A new classification of algebraic identities for linear operators on associative algebras
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866915418920189952 |
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| author | Bremner, Murray R. Elgendy, Hader A. |
| author_facet | Bremner, Murray R. Elgendy, Hader A. |
| contents | We introduce a new approach to the classification of operator identities, based on basic concepts from the theory of algebraic operads together with computational commutative algebra applied to determinantal ideals of matrices over polynomial rings. We consider operator identities of degree 2 (the number of variables in each term) and multiplicity 1 or 2 (the number of operators in each term), but our methods apply more generally. Given an operator identity with indeterminate coefficients, we use partial compositions to construct a matrix of consequences, and then use computer algebra to determine the values of the indeterminates for which this matrix has submaximal rank. For multiplicity 1 we obtain six identities, including the derivation identity. For multiplicity 2 we obtain eighteen identities and two parametrized families, including the left and right averaging identities, the Rota-Baxter identity, the Nijenhuis identity, and some new identities which deserve further study. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_04391 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A new classification of algebraic identities for linear operators on associative algebras Bremner, Murray R. Elgendy, Hader A. Rings and Algebras Operator Algebras Primary 47C05. Secondary 13P10, 13P15, 17B38, 18M70, 39B42, 47-08 We introduce a new approach to the classification of operator identities, based on basic concepts from the theory of algebraic operads together with computational commutative algebra applied to determinantal ideals of matrices over polynomial rings. We consider operator identities of degree 2 (the number of variables in each term) and multiplicity 1 or 2 (the number of operators in each term), but our methods apply more generally. Given an operator identity with indeterminate coefficients, we use partial compositions to construct a matrix of consequences, and then use computer algebra to determine the values of the indeterminates for which this matrix has submaximal rank. For multiplicity 1 we obtain six identities, including the derivation identity. For multiplicity 2 we obtain eighteen identities and two parametrized families, including the left and right averaging identities, the Rota-Baxter identity, the Nijenhuis identity, and some new identities which deserve further study. |
| title | A new classification of algebraic identities for linear operators on associative algebras |
| topic | Rings and Algebras Operator Algebras Primary 47C05. Secondary 13P10, 13P15, 17B38, 18M70, 39B42, 47-08 |
| url | https://arxiv.org/abs/2109.04391 |