Hasse–Schmidt derivation
In mathematics, a Hasse–Schmidt derivation is an extension of the notion of a derivation. The concept was introduced by Schmidt & Hasse (1937).
Definition
For a (not necessarily commutative nor associative)
taking values in the ring of formal power series with coefficients in A. This definition is found in several places, such as Gatto & Salehyan (2016, §3.4), which also contains the following example: for A being the ring of infinitely differentiable functions (defined on, say, Rn) and B=R, the map
is a Hasse–Schmidt derivation, as follows from applying the Leibniz rule iteratedly.
Equivalent characterizations
Hazewinkel (2012) shows that a Hasse–Schmidt derivation is equivalent to an action of the bialgebra
of noncommutative symmetric functions in countably many variables Z1, Z2, ...: the part of D which picks the coefficient of , is the action of the indeterminate Zi.
Applications
Hasse–Schmidt derivations on the exterior algebra of some B-module M have been studied by Gatto & Salehyan (2016, §4). Basic properties of derivations in this context lead to a conceptual proof of the Cayley–Hamilton theorem. See also Gatto & Scherbak (2015).
References
- Gatto, Letterio; Salehyan, Parham (2016), Hasse–Schmidt derivations on Grassmann algebras, Springer, MR 3524604
- Gatto, Letterio; Scherbak, Inna (2015), Remarks on the Cayley-Hamilton Theorem, arXiv:1510.03022
- Hazewinkel, Michiel (2012), "Hasse–Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions", Axioms, 1 (2): 149–154, S2CID 15969581
- Schmidt, F.K.; S2CID 120317012