Operator identities on Lie algebras, rewriting systems and Gröbner-Shirshov bases. (English) Zbl 1517.17020
Summary: Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algebras, later called Rota’s Program on Algebraic Operators. Recent progresses on this program have been achieved in the contexts of operated algebra, rewriting systems and Gröbner-Shirshov bases. These developments also suggest that Rota’s insight can be applied to determine operator identities on Lie algebras, and thus to put the various linear operators on Lie algebras in a uniform perspective. This paper carries out this approach, utilizing operated polynomial Lie algebras spanned by nonassociative Lyndon-Shirshov bracketed words. The Lie algebra analog of Rota’s program is formulated in terms convergent rewriting systems and equivalently in terms of Gröbner-Shirshov bases. The relation of this Lie algebra analog of Rota’s program with Rota’s program for associative algebras is established. Applications are given to modified Rota-Baxter operators, differential type operators and Rota-Baxter type operators.
MSC:
| 17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 16Z10 | Gröbner-Shirshov bases |
| 17A61 | Gröbner-Shirshov bases in nonassociative algebras |
| 16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |
| 05A05 | Permutations, words, matrices |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
Keywords:
operated algebra; operator identity; operated Lie algebra; Rota’s program on algebraic operators; rewriting system; Gröbner basis; Gröbner-Shirshov basis; differential operator; Rota-Baxter operatorSoftware:
TenResReferences:
| [1] | Baader, F.; Nipkow, T., Term Rewriting and All That (1998), Cambridge University Press |
| [2] | Bai, C.; Guo, L.; Ni, X., Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, Commun. Math. Phys., 297, 553-596 (2010) · Zbl 1206.17020 |
| [3] | Baxter, G., An analytic problem whose solution follows from a simple algebraic identity, Pac. J. Math., 10, 731-742 (1960) · Zbl 0095.12705 |
| [4] | Bokut, L. A.; Chen, Y., Gröbner-Shirshov bases: after A.I. Shirshov, Southeast Asian Bull. Math., 31, 1057-1076 (2007) · Zbl 1150.17008 |
| [5] | Bokut, L. A.; Chen, Y.; Qiu, J., Gröbner-Shirshov bases for associative algebras with multiple operations and free Rota-Baxter algebras, J. Pure Appl. Algebra, 214, 89-100 (2010) · Zbl 1213.16014 |
| [6] | Bordemann, M., Generalized Lax pairs, the modified classical Yang-Baxter equation, and affine geometry of Lie groups, Commun. Math. Phys., 201-216 (1990) · Zbl 0714.58025 |
| [7] | Bremner, M. R.; Elgendy, H. A., A new classification of algebraic identities for linear operators on associative algebras, J. Algebra, 596, 177-199 (2022) · Zbl 1494.13034 |
| [8] | Carinẽna, J.; Grabowski, J.; Marmo, G., Quantum bi-Hamiltonian systems, Int. J. Mod. Phys. A, 15, 4797-4810 (2000) · Zbl 1002.81026 |
| [9] | Connes, A.; Kreimer, D., Hopf algebras, renormalization, and noncommutative geometry, Commun. Math. Phys., 199, 203-242 (1998) · Zbl 0932.16038 |
| [10] | Cotlar, M., A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana, 1, Article 05 pp. (1955) |
| [11] | Das, A., Deformations of associative Rota-Baxter operators, J. Algebra, 560, 144-180 (2020) · Zbl 1458.16008 |
| [12] | Frölicher, A.; Nijenhuis, A., Theory of vector valued differential forms. Part I, Indag. Math., 18, 338-360 (1956) · Zbl 0079.37502 |
| [13] | Gao, X.; Guo, L., Rota’s Classification Problem, rewriting systems and Gröbner-Shirshov bases, J. Algebra, 470, 219-253 (2017) · Zbl 1423.16047 |
| [14] | Gao, X.; Guo, L.; Zhang, H., Rota’s program on algebraic operators, rewriting systems and Gröbner-Shirshov bases, Adv. in Math. (China), 51, 1-31 (2022) · Zbl 1524.16086 |
| [15] | Gao, X.; Guo, L.; Zhang, Y., Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras, J. Algebra, 552, 134-170 (2020) · Zbl 1444.16058 |
| [16] | Guo, L., Operated semigroups, Motzkin paths and rooted trees, J. Algebraic Comb., 29, 35-62 (2009) · Zbl 1227.05271 |
| [17] | Gubarev, V., Universal enveloping associative Rota-Baxter algebras of preassociative and postassociative algebra, J. Algebra, 516, 298-328 (2018) · Zbl 1433.16027 |
| [18] | Gubarev, V.; Kolesnikov, P., Gröbner-Shirshov basis of the universal enveloping Rota-Baxter algebra of a Lie algebra, J. Lie Theory, 27, 887-905 (2017) · Zbl 1430.17013 |
| [19] | Guo, L., An Introduction to Rota-Baxter Algebra (2012), International Press and Higher Education Press · Zbl 1271.16001 |
| [20] | Guo, L.; Gustavson, R.; Li, Y., An algebraic study of Volterra integral equations and their operator linearity, J. Algebra, 595, 398-433 (2022) · Zbl 1489.16046 |
| [21] | Guo, L.; Lang, H.; Sheng, Y., Integration and geometrization of Rota-Baxter Lie algebras, Adv. Math., 387, Article 107834 pp. (2021) · Zbl 1468.17026 |
| [22] | Guo, L.; Sit, W.; Zhang, R., Differential type operators and Gröbner-Shirshov bases, J. Symb. Comput., 52, 97-123 (2013) · Zbl 1290.16021 |
| [23] | Hossein Poor, J.; Raab, C. G.; Regensburger, G., Algorithmic operator algebras via normal forms in tensor rings, J. Symb. Comput., 85, 247-274 (2018) · Zbl 06789139 |
| [24] | Kampéde Fériet, J., L’etat actuel du probléme de la turbulaence (I and II), Sci. Aérienne. Sci. Aérienne, Sci. Aérienne, 4, 12-52 (1935) |
| [25] | Karimjanov, I.; Kaygorodov, I.; Ladra, M., Rota-type operators on null-filiform associative algebras, Linear Multilinear Algebra, 68, 205-219 (2020) · Zbl 1464.16039 |
| [26] | Kosmann-Schwarzbach, Y.; Magri, F., Poisson-Nijenhuis structures, Ann. Inst. Henri Poincaré, 53, 35-81 (1990) · Zbl 0707.58048 |
| [27] | Kupershmidt, B. A., What a classical r-matrix really is, J. Nonlinear Math. Phys., 6, 448-488 (1999) · Zbl 1015.17015 |
| [28] | Kurosh, A. G., Free sums of multiple operator algebras, Sib. Math. J., 1, 62-70 (1960) · Zbl 0096.25304 |
| [29] | Lazarev, A.; Sheng, Y.; Tang, R., Deformations and homotopy theory of relative Rota-Baxter Lie algebras, Commun. Math. Phys., 383, 595-631 (2021) · Zbl 1476.17010 |
| [30] | Malkhouf, A.; Yau, D., Rota-Baxter Hom-Lie-admissible algebras, Commun. Algebra, 42, 1231-1257 (2014) · Zbl 1367.17003 |
| [31] | Nijenhuis, A., \( X_{n - 1}\)-forming sets of eigenvectors, Indag. Math., 13, 200-212 (1951) · Zbl 0042.16001 |
| [32] | Qiu, J.; Chen, Y., Gröbner-Shirshov bases for Lie Ω-algebras and free Rota-Baxter Lie algebras, J. Algebra Appl., 16, Article 1750190 pp. (2017) · Zbl 1384.16016 |
| [33] | Qi, Z.; Qin, Y.; Wang, K.; Zhou, G., Free objects and Gröbner-Shirshov bases in operated contexts, J. Algebra, 584, 89-124 (2021) · Zbl 1467.16025 |
| [34] | Reutenauer, C., Free Lie Algebras (1993), Oxford University Press: Oxford University Press New York · Zbl 0798.17001 |
| [35] | Reyman, A. G.; Semenov-Tian-Shansky, M. A., Reduction of Hamilton systems, affine Lie algebras and Lax equations, Invent. Math., 54, 81-100 (1979) · Zbl 0403.58004 |
| [36] | Reyman, A. G.; Semenov-Tian-Shansky, M. A., Reduction of Hamilton systems, affine Lie algebras and Lax equations II, Invent. Math., 63, 423-432 (1981) · Zbl 0442.58016 |
| [37] | Reynolds, O., On the dynamic theory of incompressible viscous fluids, Philos. Trans. R. Soc. A, 136, 123-164 (1895) · JFM 26.0872.02 |
| [38] | Rota, G.-C., Baxter algebras and combinatorial identities I, II, Bull. Am. Math. Soc., 75, 325-329 (1969), 330-334 · Zbl 0192.33801 |
| [39] | Rota, G.-C., Baxter operators, an introduction, (Kung, Joseph P. S., Gian-Carlo Rota on Combinatorics, Introductory Papers and Commentaries (1995), Birkhäuser: Birkhäuser Boston), 504-512 · Zbl 0841.01031 |
| [40] | Semonov-Tian-Shansky, M., What is a classical R-matrix?, Funct. Anal. Appl., 17, 259-272 (1983) · Zbl 0535.58031 |
| [41] | Shirshov, A. I., Some algorithmic problem for Lie algebras, Sib. Mat. Zh.. Sib. Mat. Zh., SIGSAM Bull., 33, 2, 3-6 (1999), (in Russian), English translation in · Zbl 1097.17502 |
| [42] | Tang, R.; Bai, C.; Guo, L.; Sheng, Y., Deformations and their controlling cohomologies of \(\mathcal{O} \)-operators, Commun. Math. Phys., 368, 665-700 (2019) · Zbl 1440.17015 |
| [43] | Tricomi, F. G., On the finite Hilbert transformation, Q. J. Math. Oxf., 2, 199-211 (1951) · Zbl 0043.10701 |
| [44] | Uchino, K., Twisting on associative algebras and Rota-Baxter type operators, J. Noncommut. Geom., 4, 349-379 (2010) · Zbl 1248.16027 |
| [45] | Wang, J.; Zhu, Z.; Gao, X., New operated polynomial identities and Gröbner-Shirshov bases |
| [46] | Zhang, T.; Gao, X.; Guo, L., Reynolds algebras and their free objects from bracketed words and rooted trees, J. Pure Appl. Algebra, 225, Article 106766 pp. (2021) · Zbl 1477.16057 |
| [47] | Zhang, X.; Gao, X.; Guo, L., Free modified Rota-Baxter algebra and Hopf algebra, Int. Electron. J. Algebra, 25, 12-34 (2019) · Zbl 1406.16034 |
| [48] | Zhang, X.; Gao, X.; Guo, L., Modified Rota-Baxter algebras, shuffle products and Hopf algebras, Bull. Malays. Math. Sci. Soc., 42, 3047-3072 (2019) · Zbl 1466.16039 |
| [49] | Zhang, Y.; Gao, X., Free Rota-Baxter family algebras and (tri)dendriform family algebras, Pac. J. Math., 301, 741-766 (2019) · Zbl 1521.17034 |
| [50] | Zhang, Y.; Gao, X.; Manchon, D., Free (tri)dendriform family algebras, J. Algebra, 547, 456-493 (2020) · Zbl 1435.16012 |
| [51] | Zheng, S.; Gao, X.; Guo, L.; Sit, W., Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases, J. Symb. Comput., accepted |
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