On Commutative Analogues of Clifford Algebras and Their Decompositions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917158690226176 |
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| author | Sharma, Heerak Shirokov, Dmitry |
| author_facet | Sharma, Heerak Shirokov, Dmitry |
| contents | We investigate commutative analogues of Clifford algebras -- algebras whose generators square to $\pm1$ but commute, instead of anti-commuting as they do in Clifford algebras. We observe that commutativity allows for elegant results. We note that these algebras generalise multicomplex spaces -- we show that a commutative analogue of Clifford algebra is either isomorphic to a multicomplex space or to `multi split-complex space' (space defined just like multicomplex numbers but uses split-complex numbers instead of complex numbers). We do a general study of commutative analogues of Clifford algebras and use tools like operations of conjugation and idempotents to give a tensor product decomposition and a direct sum decomposition for them. Tensor product decomposition follows relatively easily from the definition. For the direct sum decomposition, we give explicit basis using new techniques. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_19763 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Commutative Analogues of Clifford Algebras and Their Decompositions Sharma, Heerak Shirokov, Dmitry Rings and Algebras Commutative Algebra 15A69, 15A66 We investigate commutative analogues of Clifford algebras -- algebras whose generators square to $\pm1$ but commute, instead of anti-commuting as they do in Clifford algebras. We observe that commutativity allows for elegant results. We note that these algebras generalise multicomplex spaces -- we show that a commutative analogue of Clifford algebra is either isomorphic to a multicomplex space or to `multi split-complex space' (space defined just like multicomplex numbers but uses split-complex numbers instead of complex numbers). We do a general study of commutative analogues of Clifford algebras and use tools like operations of conjugation and idempotents to give a tensor product decomposition and a direct sum decomposition for them. Tensor product decomposition follows relatively easily from the definition. For the direct sum decomposition, we give explicit basis using new techniques. |
| title | On Commutative Analogues of Clifford Algebras and Their Decompositions |
| topic | Rings and Algebras Commutative Algebra 15A69, 15A66 |
| url | https://arxiv.org/abs/2504.19763 |