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August  2021, 29(3): 2457-2473. doi: 10.3934/era.2020124

A generalization on derivations of Lie algebras

Hongliang Chang , Yin Chen and  Runxuan Zhang

School of Mathematics and Statistics, Northeast Normal University, Changchun, China

Received  April 2020 Revised  November 2020 Published  August 2021 Early access  December 2020

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.

Keywords: Generalized derivation, automorphism, G-derivation.
Mathematics Subject Classification: 17B40.
Citation: Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, 2021, 29 (3) : 2457-2473. doi: 10.3934/era.2020124
References:
[1]

K. I. Beida, M. Brešar and M. A. Chebotar, Generalized functional identities with (anti-) automorphisms and derivations on prime rings. Ⅰ., J. Algebra, 215 (1999), 644-665.  doi: 10.1006/jabr.1998.7751.  Google Scholar

[2]

H. E. Bell and L.-C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar., 53 (1989), 339-346.  doi: 10.1007/BF01953371.  Google Scholar

[3]

G. Benkart, A. I. Kostrikin and M. I. Kuznetsov, Finite-dimensional simple Lie algebras with a nonsingular derivation, J. Algebra, 171 (1995), 894-916.  doi: 10.1006/jabr.1995.1041.  Google Scholar

[4]

J. Bergen and P. Grzeszczuk, Invariants of skew derivations, Proc. Amer. Math. Soc., 125 (1997), 3481-3488.  doi: 10.1090/S0002-9939-97-04045-8.  Google Scholar

[5]

M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra, 156 (1993), 385-394.  doi: 10.1006/jabr.1993.1080.  Google Scholar

[6]

M. Brešar, Near-derivations in Lie algebras, J. Algebra, 320 (2008), 3765-3772.  doi: 10.1016/j.jalgebra.2008.09.007.  Google Scholar

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D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math., 4 (2006), 323-357.  doi: 10.2478/s11533-006-0014-9.  Google Scholar

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D. Burde and W. A. Moens, Periodic derivations and prederivations of Lie algebras, J. Algebra, 357 (2012), 208-221.  doi: 10.1016/j.jalgebra.2012.02.015.  Google Scholar

[9]

Y. Chen and R. Zhang, A commutative algebra approach to multiplicative Hom-Lie algebras, arXiv: 1907.02415. Google Scholar

[10]

C.-L. Chuang and T.-K. Lee, Identities with a single skew derivation, J. Algebra, 288 (2005), 59-77.  doi: 10.1016/j.jalgebra.2003.12.032.  Google Scholar

[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.  Google Scholar

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V. De Filippis and F. Wei, $b$-generalized $(\alpha, \beta)$-derivations and $b$-generalized $(\alpha, \beta)$-biderivations of prime rings, Taiwanese J. Math., 22 (2018), 313-323.  doi: 10.11650/tjm/170903.  Google Scholar

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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.  Google Scholar

[14]

R. García-Delgado, G. Salgado and O. A. Sánchez-Valenzuela, On 3-dimensional complex Hom-Lie algebras, J. Algebra, 555 (2020), 361-385.  doi: 10.1016/j.jalgebra.2020.03.005.  Google Scholar

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L. Guo and F. Li, Structure of Hochschild cohomology of path algebras and differential formulation of Euler's polyhedron formula, Asian J. Math., 18 (2014), 545-572.  doi: 10.4310/AJM.2014.v18.n3.a9.  Google Scholar

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J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations, J. Algebra, 295 (2006), 314-361.  doi: 10.1016/j.jalgebra.2005.07.036.  Google Scholar

[19]

B. Hvala, Generalized derivations in rings, Comm. Algebra, 26 (1998), 1147-1166.  doi: 10.1080/00927879808826190.  Google Scholar

[20]

N. Jacobson, A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc., 6 (1955), 281-283.  doi: 10.1090/S0002-9939-1955-0068532-9.  Google Scholar

[21]

V. K. Kharchenko and A. Z. Popov, Skew derivations of prime rings, Comm. Algebra, 20 (1992), 3321-3345.  doi: 10.1080/00927879208824517.  Google Scholar

[22]

E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54. Academic Press, New York-London, 1973.  Google Scholar

[23]

G. F. Leger and E. M. Luks, Generalized derivations of Lie algebras, J. Algebra, 228 (2000), 165-203.  doi: 10.1006/jabr.1999.8250.  Google Scholar

[24]

P. Novotný and J. Hrivnák, On $(\alpha, \beta, \gamma)$-derivations of Lie algebras and corresponding invariant functions, J. Geom. Phys., 58 (2008), 208-217.  doi: 10.1016/j.geomphys.2007.10.005.  Google Scholar

[25]

E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.  doi: 10.1090/S0002-9939-1957-0095863-0.  Google Scholar

[26]

J. F. Ritt, Differential Equations from the Algebraic Standpoint, American Mathematical Society Colloquium Publications, Vol. 14, American Mathematical Society, New York, 1932.  Google Scholar

[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.  Google Scholar

[28]

P. Zusmanovich, On $\delta$-derivations of Lie algebras and superalgebras, J. Algebra, 324 (2010), 3470-3486.  doi: 10.1016/j.jalgebra.2010.09.032.  Google Scholar

show all references

References:
[1]

K. I. Beida, M. Brešar and M. A. Chebotar, Generalized functional identities with (anti-) automorphisms and derivations on prime rings. Ⅰ., J. Algebra, 215 (1999), 644-665.  doi: 10.1006/jabr.1998.7751.  Google Scholar

[2]

H. E. Bell and L.-C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar., 53 (1989), 339-346.  doi: 10.1007/BF01953371.  Google Scholar

[3]

G. Benkart, A. I. Kostrikin and M. I. Kuznetsov, Finite-dimensional simple Lie algebras with a nonsingular derivation, J. Algebra, 171 (1995), 894-916.  doi: 10.1006/jabr.1995.1041.  Google Scholar

[4]

J. Bergen and P. Grzeszczuk, Invariants of skew derivations, Proc. Amer. Math. Soc., 125 (1997), 3481-3488.  doi: 10.1090/S0002-9939-97-04045-8.  Google Scholar

[5]

M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra, 156 (1993), 385-394.  doi: 10.1006/jabr.1993.1080.  Google Scholar

[6]

M. Brešar, Near-derivations in Lie algebras, J. Algebra, 320 (2008), 3765-3772.  doi: 10.1016/j.jalgebra.2008.09.007.  Google Scholar

[7]

D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math., 4 (2006), 323-357.  doi: 10.2478/s11533-006-0014-9.  Google Scholar

[8]

D. Burde and W. A. Moens, Periodic derivations and prederivations of Lie algebras, J. Algebra, 357 (2012), 208-221.  doi: 10.1016/j.jalgebra.2012.02.015.  Google Scholar

[9]

Y. Chen and R. Zhang, A commutative algebra approach to multiplicative Hom-Lie algebras, arXiv: 1907.02415. Google Scholar

[10]

C.-L. Chuang and T.-K. Lee, Identities with a single skew derivation, J. Algebra, 288 (2005), 59-77.  doi: 10.1016/j.jalgebra.2003.12.032.  Google Scholar

[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.  Google Scholar

[12]

V. De Filippis and F. Wei, $b$-generalized $(\alpha, \beta)$-derivations and $b$-generalized $(\alpha, \beta)$-biderivations of prime rings, Taiwanese J. Math., 22 (2018), 313-323.  doi: 10.11650/tjm/170903.  Google Scholar

[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.  Google Scholar

[14]

R. García-Delgado, G. Salgado and O. A. Sánchez-Valenzuela, On 3-dimensional complex Hom-Lie algebras, J. Algebra, 555 (2020), 361-385.  doi: 10.1016/j.jalgebra.2020.03.005.  Google Scholar

[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.  Google Scholar

[16]

L. Guo and W. Keigher, On differential Rota-Baxter algebras, J. Pure Appl. Algebra, 212 (2008), 522-540.  doi: 10.1016/j.jpaa.2007.06.008.  Google Scholar

[17]

L. Guo and F. Li, Structure of Hochschild cohomology of path algebras and differential formulation of Euler's polyhedron formula, Asian J. Math., 18 (2014), 545-572.  doi: 10.4310/AJM.2014.v18.n3.a9.  Google Scholar

[18]

J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations, J. Algebra, 295 (2006), 314-361.  doi: 10.1016/j.jalgebra.2005.07.036.  Google Scholar

[19]

B. Hvala, Generalized derivations in rings, Comm. Algebra, 26 (1998), 1147-1166.  doi: 10.1080/00927879808826190.  Google Scholar

[20]

N. Jacobson, A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc., 6 (1955), 281-283.  doi: 10.1090/S0002-9939-1955-0068532-9.  Google Scholar

[21]

V. K. Kharchenko and A. Z. Popov, Skew derivations of prime rings, Comm. Algebra, 20 (1992), 3321-3345.  doi: 10.1080/00927879208824517.  Google Scholar

[22]

E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54. Academic Press, New York-London, 1973.  Google Scholar

[23]

G. F. Leger and E. M. Luks, Generalized derivations of Lie algebras, J. Algebra, 228 (2000), 165-203.  doi: 10.1006/jabr.1999.8250.  Google Scholar

[24]

P. Novotný and J. Hrivnák, On $(\alpha, \beta, \gamma)$-derivations of Lie algebras and corresponding invariant functions, J. Geom. Phys., 58 (2008), 208-217.  doi: 10.1016/j.geomphys.2007.10.005.  Google Scholar

[25]

E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.  doi: 10.1090/S0002-9939-1957-0095863-0.  Google Scholar

[26]

J. F. Ritt, Differential Equations from the Algebraic Standpoint, American Mathematical Society Colloquium Publications, Vol. 14, American Mathematical Society, New York, 1932.  Google Scholar

[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.  Google Scholar

[28]

P. Zusmanovich, On $\delta$-derivations of Lie algebras and superalgebras, J. Algebra, 324 (2010), 3470-3486.  doi: 10.1016/j.jalgebra.2010.09.032.  Google Scholar

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