In the early days [ 1,2] a was taken to be A itself. Later [9, Chap. 31 examples where Z? formed a Lie algebra, or some other algebraic relationship such as [p, X] = 1 as in quantum mechanics, or xy = qyx as in q-deformed algebras, were studied.
For each subspace B one could construct a co-frame. This co-frame is loosely analogous to the orthonormal co-frame used in normal differential geometry. By quotienting the universal calculus one could then construct the set of 2-forms and higher-order forms.
It was discovered that if B = A, or B formed a Lie algebra or a quantum algebra then one could consistently impose the condition that the co-frame basis elements of the exterior algebra anticommute. Whilst for q-deformed algebras the basis elements of the exterior algebra q-anticommute.
It is only recently [3,4] that people have looked at a general t3. They showed that in order for forms of order 2 and above to exist puts constraints on the elements of B. However, these constraints have not been pursued.
In this paper we impose the condition A = M,(C) and that all the elements in B are traceless. In Section 3, we show that this is a sufficient condition for the co-frame to exist.
However, this condition is not a necessary condition for the co-frame to exist. To show this we give some examples where A # M,(C), some of which have a co-frame and others which do not. Since M,(C) is a finite approximation to the infinite-dimensional space of functions it is hoped that this procedure can be used as an alternative to the theory of renormalisation or lattice QFT.
In Section 2 we introduce the concept of a "generalised algebra". This is an algebraic structure that includes commutative algebras, anti-commutative algebras, Lie algebras, Clifford algebras and q-deformed algebras as examples. Each generalised algebra has a specific rank and the space of 2-forms is a free module over A of rank equal to the rank of the generalised algebra. In Section 4 we show that for 2-forms and higher forms to exist B must form a generalised algebra. In Section 5, we then give the structure of the higher-order forms, all of which are also free modules over A, and an explicit expression for the exterior derivative. In Section 7 we give a couple of simple examples of maps between generalised algebras which are d-homomorphism, i.e. they preserve the differentiable structure. To elucidate the relationship between the generalised algebra of B and the space of 2forms we give, in Section 8, four examples: Much emphasis has been placed on the case that f3 form a Lie algebra. Especially since su(2) corresponds to the fuzzy or noncommutative version of the sphere [9, Chap. 7.21 and su ( 4) is an analogue of the Euclidianised compactified Minkowski space [8]. Another example is that of the q-deformed algebra, this has a finite-dimensional representation only if there exists an nz E Z such that qm = 1. Finally a t3 is given of dimension 3 and rank 1 which may be thought of as the fuzzy ellipse.
For further references, and history of this subject the reader is asked to read the book [9].
1.1. Note on notation
Unless otherwise stated A = M,(C). L3 c A is a subspace of dimension n of traceless matrices and h, is a basis for B. Early Roman letters used as indices a, b, . . . run over 1 , . . . . n, and we use the Einstein summation convention so that the summation is implicit