Spectrum of Rota-Baxter operators
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866915450375372800 |
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| author | Gubarev, Vsevolod |
| author_facet | Gubarev, Vsevolod |
| contents | We prove that the spectrum of every Rota-Baxter operator of weight $λ$ on a unital algebraic (not necessarily associative) algebra over a field of characteristic zero is a subset of $\{0,-λ\}$. For a finite-dimensional unital algebra the same statement is shown to hold without a restriction on the characteristic of the ground field. Based on these results, we define the Rota-Baxter $λ$-index $\mathrm{rb}_λ(A)$ of an algebra $A$ as the infimum of the degrees of minimal polynomials of all Rota-Baxter operators of weight $λ$ on $A$. We calculate the Rota-Baxter $λ$-index for the matrix algebra $M_n(F)$, $\mathrm{char}\,F = 0$: it is shown that $\mathrm{rb}_λ(M_n(F)) = 2n-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_02654 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Spectrum of Rota-Baxter operators Gubarev, Vsevolod Rings and Algebras We prove that the spectrum of every Rota-Baxter operator of weight $λ$ on a unital algebraic (not necessarily associative) algebra over a field of characteristic zero is a subset of $\{0,-λ\}$. For a finite-dimensional unital algebra the same statement is shown to hold without a restriction on the characteristic of the ground field. Based on these results, we define the Rota-Baxter $λ$-index $\mathrm{rb}_λ(A)$ of an algebra $A$ as the infimum of the degrees of minimal polynomials of all Rota-Baxter operators of weight $λ$ on $A$. We calculate the Rota-Baxter $λ$-index for the matrix algebra $M_n(F)$, $\mathrm{char}\,F = 0$: it is shown that $\mathrm{rb}_λ(M_n(F)) = 2n-1$. |
| title | Spectrum of Rota-Baxter operators |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2006.02654 |