Ore algebra

In computer algebra, an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear differential and/or recurrence operators.{{Cite journal | last1 = Chyzak | first1 = Frédéric | last2 = Salvy | first2 = Bruno | year = 1998 | title = Non-commutative Elimination in Ore Algebras Proves Multivariate Identities | journal = Journal of Symbolic Computation | volume = 26 | issue = 2 | pages = 187–227 | publisher = Elsevier | doi = 10.1006/jsco.1998.0207 | url = https://hal.inria.fr/hal-01069833/file/holonomy.pdf }} The concept is named after Øystein Ore.

Definition

Let $K$ be a (commutative) field and $A = K[x_1, \ldots, x_s]$ be a commutative polynomial ring (with $A = K$ when $s = 0$ ). The iterated skew polynomial ring $A[\partial_1; \sigma_1, \delta_1] \cdots [\partial_r; \sigma_r, \delta_r]$ is called an Ore algebra when the $\sigma_i$ and $\delta_j$ commute for $i \neq j$ , and satisfy $\sigma_i(\partial_j) = \partial_j$ , $\delta_i(\partial_j) = 0$ for $i > j$ .

Properties

Ore algebras satisfy the Ore condition, and thus can be embedded in a (skew) field of fractions.

The constraint of commutation in the definition makes Ore algebras have a non-commutative generalization theory of Gröbner basis for their left ideals.

References

Category:Computer algebra

Category:Ring theory