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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.14995 |
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| _version_ | 1866915974514475008 |
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| author | Greensite, Fred |
| author_facet | Greensite, Fred |
| contents | We show that a real finite-dimensional unital associative algebra is naturally associated with a vector space of pseudo-Finsler norms whose members are linked to the algebra's space of normalized trace forms through an integral transform. Since components of the space of trace forms act as parameters controlling the implied pseudo-Finsler indicatrices, successive application of the usual Hamiltonian formalism can lead to caustics and bifurcations of caustics (i.e., catastrophes and catastrophes of catastrophes) by continuous variation of these parameters. In order to capture influence from the entirety of the Jacobson radical, the transform procedure can be applied at all appropriate levels of a Cuntz-Quillen tower defining the algebra's formal neighborhood. The latter procedure leads to a trove of algebra invariants, whose intricacy reflects the wildness of the algebra isomorphism problem that appears when dimensions of the algebras in question are not small. In that respect, description of algebra structure using this methodology compares favorably with programs whose content is primarily homological. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14995 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The catastrophes of algebras Greensite, Fred Rings and Algebras We show that a real finite-dimensional unital associative algebra is naturally associated with a vector space of pseudo-Finsler norms whose members are linked to the algebra's space of normalized trace forms through an integral transform. Since components of the space of trace forms act as parameters controlling the implied pseudo-Finsler indicatrices, successive application of the usual Hamiltonian formalism can lead to caustics and bifurcations of caustics (i.e., catastrophes and catastrophes of catastrophes) by continuous variation of these parameters. In order to capture influence from the entirety of the Jacobson radical, the transform procedure can be applied at all appropriate levels of a Cuntz-Quillen tower defining the algebra's formal neighborhood. The latter procedure leads to a trove of algebra invariants, whose intricacy reflects the wildness of the algebra isomorphism problem that appears when dimensions of the algebras in question are not small. In that respect, description of algebra structure using this methodology compares favorably with programs whose content is primarily homological. |
| title | The catastrophes of algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2306.14995 |