Weakly primitive axial algebras
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916390350356480 |
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| author | Rowen, Louis Halle Segev, Yoav |
| author_facet | Rowen, Louis Halle Segev, Yoav |
| contents | In earlier work we studied the structure of primitive axial algebras of Jordan type (PAJ's), not necessarily commutative, in terms of their primitive axes. In this paper we weaken primitivity and permit several pairs of (left and right) eigenvalues satisfying a more general fusion rule, bringing in interesting new examples such as the band semigroup algebras and other commutative and noncommutative examples. Also we broaden our investigation and describe 2-generated algebras in which only one of the generating axes is weakly primitive and satisfies the fusion rules, on condition that its zero-eigenspace is one dimensional. We also characterize when both axes satisfy the fusion rules (weak PAJ's), and describe precisely the 2-dimensional axial algebras. In contrast to the previous situation, there are weak PAJ's of dimension~$> 3$ generated by two axes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08606 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weakly primitive axial algebras Rowen, Louis Halle Segev, Yoav Rings and Algebras 17C27 In earlier work we studied the structure of primitive axial algebras of Jordan type (PAJ's), not necessarily commutative, in terms of their primitive axes. In this paper we weaken primitivity and permit several pairs of (left and right) eigenvalues satisfying a more general fusion rule, bringing in interesting new examples such as the band semigroup algebras and other commutative and noncommutative examples. Also we broaden our investigation and describe 2-generated algebras in which only one of the generating axes is weakly primitive and satisfies the fusion rules, on condition that its zero-eigenspace is one dimensional. We also characterize when both axes satisfy the fusion rules (weak PAJ's), and describe precisely the 2-dimensional axial algebras. In contrast to the previous situation, there are weak PAJ's of dimension~$> 3$ generated by two axes. |
| title | Weakly primitive axial algebras |
| topic | Rings and Algebras 17C27 |
| url | https://arxiv.org/abs/2408.08606 |