It is well known that the derivative is an operation on functions which is linear
and satisfies the Leibniz rule
In fact, these two properties are purely algebraic and can be generalized with no reference to infinitesimal calculus. This approach has become common in modern mathematics since it allows profitable interactions between multiples domains such as algebra, analysis and geometry.
Let A be an algebra over a commutative field K. A derivation of A is any linear map D satisfying the Leibniz rule for all
If A is unital then because .
We denote by the set of all derivations of A.
Clearly, the sum of two derivations is a derivation. We could be tempted to consider that it has a module structure over A. Unfortunately, it is not true in general. Indeed, if and we have for every
so the Leibniz rule is satisfied by aD only if . Thus is not an A-module in general unless A is commutative.
The product of two derivations is not a derivation apart trivial case, but it is straightforward to show that the commutator of two derivations is a derivation. Thus has a Lie algebra structure
The commutator also allows to associate an operator to each
It is easy to check that is a derivation
Such a derivation is called an inner derivation as it is automatically generated by the algebra.
- Let be the algebra of smooth functions on . Then the operator is a derivation.
- Let be the algebra of square matrices of order n over a field K. Then given a matrix the operator is an inner derivation.
- Let be the algebra of smooth functions over a differentiable manifold M. Then a vector field X gives rise to a derivation defined by where f is any smooth function and is the directionnal derivative at point . The Leibniz rule is satisfied as .