Subsections

• Construction of a Lie Structure Constant Algebra

The construction of a Lie structure constant algebra is identical to that of a general structure constant algebra, with the exception that an additional parameter is provided which may be used to avoid checking that the algebra is Lie.

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 A over the free module M = R^n, with standard basis (e_1, e_2, ..., e_n), and with the structure constants a_(ij)^k 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: it is only their division that varies.

• A sequence of n sequences of n sequences of length n. The j-th element of the i-th sequence is the sequence [ a_(ij)^1, ..., a_(ij)^n ], or the element (a_(ij)^1, ..., a_(ij)^n) of M, giving the coefficients of the product e_i * e_j.
• A sequence of n^2 sequences of length n, or n^2 elements of M. Here the coefficients of e_i * e_j are given by position (i - 1) * n + j of Q.
• A sequence of n^3 elements of the ring R. Here the sequence elements are the structure constants themselves, with the ordering a_(11)^1, a_(11)^2, ..., a_(11)^n, a_(12)^1, a_(12)^2, ..., a_(nn)^n. So a_(ij)^k lies in position (i - 1) * n^2 + (j - 1) * n + k of Q.

By default the algebra is checked to be a Lie algebra; this can be overruled by setting the parameter Check to false.

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 n^3 structure constants are stored as n^2 vectors of length n, similarly to (ii) above. 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 | T : parameters > : Rng, RngIntElt, SeqEnum -> AlgLie

    Check: BoolElt                      Default: true

    Rep: MonStgElt                      Default: "Sparse"

This function creates the Lie structure constant algebra A with standard basis (e_1, e_2, ..., e_n) over R. The sequence T contains quadruples < i, j, k, a_(ij)^k> giving the non-zero structure constants. All other structure constants are defined to be 0.

The optional parameters are as above.

LieAlgebra(A) : AlgAss -> AlgLie

Given an associative algebra A, create the Lie algebra L generated by the elements in A using the induced Lie product (x, y) -> x * y - y * x. As a second value the map identifying the elements of A and L is returned.