38.1 More about Algebras

Let F be a field. A ring A is called an F-algebra if A is an F-vector space. All algebras in GAP are associative, that is, the multiplication is associative.

An algebra always contains a zero element that can be obtained by subtracting an arbitrary element from itself. A discussion of identity elements of algebras (and of the consequences for the implementation in GAP) can be found in Algebras and Unital Algebras.

Elements of the field F are not regarded as elements of A. The practical reason (besides the obvious mathematical one) for this is that even if the identity matrix is contained in the matrix algebra A it is not possible to write 1 + a for adding the identity matrix to the algebra element a, since independent of the algebra A the meaning in GAP is already defined as to add 1 to all positions of the matrix a. Thus one has to write One( A ) + a or a^0 + a instead.

The natural operation domains for algebras are modules (see Operation for Algebras, and chapter Modules).

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GAP 3.4.4
April 1997