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Year 2023, , 247 - 269, 09.01.2023

Vsevolod Gubarev Roman Kozlov

https://doi.org/10.24330/ieja.1218727

Abstract

Keywords

Lie conformal algebra Rota---Baxter operator conformal classical Yang---Baxter equation

References

  • M. Aguiar, Pre-Poisson algebras, Lett. Math. Phys., 54(4) (2000), 263-277.
  • A. D'Andrea and V. G. Kac, Structure theory of finite conformal algebras, Selecta Math., 4(3) (1998), 377-418.
  • C. Bai, O. Bellier, L. Guo and X. Ni, Splitting of operations, Manin products, and Rota---Baxter operators, Int. Math. Res. Not., 3 (2013), 485-524.
  • B. Bakalov, A. D'Andrea and V. G. Kac, Theory of finite pseudoalgebras, Adv. Math., 162(1) (2001), 1-140.
  • G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 10 (1960), 731-742.
  • A. A. Belavin, V. G. Drinfel'd, Solutions of the classical Yang---Baxter equation for simple Lie algebras, Funktsional. Anal. i Prilozhen., 16(3) (1982), 1-29.
  • A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B, 241(2) (1984), 333-380.
  • P. Benito, V. Gubarev and A. Pozhidaev, Rota---Baxter operators on quadratic algebras, Mediterr. J. Math., 15(5) (2018), 189 (23 pp).
  • R. E. Borcherds, Vertex algebras, Kac–Moody algebras, and the monster, Proc. Nat. Acad. Sci. U.S.A., 83(10) (1986), 3068-3071.
  • C. Boyallian and J. I. Liberati, On pseudo-bialgebras, J. Algebra, 372 (2012), 1-34.
  • D. Burde and V. Gubarev, Rota---Baxter operators and post-Lie algebra structures on semisimple Lie algebras, Comm. Algebra, 47(5) (2019), 2280-2296.
  • E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs, 88. American Mathematical Society, Providence, RI, 2001.
  • M. Goncharov and V. Gubarev, Double Lie algebras of nonzero weight, Adv. Math., 409 (2022), 108680 (30 pp).
  • M. E. Goncharov and P. S. Kolesnikov, Simple finite-dimensional double algebras, J. Algebra, 500 (2018), 425-438.
  • V. Gubarev and P. Kolesnikov, Embedding of dendriform algebras into Rota---Baxter algebras, Cent. Eur. J. Math., 11(2) (2013), 226-245.
  • V. Gubarev and R. Kozlov, Conformal Yang---Baxter equation on $\Cur(\sl_2(\mathbb{C}))$, arXiv:2209.12431, (20~pp).
  • L. Guo, An Introduction to Rota---Baxter Algebra, Surveys of Modern Mathematics, 4. International Press, Somerville, MA; Higher Education Press, Beijing, 2012.
  • Y. Hong and C. Bai, Conformal classical Yang-Baxter equation, $S$-equation and $\mathcal{O}$-operators, Lett. Math. Phys., 110(5) (2019), 885-909.
  • V. G. Kac, Vertex Algebras for Beginners, University Lecture Series, 10. American Mathematical Society, Providence, RI, 1997.
  • P. Kolesnikov, Homogeneous averaging operators on simple finite conformal Lie algebras, J. Math. Phys., 56(7) (2015), 071702 (10 pp).
  • E. I. Konovalova, Double Lie Algebras, PhD thesis, Samara State University, 2009, 189 pp. (in Russian).
  • J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Mathematics, 227. Birkhäuser Boston, Inc., Boston, MA, 2004.
  • J. I. Liberati, conformal bialgebras, J. Algebra, 319(6) (2008), 2295-2318.
  • L. Liu and S. Wang, Rota---Baxter $H$-operators and pre-Lie $H$-pseudoalgebras over a~cocommutative Hopf algebra $H$, Linear Multilinear Algebra, 68(11) (2020), 2170-2184.
  • J. Liu, S. Zhou and L. Yuan, Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O}N$-structures, J. Math. Phys., 63(10) (2022), 101701 (22 pp).
  • Y. Pan, Q. Liu, C. Bai and L. Guo, PostLie algebra structures on the Lie algebra $\mathrm{sl}(2,\mathbb{C})$, Electron. J. Linear Algebra, 23 (2012), 180-197.
  • L. Yuan, $\mathcal{O}$-operators and Nijenhius operators of associative conformal algebras, J. Algebra, 609 (2022), 245-291.
  • J. Zhao, L. Chen and B. Sun, Representations and cohomology of Rota---Baxter Lie conformal algebras, preprint (researchgate), 2021, 18 pp.

Rota---Baxter operators on $Cur(sl_2(\mathbb{C}))$

Year 2023, , 247 - 269, 09.01.2023

Vsevolod Gubarev Roman Kozlov

https://doi.org/10.24330/ieja.1218727

Abstract

We classify all Rota---Baxter operators on the simple Lie conformal algebra $\Cur(\sl_2(\mathbb{C}))$ and clarify which of them arise from the solutions to the conformal classical Yang---Baxter equation due to the connection discovered by Y. Hong and C. Bai in 2020.

Keywords

Lie conformal algebra Rota---Baxter operator conformal classical Yang---Baxter equation

References

  • M. Aguiar, Pre-Poisson algebras, Lett. Math. Phys., 54(4) (2000), 263-277.
  • A. D'Andrea and V. G. Kac, Structure theory of finite conformal algebras, Selecta Math., 4(3) (1998), 377-418.
  • C. Bai, O. Bellier, L. Guo and X. Ni, Splitting of operations, Manin products, and Rota---Baxter operators, Int. Math. Res. Not., 3 (2013), 485-524.
  • B. Bakalov, A. D'Andrea and V. G. Kac, Theory of finite pseudoalgebras, Adv. Math., 162(1) (2001), 1-140.
  • G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 10 (1960), 731-742.
  • A. A. Belavin, V. G. Drinfel'd, Solutions of the classical Yang---Baxter equation for simple Lie algebras, Funktsional. Anal. i Prilozhen., 16(3) (1982), 1-29.
  • A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B, 241(2) (1984), 333-380.
  • P. Benito, V. Gubarev and A. Pozhidaev, Rota---Baxter operators on quadratic algebras, Mediterr. J. Math., 15(5) (2018), 189 (23 pp).
  • R. E. Borcherds, Vertex algebras, Kac–Moody algebras, and the monster, Proc. Nat. Acad. Sci. U.S.A., 83(10) (1986), 3068-3071.
  • C. Boyallian and J. I. Liberati, On pseudo-bialgebras, J. Algebra, 372 (2012), 1-34.
  • D. Burde and V. Gubarev, Rota---Baxter operators and post-Lie algebra structures on semisimple Lie algebras, Comm. Algebra, 47(5) (2019), 2280-2296.
  • E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs, 88. American Mathematical Society, Providence, RI, 2001.
  • M. Goncharov and V. Gubarev, Double Lie algebras of nonzero weight, Adv. Math., 409 (2022), 108680 (30 pp).
  • M. E. Goncharov and P. S. Kolesnikov, Simple finite-dimensional double algebras, J. Algebra, 500 (2018), 425-438.
  • V. Gubarev and P. Kolesnikov, Embedding of dendriform algebras into Rota---Baxter algebras, Cent. Eur. J. Math., 11(2) (2013), 226-245.
  • V. Gubarev and R. Kozlov, Conformal Yang---Baxter equation on $\Cur(\sl_2(\mathbb{C}))$, arXiv:2209.12431, (20~pp).
  • L. Guo, An Introduction to Rota---Baxter Algebra, Surveys of Modern Mathematics, 4. International Press, Somerville, MA; Higher Education Press, Beijing, 2012.
  • Y. Hong and C. Bai, Conformal classical Yang-Baxter equation, $S$-equation and $\mathcal{O}$-operators, Lett. Math. Phys., 110(5) (2019), 885-909.
  • V. G. Kac, Vertex Algebras for Beginners, University Lecture Series, 10. American Mathematical Society, Providence, RI, 1997.
  • P. Kolesnikov, Homogeneous averaging operators on simple finite conformal Lie algebras, J. Math. Phys., 56(7) (2015), 071702 (10 pp).
  • E. I. Konovalova, Double Lie Algebras, PhD thesis, Samara State University, 2009, 189 pp. (in Russian).
  • J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Mathematics, 227. Birkhäuser Boston, Inc., Boston, MA, 2004.
  • J. I. Liberati, conformal bialgebras, J. Algebra, 319(6) (2008), 2295-2318.
  • L. Liu and S. Wang, Rota---Baxter $H$-operators and pre-Lie $H$-pseudoalgebras over a~cocommutative Hopf algebra $H$, Linear Multilinear Algebra, 68(11) (2020), 2170-2184.
  • J. Liu, S. Zhou and L. Yuan, Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O}N$-structures, J. Math. Phys., 63(10) (2022), 101701 (22 pp).
  • Y. Pan, Q. Liu, C. Bai and L. Guo, PostLie algebra structures on the Lie algebra $\mathrm{sl}(2,\mathbb{C})$, Electron. J. Linear Algebra, 23 (2012), 180-197.
  • L. Yuan, $\mathcal{O}$-operators and Nijenhius operators of associative conformal algebras, J. Algebra, 609 (2022), 245-291.
  • J. Zhao, L. Chen and B. Sun, Representations and cohomology of Rota---Baxter Lie conformal algebras, preprint (researchgate), 2021, 18 pp.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Vsevolod Gubarev This is me

Roman Kozlov This is me

Publication Date January 9, 2023
Published in Issue Year 2023

Cite

APA Gubarev, V., & Kozlov, R. (2023). Rota---Baxter operators on $Cur(sl_2(\mathbb{C}))$. International Electronic Journal of Algebra, 33(33), 247-269. https://doi.org/10.24330/ieja.1218727
AMA Gubarev V, Kozlov R. Rota---Baxter operators on $Cur(sl_2(\mathbb{C}))$. IEJA. January 2023;33(33):247-269. doi:10.24330/ieja.1218727
Chicago Gubarev, Vsevolod, and Roman Kozlov. “Rota---Baxter Operators on $Cur(sl_2(\mathbb{C}))$”. International Electronic Journal of Algebra 33, no. 33 (January 2023): 247-69. https://doi.org/10.24330/ieja.1218727.
EndNote Gubarev V, Kozlov R (January 1, 2023) Rota---Baxter operators on $Cur(sl_2(\mathbb{C}) $. International Electronic Journal of Algebra 33 33 247–269.
IEEE V. Gubarev and R. Kozlov, “Rota---Baxter operators on $Cur(sl_2(\mathbb{C}))$”, IEJA, vol. 33, no. 33, pp. 247–269, 2023, doi: 10.24330/ieja.1218727.
ISNAD Gubarev, Vsevolod - Kozlov, Roman. “Rota---Baxter Operators on $Cur(sl_2(\mathbb{C}))$”. International Electronic Journal of Algebra 33/33 (January 2023), 247-269. https://doi.org/10.24330/ieja.1218727.
JAMA Gubarev V, Kozlov R. Rota---Baxter operators on $Cur(sl_2(\mathbb{C}))$. IEJA. 2023;33:247–269.
MLA Gubarev, Vsevolod and Roman Kozlov. “Rota---Baxter Operators on $Cur(sl_2(\mathbb{C}))$”. International Electronic Journal of Algebra, vol. 33, no. 33, 2023, pp. 247-69, doi:10.24330/ieja.1218727.
Vancouver Gubarev V, Kozlov R. Rota---Baxter operators on $Cur(sl_2(\mathbb{C}))$. IEJA. 2023;33(33):247-69.