60.2 Constructing Lie algebras

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>

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>

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)

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>

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 [ d_i,j ] which means that D maps the i-th basis element b_i to å_{j = 1}ⁿ d_ij b_j.

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>

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 A_n over a field of characteristic p > 0 where p divides n+1, and the Lie algebras of type K_n 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)>

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