The criteria for the uniqueness of a weight homomorphism of a baric algebra
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913670575947776 |
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| author | Zangurashvili, Dali |
| author_facet | Zangurashvili, Dali |
| contents | The criteria for a baric algebra $A$ (over a field $K$) to have a unique weight homomorphism are found. One of them requires a certain system of equations to have a unique non-trivial solution in the field $K$. Applying this criterion, we provide an example showing that Holgate's well-known sufficient condition for the uniqueness of a weight homomorphism is not necessary, and give also a new example of a baric algebra with two weight homomorphisms. Another criterion found in this paper asserts that a baric algebra has a unique weight homomorphism if and only if the transition matrix from any semi-natural basis $B_1$ to any semi-natural basis $B_2$ is stochastic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04612 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The criteria for the uniqueness of a weight homomorphism of a baric algebra Zangurashvili, Dali Rings and Algebras 17D92, 17D99 The criteria for a baric algebra $A$ (over a field $K$) to have a unique weight homomorphism are found. One of them requires a certain system of equations to have a unique non-trivial solution in the field $K$. Applying this criterion, we provide an example showing that Holgate's well-known sufficient condition for the uniqueness of a weight homomorphism is not necessary, and give also a new example of a baric algebra with two weight homomorphisms. Another criterion found in this paper asserts that a baric algebra has a unique weight homomorphism if and only if the transition matrix from any semi-natural basis $B_1$ to any semi-natural basis $B_2$ is stochastic. |
| title | The criteria for the uniqueness of a weight homomorphism of a baric algebra |
| topic | Rings and Algebras 17D92, 17D99 |
| url | https://arxiv.org/abs/2412.04612 |