We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.
Citation: Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras[J]. Electronic Research Archive, 2021, 29(3): 2457-2473. doi: 10.3934/era.2020124
We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.
[1] | Generalized functional identities with (anti-) automorphisms and derivations on prime rings. Ⅰ.. J. Algebra (1999) 215: 644-665. |
[2] | Rings in which derivations satisfy certain algebraic conditions. Acta Math. Hungar. (1989) 53: 339-346. |
[3] | Finite-dimensional simple Lie algebras with a nonsingular derivation. J. Algebra (1995) 171: 894-916. |
[4] | Invariants of skew derivations. Proc. Amer. Math. Soc. (1997) 125: 3481-3488. |
[5] | Centralizing mappings and derivations in prime rings. J. Algebra (1993) 156: 385-394. |
[6] | Near-derivations in Lie algebras. J. Algebra (2008) 320: 3765-3772. |
[7] | Left-symmetric algebras, or pre-Lie algebras in geometry and physics. Cent. Eur. J. Math. (2006) 4: 323-357. |
[8] | Periodic derivations and prederivations of Lie algebras. J. Algebra (2012) 357: 208-221. |
[9] | Y. Chen and R. Zhang, A commutative algebra approach to multiplicative Hom-Lie algebras, arXiv: 1907.02415. |
[10] | Identities with a single skew derivation. J. Algebra (2005) 288: 59-77. |
[11] | D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Third edition., Undergraduate Texts in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-35651-8 |
[12] | $b$-generalized $(\alpha, \beta)$-derivations and $b$-generalized $(\alpha, \beta)$-biderivations of prime rings. Taiwanese J. Math. (2018) 22: 313-323. |
[13] | K. Erdmann and M. J. Wildon, Introduction to Lie Algebras, Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London, 2006. doi: 10.1007/1-84628-490-2 |
[14] | On 3-dimensional complex Hom-Lie algebras. J. Algebra (2020) 555: 361-385. |
[15] | L. Guo, P. J. Cassidy, W. F. Keigher and W. Y. Sit, Differential Algebra and Related Topics, Proceedings of the International Workshop held at Rutgers University, Newark, NJ. World Scientific Publishing Co., Inc., River Edge, NJ, 2002. doi: 10.1142/4768 |
[16] | On differential Rota-Baxter algebras. J. Pure Appl. Algebra (2008) 212: 522-540. |
[17] | Structure of Hochschild cohomology of path algebras and differential formulation of Euler's polyhedron formula. Asian J. Math. (2014) 18: 545-572. |
[18] | Deformations of Lie algebras using $\sigma$-derivations. J. Algebra (2006) 295: 314-361. |
[19] | Generalized derivations in rings. Comm. Algebra (1998) 26: 1147-1166. |
[20] | A note on automorphisms and derivations of Lie algebras. Proc. Amer. Math. Soc. (1955) 6: 281-283. |
[21] | Skew derivations of prime rings. Comm. Algebra (1992) 20: 3321-3345. |
[22] | E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54. Academic Press, New York-London, 1973. |
[23] | Generalized derivations of Lie algebras. J. Algebra (2000) 228: 165-203. |
[24] | On $(\alpha, \beta, \gamma)$-derivations of Lie algebras and corresponding invariant functions. J. Geom. Phys. (2008) 58: 208-217. |
[25] | Derivations in prime rings. Proc. Amer. Math. Soc. (1957) 8: 1093-1100. |
[26] | J. F. Ritt, Differential Equations from the Algebraic Standpoint, American Mathematical Society Colloquium Publications, Vol. 14, American Mathematical Society, New York, 1932. |
[27] | M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-55750-7 |
[28] | On $\delta$-derivations of Lie algebras and superalgebras. J. Algebra (2010) 324: 3470-3486. |