The construction of a Lie structure constant algebrais identical to that of a general structure constant algebra. By default, the algebra is checked to be Lie. If the user decides to omit this check and the algebra is not actually Lie, the functions in this section will fail or give incorrect answers.
This function creates the Lie structure constant algebra L over the free module M = R^n, with standard basis (e_1, e_2, ..., e_n), and 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.
- (i)
- 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.
- (ii)
- 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.
- (iii)
- 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.
Check: BoolElt Default: trueBy default the algebra is checked to be a Lie algebra; this can be overruled by setting the parameter Check to false.
Rep: MonStgElt Default: "Dense"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.
This function creates the Lie structure constant algebra L 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.
Check: BoolElt Default: trueBy default the algebra is checked to be a Lie algebra; this can be overruled by setting the parameter Check to false.
Rep: MonStgElt Default: "Dense"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.
Given an associative algebra L, create the Lie algebra L generated by the elements in L using the induced Lie product (x, y) -> x * y - y * x. As a second value the map identifying the elements of L and L is returned.
The radical of a Lie algebra is the maximal soluble ideal. A Lie algebra is called reductiveif its radical is equal to its centre, and semisimpleif its radical is trivial. The commands in the section construct the split reductive Lie algebras over an arbitrary field. Such Lie algebras have a corresponding root datum.
The semisimple Lie algebra over the ring k with Cartan type N given as a string.
The reductive Lie algebra with root datum R over the ring k (see resp. ROOT DATA).
The semisimple Lie algebra with crystallographic root system R over the ring k (see Chapter ROOT SYSTEMS).
> SemisimpleLieAlgebra( "G2 B3", Rationals() ); Lie Algebra of dimension 35 with base ring Rational Field > SemisimpleLieAlgebra( "E8", GF(2) ); Lie Algebra of dimension 248 with base ring GF(2)
The semisimple Lie algebra with crystallographic Cartan matrix C over the ring k (see Section Cartan Matrices).
The semisimple Lie algebra with Dynkin digraph D over the ring k (see Section Dynkin Digraphs).
This function constructs the simple Lie algebra of Cartan type X_n over the ring k. The result is a Lie algebra defined by a multiplication table. Here X is a string which can be one of "A", "B", "C", "D", "E", "F" or "G" and n is a positive integer. In a few 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 D_1 and D_2.
> SimpleLieAlgebra("D", 7, RationalField());
Lie Algebra of dimension 91 with base ring Rational Field
> SimpleLieAlgebra("G", 2, GF(5));
Lie Algebra of dimension 14 with base ring GF(5)