Construction of Associative Algebras

Construction of an Associative Structure Constant Algebra

The construction of an associative 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 associative.

AssociativeAlgebra< R, n | Q : parameters > : Rng, RngIntElt, SeqEnum -> AlgAss

AssociativeAlgebra< M | Q : parameters > : ModTupRng, SeqEnum -> AlgAss

    Check: BoolElt                      Default: true

    Rep: MonStgElt                      Default: "Dense"

This function creates the associative structure constant algebra A over the free module M = Rⁿ, with standard basis (e₁, e₂, ..., 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.

(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_ij¹, ..., a_ijⁿ ], or the element (a_ij¹, ..., a_ijⁿ) of M, giving the coefficients of the product e_i * e_j.

(ii)

A sequence of n² sequences of length n, or n² 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 n³ elements of the ring R. Here the sequence elements are the structure constants themselves, a₁₁¹, a₁₁², ..., a₁₁ⁿ, a₁₂¹, a₁₂², ..., a_nnⁿ. So a_ij^k lies in position (i - 1) * n² + (j - 1) * n + k of Q.

By default the algebra is checked to be associative; 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 n³ structure constants are stored as n² 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.

AssociativeAlgebra< R, n | T : parameters > : Rng, RngIntElt, SeqEnum -> AlgAss

    Check: BoolElt                      Default: true

    Rep: MonStgElt                      Default: "Sparse"

This function creates the associative structure constant algebra A with standard basis (e₁, e₂, ..., 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.

AssociativeAlgebra(A) : AlgGen -> AlgAss

Given a structure constant algebra A of type AlgGen, construct an isomorphic associative structure constant algebra of type AlgAss. If it is not known whether or not A is associative, this will be checked and an error occurs if it is not. The elements of the resulting algebra can be coerced into A and vice versa.

ChangeBasis(A, B) : AlgAss, [AlgAssElt] -> AlgAss

ChangeBasis(A, B) : AlgAss, [ModTupFldElt] -> AlgAss

ChangeBasis(A, B) : AlgAss, Mtrx -> AlgAss

    Rep: MonStgElt                      Default: "Dense"

Create a new associative structure constant algebra A', isomorphic to A, 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 A, a set or sequence of vectors, or a matrix. The second returned value is the isomorphism from A to A'.

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 A.

Associative Structure Constant Algebras from other Algebras

Algebra(A) : AlgGrp -> AlgAss, Map

If A is either a group algebra of type AlgGrp given in vector representation or a matrix algebra of type AlgMat, construct the associative structure constant algebra B isomorphic to A together with the isomorphism A -> B.

Algebra(F, E) : FldFin, FldFin -> AlgAss, Map;

Algebra(F, E) : FldNum, FldNum -> AlgAss, Map;

Let E and F be either finite fields or algebraic number fields such that E is a subfield of F. This function returns the associative algebra A of dimension [F:E] over E which is isomorphic to F, together with the isomorphism from F to A such that the (i - 1)-th power of the generator of F over E is mapped to the i-th basis vector of A.

AlgebraOverCenter(A) : Alg -> AlgAss, Map;

Given a simple algebra A of type AlgMat or AlgAss with center K, this function returns a K-algebra B which is K-isomorphic to A as well as an isomorphism from A to B.