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The construction of a Lie algebra defined by structure constants
is identical to that of a
general structure constant algebra. Most constructors take two optional
parameters: Check and Rep.
By default, the conditions for the algebra to be a Lie algebra are checked.
If the user decides to omit this check, by setting the parameter
Check to false, and the algebra is not actually Lie
then functions in this section will fail or give incorrect answers.
The optional parameter Rep can be used to select the internal
representation of the structure constants. The possible values for
Rep are "Dense", "Sparse" and "Partial", with the
default being "Dense".
In the dense format, the n3 structure constants are stored as n2 vectors
of length n. This is the best representation if most
of the structure constants are non-zero.
The sparse format, intended for use when most structure constants are zero,
stores the positions and values of the non-zero structure constants.
The partial format stores the vectors,
but records for efficiency the positions of the non-zero structure constants.
LieAlgebra<R, n | Q : parameters > : Rng, RngIntElt, SeqEnum -> AlgLie
LieAlgebra<M | Q : parameters > : ModTupRng, SeqEnum -> AlgLie
Check: BoolElt Default: true
Rep: MonStgElt Default: "Dense"
This function creates the Lie structure constant algebra L over
the free module M = Rn, with standard basis {e1, e2, ..., en},
and structure constants aijk being given by the sequence Q.
The sequence Q can be of any of the following three forms.
Note that in all cases the actual ordering of the structure constants is
the same: the only difference is that their partitioning into blocks varies.
- (i)
- A sequence of n sequences of n sequences of length n.
The j-th element of the i-th sequence is the sequence
[ aij1, ..., aijn ], or the element (aij1, ..., aijn) of M, giving the coefficients of the product
ei * ej.
- (ii)
- A sequence of n2 sequences of length n, or n2 elements
of M. Here the coefficients of ei * ej are given by position
(i - 1)n + j of Q.
- (iii)
- A sequence of n3 elements of the ring R. The
sequence elements are the structure constants themselves, in the
order a111, a112, ..., a11n, a121, a122, ...,
annn. So aijk lies in position (i - 1)n2 + (j - 1)n + k
of Q.
Check: BoolElt Default: true
Rep: MonStgElt Default: "Dense"
This function creates the Lie structure constant algebra L with
standard basis
{e1, e2, ..., en} over the ring R. The sequence T contains quadruples
< i, j, k, aijk> giving the non-zero structure constants. All other
structure constants are defined to be 0.
Check: BoolElt Default: true
Rep: MonStgElt Default: "Dense"
This function creates the Lie structure constant algebra L over the
integers, with standard basis {e1, e2, ..., en}.
The sequence T contains quadruples < i, j, k, aijk> (where
the aijk are integers) giving the non-zero structure constants.
All other structure constants are defined to be 0.
The argument t is a sequence of length n consisting of nonnegative
integers giving the moduli of the basis elements. Thus let ti denote
the i-th element of t; then tiei =0. So if ti=0, then
kei != 0 for all integers k.
Given an associative structure-constant algebra A, create the Lie
algebra L consisting of the elements in A with the induced Lie product
(x, y) -> x * y - y * x.
As a second value the map identifying the elements of L and A is returned.
Given an associative matrix algebra A, create a structure-constant Lie
algebra L isomorphic to A with the induced Lie product
(x, y) -> x * y - y * x.
Rep: MonStgElt Default: "Sparse"
Create the abelian Lie algebra of dimension n over the ring R.
We construct the Heisenberg Lie algebra, then a Lie algebra from an
associative algebra, and finally a Lie algebra over the integers (also
called a Lie ring).
> T:= [ <1,2,3,1>, <2,1,3,-1> ];
> LieAlgebra< Rationals(), 3 | T >;
Lie Algebra of dimension 3 with base ring Rational Field
> A:= Algebra( GF(27), GF(3) );
> LieAlgebra(A);
Lie Algebra of dimension 3 with base ring GF(3)
> T:= [ <1,2,2,2>, <2,1,2,2> ];
> t:= [0,4];
> K:= LieAlgebra< t | T : Rep:= "Dense" >; K;
Lie Algebra of dimension 2 with base ring Integer Ring
Column moduli: [0, 4]
> LowerCentralSeries( K );
[
Lie Algebra of dimension 2 with base ring Integer Ring
Column moduli: [0, 4],
Lie Algebra of dimension 1 with base ring Integer Ring
Column moduli: [2],
Lie Algebra of dimension 0 with base ring Integer Ring
]
ChangeBasis(L, B) : AlgLie, [ModTupFldElt] -> AlgLie
ChangeBasis(L, B) : AlgLie, Mtrx -> AlgLie
Rep: MonStgElt Default: "Dense"
Create a new Lie structure constant algebra L', isomorphic to L, by recomputing the structure constants with respect to the basis B.
The basis B can be specified as a set or sequence of elements of L, a set or sequence of vectors, or a matrix.
The second returned value is the isomorphism from L to L'.
As above, the optional parameter Rep can be used to select the internal
representation of the structure constants.
Note that the default is dense representation, regardless of the representation used by L.
Given a ring R and an integer n, create the full Lie algebra of matrices of degree
d over R.
Given an associative matrix algebra A, create the matrix Lie
algebra L consisting of the elements in A with the induced Lie product
(x, y) -> x * y - y * x.
LieAlgebra(M) : AlgMatLie -> AlgLie, Map
Return a structure-constant Lie algebra isomorphic to the matrix Lie algebra
M.
We construct the subalgebra of the matrix Lie algebra of 2 x 2 matrices,
consisting of upper triangular matrices.
> L:= MatrixLieAlgebra( Rationals(), 2 );
> a:= L!Matrix( [[1,0],[0,0]] );
> b:= L!Matrix( [[0,0],[1,0]] );
> c:= L!Matrix( [[0,0],[0,1]] );
> K:= sub< L | [ a, b, c ] >;
> Dimension(K);
3
> IsSolvable(K);
true
> IsNilpotent(K);
false
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