Normal Forms for Operators via Gröbner Bases in Tensor Algebras
We propose a general algorithmic approach to noncommutative operator algebras generated by linear operators using quotients of tensor algebras. In order to work with reduction systems in tensor algebras, Bergman’s setting provides a tensor analog of Gröbner bases. We discuss a modification of Bergman’s setting that allows for smaller reduction systems and tends to make computations more efficient. Verification of the confluence criterion based on S-polynomials has been implemented as a Mathematica package. Our implementation can also be used for computer-assisted construction of Gröbner bases starting from basic identities of operators. We illustrate our approach and the software using differential and integro-differential operators as examples.
KeywordsOperator algebra Tensor algebra Noncommutative Gröbner basis Reduction systems
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