L, and if U is an ideal of L then U^ is a subalgebra of L.

      gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
      gap> b:= BasisVectors( Basis( L ) );;
      gap> V:= VectorSpace( Rationals, [b[1],b[2]] );;
      gap> KappaPerp( L, V );
      <vector space of dimension 1 over Rationals>

- `IsNilpotentElement(` L`,` x`) O`

  x is nilpotent in L if its adjoint matrix is a nilpotent matrix.

      gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
      gap> IsNilpotentElement( L, Basis( L )[1] );
      true

- `NonNilpotentElement(` L`) A`

  A non-nilpotent element of a Lie algebra L is an element x such that ad x is not nilpotent. If L is not nilpotent, then by Engel's theorem non nilpotent elements exist in L. In this case this function returns a non nilpotent element of L, otherwise (if L is nilpotent) `fail` is returned.

      gap> L:= SimpleLieAlgebra( "G", 2, Rationals );;
      gap> NonNilpotentElement( L );
      v.13
      gap> IsNilpotentElement( L, last );
      false

- `FindSl2(` L`,` x`) O`

  This function tries to find a subalgebra S of the Lie algebra L with S isomorphic to sl2 and such that the nilpotent element x of L is contained in S. If such an algebra exists then it is returned, otherwise `fail` is returned.

      gap> L:= SimpleLieAlgebra( "G", 2, Rationals );;
      gap> b:= BasisVectors( Basis( L ) );;
      gap> IsNilpotentElement( L, b[1] );
      true
      gap> FindSl2( L, b[1] );
      <Lie algebra of dimension 3 over Rationals>

  ## 61.10 Universal Enveloping Algebras

- `UniversalEnvelopingAlgebra(` L`) A`
- `UniversalEnvelopingAlgebra(` L`,` B`) O`

  Returns the universal enveloping algebra of the Lie algebra L. The elements of this algebra are written on a Poincare-Birkhoff-Witt basis.

  In the second form B must be a basis of L. If this second argument is given, then an isomorphic copy of the universal enveloping algebra is returned, generated by the images (in the universal enveloping algebra) of the elements of B.

      gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
      gap> UL:= UniversalEnvelopingAlgebra( L );
      <algebra-with-one of dimension infinity over Rationals>
      gap> g:= GeneratorsOfAlgebraWithOne( UL );
      [ [(1)*x.1], [(1)*x.2], [(1)*x.3] ]
      gap> g[3]^2*g[2]^2*g[1]^2;
      [(-4)*x.1*x.2*x.3^3+(1)*x.1^2*x.2^2*x.3^2+(2)*x.3^3+(2)*x.3^4]

  ## 61.11 Finitely Presented Lie Algebras

  Finitely presented Lie algebras can be constructed from free Lie algebras by using the `/` constructor, i.e., `FL/[r1...rk]` is the quotient of the free Lie algebra `FL` by the ideal generated by the elements `r1...rk` of `FL`. If the finitely presented Lie algebra `K` happens to be finite dimensional then an isomorphic structure constants Lie algebra can be constructed by `NiceAlgebraMonomorphism(K)`, which returns a surjective homomorphism. The structure constants Lie algebra can then be accessed by calling `Range` for this map. Also limited computations with elements of the finitely presented Lie algebra are possible.

      gap> L:= FreeLieAlgebra( Rationals, "s", "t" );
      <Lie algebra over Rationals, with 2 generators>
      gap> gL:= GeneratorsOfAlgebra( L );; s:= gL[1];; t:= gL[2];;
      gap> K:= L/[ s*(s*t), t*(t*(s*t)), s*(t*(s*t))-t*(s*t) ];
      <Lie algebra over Rationals, with 2 generators>
      gap> h:= NiceAlgebraMonomorphism( K );
      [ [(1)*s], [(1)*t] ] -> [ v.1, v.2 ]
      gap> U:= Range( h );
      <Lie algebra of dimension 3 over Rationals>
      gap> IsLieNilpotent( U );
      true
      gap> gK:= GeneratorsOfAlgebra( K );
      [ [(1)*s], [(1)*t] ]
      gap> gK[1]*(gK[2]*gK[1]) = Zero( K );
      true

- `FpLieAlgebraByCartanMatrix( C ) F`

  Here C must be a Cartan matrix. The function returns the finitely-presented Lie algebra over the field of rational numbers defined by this Cartan matrix. By Serre's theorem, this Lie algebra is a semisimple Lie algebra, and its root system has Cartan matrix C.

      gap> C:= [ [ 2, -1 ], [ -3, 2 ] ];;
      gap> K:= FpLieAlgebraByCartanMatrix( C );
      <Lie algebra over Rationals, with 6 generators>
      gap> h:= NiceAlgebraMonomorphism( K );
      [ [(1)*x1], [(1)*x2], [(1)*x3], [(1)*x4], [(1)*x5], [(1)*x6] ] ->
      [ v.1, v.2, v.3, v.4, v.5, v.6 ]
      gap> SemiSimpleType( Range( h ) );
      "G2"

- `NilpotentQuotientOfFpLieAlgebra(` FpL`,` max`) F`
- `NilpotentQuotientOfFpLieAlgebra(` FpL`,` max`,` weights`) F`

  Here FpL is a finitely presented Lie algebra. Let K be the quotient of FpL by the max+1-th term of its lower central series. This function calculates a surjective homomorphism of FpL onto K. When called with the third argument weights, the k-th generator of FpL gets assigned the k-th element of the list weights. In that case a quotient is calculated of FpL by the ideal generated by all elements of weight max+1. If the list weights only consists of 1's then the two calls are equivalent. The default value of weights is a list (of length equal to the number of generators of FpL) consisting of 1's.

  If the relators of FpL are homogeneous, then the resulting algebra is naturally graded.

      gap> L:= FreeLieAlgebra( Rationals, "x", "y" );;
      gap> g:= GeneratorsOfAlgebra(L);; x:= g[1]; y:= g[2];
      (1)*x
      (1)*y
      gap> rr:=[((y*x)*x)*x-6*(y*x)*y, 3*((((y*x)*x)*x)*x)*x-20*(((y*x)*x)*x)*y ];
      [ (-1)*(x*(x*(x*y)))+(6)*((x*y)*y),
        (-3)*(x*(x*(x*(x*(x*y)))))+(20)*(x*(x*((x*y)*y)))+(-20)*((x*(x*y))*(x*y)) ]
      gap> K:= L/rr;
      <Lie algebra over Rationals, with 2 generators>
      gap> h:=NilpotentQuotientOfFpLieAlgebra(K, 50, [1,2] );
      [ [(1)*x], [(1)*y] ] -> [ v.1, v.2 ]
      gap> L:= Range( h );
      <Lie algebra of dimension 50 over Rationals>
      gap> Grading( L );
      rec( min_degree := 1, max_degree := 50, source := Integers,
        hom_components := function( d ) ... end )

  ## 61.12 Modules over Lie Algebras and Their Cohomology

  Representations of Lie algebras are dealt with in the same way as representations of ordinary algebras (see Representations of Algebras). In this section we mainly deal with modules over general Lie algebras and their cohomology. The next section is devoted to modules over semisimple Lie algebras.

- `FaithfulModule(` A`) A`

  returns a faithful finite-dimensional left-module over the algebra A. This is only implemented for associative algebras, and for Lie algebras of characteristic 0. (It may also work for certain Lie algebras of characteristic p > 0.)

      gap> T:= EmptySCTable( 3, 0, "antisymmetric" );;
      gap> SetEntrySCTable( T, 1, 2, [ 1, 3 ]);
      gap> L:= LieAlgebraByStructureConstants( Rationals, T );
      <Lie algebra of dimension 3 over Rationals>
      gap> V:= FaithfulModule( L );
      <3-dimensional left-module over <Lie algebra of dimension 3 over Rationals>>
      gap> vv:= BasisVectors( Basis( V ) );
      [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
      gap> x:= Basis( L )[3];
      v.3
      gap> List( vv, v -> x^v );
      [ [ 0, 0, 0 ], [ 1, 0, 0 ], [ 0, 0, 0 ] ]

  An s-cochain of a module V over a Lie algebra L, is an s-linear map

  c: L×¼×L ® V (s factors L)

  that is skew-symmetric (meaning that if any of the arguments are interchanged, c changes to -c).

  Let {x1,¼,xn} be a basis of L. Then any s-cochain is determined by the values c( xi1,¼, xis ), where 1 £ i1 < i2 < ¼ < is £ dimL. Now this value again is a linear combination of basis elements of V: c( xi1,¼, xis ) = å lki1,¼, is vk. Denote the dimension of V by r. Then we represent an s-cocycle by a list of r lists. The j-th of those lists consists of entries of the form

  [ [i1,i2,¼,is], lji1,¼, is ]

  where the coefficient on the second position is non-zero. (We only store those entries for which this coefficient is non-zero.) It follows that every s-tuple (i1,¼,is) gives rise to r basis elements.

  So the zero cochain is represented by a list of the form [ [ ], [ ], ¼, [ ] ]. Furthermore, if V is, e.g., 4-dimensional, then the 2-cochain represented by

      [  [ [ [1,2], 2] ], [ ], [ [ [1,2], 1/2 ] ], [ ] ]

  maps the pair (x1,x2) to 2v1+1/2 v3 (where v1 is the first basis element of V, and v3 the third), and all other pairs to zero.

  By definition, 0-cochains are constant maps c( x ) = vc Î V for all x Î L. So 0-cochains have a different representation: they are just represented by the list [ vc ].

  Cochains are constructed using the function `Cochain` (see Cochain), if c is a cochain, then its corresponding list is returned by `ExtRepOfObj(` c`)`.

- `IsCochain(` obj`) C`
- `IsCochainCollection(` obj`) C`

  Categories of cochains and of collections of cochains.

- `Cochain(` V`,` s`,` obj`) O`

  Constructs a s-cochain given by the data in obj, with respect to the Lie algebra module V. If s is non-zero, then obj must be a list.

      gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
      gap> V:= AdjointModule( L );
      <3-dimensional left-module over <Lie algebra of dimension 3 over Rationals>>
      gap> c1:= Cochain( V, 2, [ [ [ [ 1, 3 ], -1 ] ], [ ], [ [ [ 2, 3 ], 1/2 ] ] ]);
      <2-cochain>
      gap> ExtRepOfObj( c1 );
      [ [ [ [ 1, 3 ], -1 ] ], [  ], [ [ [ 2, 3 ], 1/2 ] ] ]
      gap> c2:= Cochain( V, 0, Basis( V )[1] );
      <0-cochain>
      gap> ExtRepOfObj( c2 );
      v.1
      gap> IsCochain( c2 );
      true

- `CochainSpace(` V`,` s`) O`

  Returns the space of all s-cochains with respect to V.

      gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
      gap> V:= AdjointModule( L );;
      gap> C:=CochainSpace( V, 2 );
      <vector space of dimension 9 over Rationals>
      gap> BasisVectors( Basis( C ) );
      [ <2-cochain>, <2-cochain>, <2-cochain>, <2-cochain>, <2-cochain>,
        <2-cochain>, <2-cochain>, <2-cochain>, <2-cochain> ]
      gap> ExtRepOfObj( last[1] );
      [ [ [ [ 1, 2 ], 1 ] ], [  ], [  ] ]

- `ValueCochain(` c`,` y1`,` y2`, ...,` ys`) F`

  Here c is an s-cochain. This function returns the value of c when applied to the s elements y1 to ys (that lie in the Lie algebra acting on the module corresponding to c). It is also possible to call this function with two arguments: first c and then the list containing ``y1`,...,`ys``.

      gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
      gap> V:= AdjointModule( L );;
      gap> C:= CochainSpace( V, 2 );;
      gap> c:= Basis( C )[1];
      <2-cochain>
      gap>  ValueCochain( c, Basis(L)[2], Basis(L)[1] );
      (-1)*v.1

- `LieCoboundaryOperator(` c`) V`

  This is a function that takes an s-cochain, and returns an s+1-cochain. The coboundary operator is applied.

      gap> L:= SimpleLieAlgebra( "A", 1, Rationals );;
      gap> V:= AdjointModule( L );;
      gap> C:= CochainSpace( V, 2 );;
      gap> c:= Basis( C )[1];;
      gap> c1:= LieCoboundaryOperator( c );
      <3-cochain>
      gap> c2:= LieCoboundaryOperator( c1 );
      <4-cochain>

- `Cocycles(` V`,` s`) O`

  is the space of all s-cocycles with respect to the Lie algebra module V. That is the kernel of the coboundary operator when restricted to the space of s-cochains.

- `Coboundaries(` V`,` s`) O`

  is the space of all s-coboundaries with respect to the Lie algebra module V. That is the image of the coboundary operator, when applied to the space of s-1-cochains. By definition the space of all 0-coboundaries is zero.

      gap> T:= EmptySCTable( 3, 0, "antisymmetric" );;
      gap> SetEntrySCTable( T, 1, 2, [ 1, 3 ] );
      gap> L:= LieAlgebraByStructureConstants( Rationals, T );;
      gap> V:= FaithfulModule( L );
      <3-dimensional left-module over <Lie algebra of dimension 3 over Rationals>>
      gap> Cocycles( V, 2 );
      <vector space of dimension 7 over Rationals>
      gap> Coboundaries( V, 2 );
      <vector space over Rationals, with 9 generators>
      gap> Dimension( last );
      5

  ## 61.13 Modules over Semisimple Lie Algebras

  This section contains functions for calculating information on representations of semisimple Lie algebras. First we have some functions for calculating some combinatorial data (set of dominant weights, the dominant character, the decomposition of a tensor product, the dimension of a highest-weight module). Then there is a function for creating an admissible lattice in the universal enveloping algebra of a semisimple Lie algebra. Finally we have a function for constructing a highest-weight module over a semisimple Lie algebra.

- `DominantWeights(` R`,` maxw`) O`

  Returns a list consisting of two lists. The first of these contains the dominant weights (written on the basis of fundamental weights) of the irreducible highest-weight module over the Lie algebra with root system R. The i-th element of the second list is the level of the i-th dominant weight. (Where level is defined as follows. For a weight m we write m = l-åi ki ai, where the ai are the simple roots, and l the highest weight. Then the level of m is åi ki.

- `DominantCharacter(` L`,` maxw`) O`
- `DominantCharacter(` R`,` maxw`) O`

  For a highest weight maxw and a semisimple Lie algebra L, this returns the dominant weights of the highest-weight module over L, with highest weight maxw. The output is a list of two lists, the first list contains the dominant weights; the second list contains their multiplicities.

  The first argument can also be a root system, in which case the dominant character of the highest-weight module over the corresponding semisimple Lie algebra is returned.

- `DecomposeTensorProduct(` L`,` w1`,` w2`) O`

  Here L is a semisimple Lie algebra and w1, w2 are dominant weights. Let Vi be the irreducible highest-weight module over L with highest weight wi for i=1,2. Let W=V1Ä V2. Then in general W is a reducible L-module. Now this function returns a list of two lists. The first of these is the list of highest weights of the irreducible modules occurring in the decomposition of W as a direct sum of irreducible modules. The second list contains the multiplicities of these weights (i.e., the number of copies of the irreducible module with the corresponding highest weight that occur in W). The algorithm uses Klimyk's formula (see Klimyk68 or Klimyk66 for the original Russian version).

- `DimensionOfHighestWeightModule(` L`,` w`) O`

  Here L is a semisimple Lie algebra, and w a dominant weight. This function returns the dimension of the highest-weight module over L with highest weight w. The algorithm uses Weyl's dimension formula.

      gap> L:= SimpleLieAlgebra( "F", 4, Rationals );;
      gap> R:= RootSystem( L );;
      gap> DominantWeights( R, [ 1, 1, 0, 0 ] );
      [ [ [ 1, 1, 0, 0 ], [ 2, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ],
            [ 1, 0, 0, 0 ], [ 0, 0, 0, 0 ] ], [ 0, 3, 4, 8, 11, 19 ] ]
      gap> DominantCharacter( L, [ 1, 1, 0, 0 ] );
      [ [ [ 1, 1, 0, 0 ], [ 2, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ],
            [ 1, 0, 0, 0 ], [ 0, 0, 0, 0 ] ], [ 1, 1, 4, 6, 14, 21 ] ]
      gap> DecomposeTensorProduct( L, [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ] );
      [ [ [ 1, 0, 1, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ],
            [ 2, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 1, 1, 0, 0 ] ],
        [ 1, 1, 1, 1, 1, 1, 1 ] ]
      gap> DimensionOfHighestWeightModule( L, [ 1, 2, 3, 4 ] );
      79316832731136

  Let L be a semisimple Lie algebra over a field of characteristic 0, and let R be its root system. For a positive root a we let xa and ya be positive and negative root vectors respectively, both from a fixed Chevalley basis of L. Furthermore, h1,¼, hl are the Cartan elements from the same Chevalley basis. Also we set

  xa(n) =

  xan

  n!

  , ya(n) =

  yan

  n!

  .

  Furthermore, let a1,¼, as denote the positive roots of R. For multi-indices N=(n1,¼, ns), M=(m1,¼, ms) and K=(k1,¼, ks) (where ni,mi,ki ³ 0) set

  xN

  =

  xa1(n1)¼ xas(ns),

  yM

  =

  ya1(m1)¼ yas(ms),

  hK

  =

  | æ ç è | h1 k1 | ö ÷ ø | ¼ | æ ç è | hl kl | ö ÷ ø |

  Then by a theorem of Kostant, the xa(n) and ya(n) generate a subring of the universal enveloping algebra U(L) spanned (as a free Z-module) by the elements

  yMhKxN

  (see, e.g., Hum72 or Hum78, Section 26) So by the Poincare-Birkhoff-Witt theorem this subring is a lattice in U(L). Furthermore, this lattice is invariant under the xa(n) and ya(n). Therefore, it is called an admissible lattice in U(L).

  The next functions enable us to construct the generators of such an admissible lattice.

- `IsUEALatticeElement(` obj`) C`
- `IsUEALatticeElementCollection(` obj`) C`
- `IsUEALatticeElementFamily(` fam`) C`

  is the category of elements of an admissible lattice in the universal enveloping algebra of a semisimple Lie algebra `L`.

- `LatticeGeneratorsInUEA(` L`) A`

  Here L must be a semisimple Lie algebra of characteristic 0. This function returns a list of generators of an admissible lattice in the universal enveloping algebra of L, relative to the Chevalley basis contained in `ChevalleyBasis(` L`)`. First are listed the negative root vectors (denoted by y1,¼, ys), then the positive root vectors (denoted by x1,¼, xs). At the end of the list there are the Cartan elements. They are printed as `( hi/1 )`, which means

  | æ ç è | hi 1 | ö ÷ ø | .

  In general the printed form `( hi/ k )` means

  | æ ç è | hi k | ö ÷ ø | .

  Also yi(m) is printed as `yi^(m)`, which means that entering `yi^m` at the GAP prompt results in the output `m!*yi^(m)`.

  Products of lattice generators are collected using the following order: first come the yi(mi) (in the same order as the positive roots), then the

  | æ ç è | hi ki | ö ÷ ø | ,

  and then the xi(ni) (in the same order as the positive roots).

- `ObjByExtRep(` F`,` descr`) O`

  creates an object in the family F which has the external representation descr.

  An UEALattice element is represented by a list of the form

  [ m1, c1, m2, c2,.....]

  where the c1,c2 etc. are coefficients, and the m1, m2 etc. monomials. A monomial is a list of the form [ ind1, e1, ind2, e2, ....] where ind1, ind2 are indices, and e1, e2 etc. are exponents. Let N be the number of positive roots of the underlying Lie algebra. The indices lie between 1 and dimL. If an index lies between 1 and N, then it represents a negative root vector (corresponding to the root `NegativeRoots( R )[ind]`, where R is the root system of L). This leads to a factor `yind1^(e1)` in the printed form of the monomial (which equals z^e1/e1!, where z is a basis element of L). If an index lies between N+1 and 2N, then it represents a positive root vector. Finally, if ind lies between 2N+1 and 2N+rank, then it represents an element of the Cartan subalgebra. This is printed as ( h_1/ e_1 ), meaning h_1 choose e_1 (h_1,...,h_rank are the canonical Cartan generators).

  The zero element is represented by the empty list, the identity element by the list [ [ ], 1 ].

      gap> L:= SimpleLieAlgebra( "G", 2, Rationals );;
      gap> g:=LatticeGeneratorsInUEA( L );
      [ y1, y2, y3, y4, y5, y6, x1, x2, x3, x4, x5, x6, ( h13/1 ), ( h14/1 ) ]
      gap> IsUEALatticeElement( g[1] );
      true
      gap> g[1]^3;
      6*y1^(3)
      gap> q:= g[7]*g[1]^2;
      -2*y1+2*y1*( h13/1 )+2*y1^(2)*x1
      gap> ExtRepOfObj( q );
      [ [ 1, 1 ], -2, [ 1, 1, 13, 1 ], 2, [ 1, 2, 7, 1 ], 2 ]

- `IsWeightRepElement(` obj`) C`
- `IsWeightRepElementCollection(` obj`) C`
- `IsWeightRepElementFamily(` fam`) C`

  Is a category of vectors, that is used to construct elements of highest-weight modules (by `HighestWeightModule`).

  WeightRepElements are represented by a list of the form `[ v1, c1, v2, c2, ....]`, where the `v` i`` are basis vectors, and the `c` i`` coefficients. Furthermore a basis vector `v` is a weight vector. It is represented by a list of form `[` k`,` mon`,` wt`]`, where k is an integer (the basis vectors are numbered from 1 to dimV, where V is the highest weight module), mon is an UEALatticeElement (which means that the result of applying mon to a highest weight vector is `v`) and wt is the weight of v. A WeightRepElement is printed as ``mon`*v0`, where `v0` denotes a fixed highest weight vector.

  If v is a WeightRepElement, then `ExtRepOfObj(` v`)` returns the corresponding list, and if list is such a list and fam a WeightRepElementFamily, then `ObjByExtRep(` list`,` fam`)` returns the corresponding WeightRepElement.

- `HighestWeightModule(` L`,` wt`) F`

  returns the highest weight module with highest weight wt of the semisimple Lie algebra L of characteristic 0.

  Note that the elements of such a module lie in the category `IsLeftAlgebraModuleElement` (and in particular they do not lie in the category `IsWeightRepElement`). However, if `v` is an element of such a module, then `ExtRepOfObj( v )` is a WeightRepElement.

  Note that for the following examples of this chapter we increase the line length limit from its default value 80 to 81 in order to make some long output expressions fit into the lines.

      gap> SizeScreen([ 81, ]);;
      gap> K1:= SimpleLieAlgebra( "G", 2, Rationals );;
      gap> K2:= SimpleLieAlgebra( "B", 2, Rationals );;
      gap> L:= DirectSumOfAlgebras( K1, K2 );
      <Lie algebra of dimension 24 over Rationals>
      gap> V:= HighestWeightModule( L, [ 0, 1, 1, 1 ] );
      <224-dimensional left-module over <Lie algebra of dimension 24 over Rationals>>
      gap> vv:= GeneratorsOfLeftModule( V );;
      gap> vv[100];
      y5*y7*y10*v0
      gap> e:= ExtRepOfObj( vv[100] );
      y5*y7*y10*v0
      gap> ExtRepOfObj( e );
      [ [ 100, y5*y7*y10, [ -3, 2, -1, 1 ] ], 1 ]
      gap> Basis(L)[17]^vv[100];
      -1*y5*y7*y8*v0-1*y5*y9*v0

  ## 61.14 Tensor Products and Exterior and Symmetric Powers

- `TensorProductOfAlgebraModules(` list`) O`
- `TensorProductOfAlgebraModules(` V`,` W`) O`

  Here the elements of list must be algebra modules. The tensor product is returned as an algebra module.

      gap> L:= SimpleLieAlgebra("G",2,Rationals);;
      gap> V:= HighestWeightModule( L, [ 1, 0 ] );;
      gap> W:= TensorProductOfAlgebraModules( [ V, V, V ] );
      <343-dimensional left-module over <Lie algebra of dimension 14 over Rationals>>
      gap> w:= Basis(W)[1];
      1*(1*v0<x>1*v0<x>1*v0)
      gap> Basis(L)[1]^w;
      <0-tensor>
      gap> Basis(L)[7]^w;
      1*(1*v0<x>1*v0<x>y1*v0)+1*(1*v0<x>y1*v0<x>1*v0)+1*(y1*v0<x>1*v0<x>1*v0)

- `ExteriorPowerOfAlgebraModule(` V`,` k`) O`

  Here V must be an algebra module, defined over a Lie algebra. This function returns the k-th exterior power of V as an algebra module.

      gap> L:= SimpleLieAlgebra("G",2,Rationals);;
      gap> V:= HighestWeightModule( L, [ 1, 0 ] );;
      gap> W:= ExteriorPowerOfAlgebraModule( V, 3 );
      <35-dimensional left-module over <Lie algebra of dimension 14 over Rationals>>
      gap> w:= Basis(W)[1];
      1*(1*v0/\y1*v0/\y3*v0)
      gap> Basis(L)[10]^w;
      1*(1*v0/\y1*v0/\y6*v0)+1*(1*v0/\y3*v0/\y5*v0)+1*(y1*v0/\y3*v0/\y4*v0)

- `SymmetricPowerOfAlgebraModule(` V`,` k`) O`

  Here V must be an algebra module. This function returns the k-th symmetric power of V (as an algebra module).

      gap> L:= SimpleLieAlgebra("G",2,Rationals);;
      gap> V:= HighestWeightModule( L, [ 1, 0 ] );;
      gap> W:= SymmetricPowerOfAlgebraModule( V, 3 );
      <84-dimensional left-module over <Lie algebra of dimension 14 over Rationals>>
      gap> w:= Basis(W)[1];
      1*(1*v0.1*v0.1*v0)
      gap> Basis(L)[2]^w;
      <0-symmetric element>
      gap> Basis(L)[7]^w;
      3*(1*v0.1*v0.y1*v0)

- `DirectSumOfAlgebraModules(` list`) O`
- `DirectSumOfAlgebraModules(` V`,` W`) O`

  Here list must be a list of algebra modules. This function returns the direct sum of the elements in the list (as an algebra module). The modules must be defined over the same algebras.

  In the second form is short for `DirectSumOfAlgebraModules( [` V`,` W`] )`

      gap> L:= SimpleLieAlgebra( "C", 3, Rationals );;
      gap> V:= HighestWeightModule( L, [ 1, 1, 0 ] );
      <64-dimensional left-module over <Lie algebra of dimension 21 over Rationals>>
      gap> W:= HighestWeightModule( L, [ 0, 0, 2 ] );
      <84-dimensional left-module over <Lie algebra of dimension 21 over Rationals>>
      gap> U:= DirectSumOfAlgebraModules( V, W );
      <148-dimensional left-module over <Lie algebra of dimension 21 over Rationals>>
      gap> SizeScreen([ 80, ]);;

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  GAP 4 manual
  May 2002