Abstract
Rota–Baxter operators present a natural generalization of integration by parts formula for the integral operator. In 2015, Zheng, Guo, and Rosenkranz conjectured that every injective Rota–Baxter operator of weight zero on the polynomial algebra \(\mathbb {R}[x]\) is a composition of the multiplication by a nonzero polynomial and a formal integration at some point. We confirm this conjecture over any field of characteristic zero. Moreover, we establish a structure of an ind-variety on the moduli space of these operators and describe an additive structure of generic modality two on it. Finally, we provide an infinitely transitive action on codimension one subsets.
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Vsevolod Gubarev is supported by the Program of fundamental scientific researches of the Siberian Branch of Russian Academy of Sciences, I.1.1, project 0314-2019-0001. The research of Alexander Perepechko was supported by the grant RSF-19-11-00172.
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Gubarev, V., Perepechko, A. Injective Rota–Baxter Operators of Weight Zero on F[x]. Mediterr. J. Math. 18, 267 (2021). https://doi.org/10.1007/s00009-021-01909-z
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DOI: https://doi.org/10.1007/s00009-021-01909-z
Keywords
- Rota–Baxter operator
- Formal integration operator
- Polynomial algebra
- Additive action
- Infinite transitivity