Abstract
The present paper is devoted to study local derivations on the conformal Galilei algebra. We prove that every local derivation on the conformal Galilei algebra is a derivation.
1 Introduction
Let A be an algebra (not necessary associative). Recall that a linear mapping is said to be a derivation, if for all A linear mapping Δ is said to be a local derivation, if for every there exists a derivation Dx on A (depending on x) such that
This notion was introduced and investigated independently by R.V. Kadison [Citation17] and D.R. Larson and A.R. Sourour [Citation18]. The above papers gave rise to a series of works devoted to the description of mappings which are close to automorphisms and derivations of -algebras and operator algebras. R.V. Kadison set out a program of study for local maps in [Citation17], suggesting that local derivations could prove useful in building derivations with particular properties. R.V.Kadison proved in [Citation17, Theorem A] that each continuous local derivation of a von Neumann algebra M into a dual Banach M-bimodule is a derivation. This theorem gave way to studies on derivations on -algebras, culminating with a result due to B.E. Johnson, which asserts that every local derivation of a -algebra A into a Banach A-bimodule is automatically continuous, and hence is a derivation [Citation15, Theorem 5.3].
Let us present a list of finite or infinite dimensional algebras for which all local derivations are derivations:
-algebras, in particular, the algebra of all square matrices of order n over the field of complex numbers [Citation15, Citation17];
the complex polynomial algebra [Citation17];
finite dimensional simple Lie algebras over an algebraically closed field of characteristic zero [Citation7];
Borel subalgebras of finite-dimensional simple Lie algebras [Citation22];
infinite dimensional Witt algebras over an algebraically closed field of characteristic zero [Citation12];
Witt algebras over a field of prime characteristic [Citation23];
solvable Lie algebras of maximal rank [Citation20];
Cayley algebras [Citation5];
finite dimensional semi-simple Leibniz algebras over an algebraically closed field of characteristic zero [Citation21];
locally finite split simple Lie algebras over a field of characteristic zero [Citation10];
finite-dimensional semisimple Filippov n-ary (n > 1) algebras [Citation14];
the Schrödinger algebras [Citation2].
On the other hand, some algebras (in most cases close to nilpotent algebras) admit pure local derivations, that is, local derivations which are not derivations. Below a short list of some classes of algebras which admit pure local derivations:
the algebra of rational functions [Citation17];
finite dimensional filiform Lie algebras [Citation7];
p-filiform Leibniz algebras [Citation9];
direct sum null-filiform Leibniz algebras [Citation1];
solvable Leibniz algebras with abelian nilradicals, which have a one dimensional complementary space [Citation6];
the algebra of lower triangular n × n-matrices [Citation13];
real octonion algebra [Citation8];
the ternary Malcev algebra M8 [Citation14].
In this paper, we will study local derivations on the conformal Galilei algebra. We prove that every local derivation on the conformal Galilei algebra is a derivation.
2 Preliminaries
In this paper, we denoted by , and the sets of all integers, positive integers, and complex numbers, respectively. In this section we give some necessary notations, definitions and preliminary results.
A derivation on a Lie algebra is a linear map satisfying for all . Denote by the set of all derivations of g. For all , the map ad(a) on defined as is a derivation and derivations of this form are called inner derivation and denote by . Derivations which are not inner are called outer derivations.
A linear operator Δ is called a local derivation if for any there exists a derivation (depending on x) such that we denote by the set of all local derivations on
In recent years Galilei groups and their Lie algebras have been intensively studied. The interest is due to an appearance of this kind of symmetries in very different areas of physics and mathematics [Citation3, Citation16]. The conformal extension of the Galilei algebra is parameterized by a positive half-integer number l and is called the l-conformal Galilei algebra. For , we denote the conformal Galilei algebra by , which has a basis given by and the Lie bracket given by: where δ is the Kronecker Delta.
The conformal Galilei algebra can be viewed as a semidirect product of two subalgebras. and Heisenberg subalgebra . The irreducible representations of the conformal Galilei algebra are studied in [Citation4, Citation11, Citation19, Citation25].
The following lemma can be used to determine the derivation of .
Let τ be an outer derivation of determined by
Lemma 2.1.
[Citation24] .
3 Local derivations of conformal Galilei algebra
In this section we investigate local derivations on conformal Galilei algebra.
The following theorem is the main result of this section.
Theorem 3.1.
Every local derivation on l-conformal Galilei algebra is a derivation, where .
For any local derivation Δ on by Lemma 2.1 we have
Put
Then we obtain (3.1) (3.1) where and
For the proof of this Theorem we need several Lemmata.
There exists an element such that
In the following lemmas, we assume that Δ be a local derivation on such that
Lemma 3.2.
Proof.
We consider, which implies where Then □
Lemma 3.3.
(3.2) (3.2)
Proof.
We consider
On the other hand,
Comparing the coefficients at the basis elements f and h, we get
Similarly, considering
On the other hand,
Comparing the coefficients at the basis elements e and h, we get □
Setting
Lemma 3.4.
(3.3) (3.3) where
Proof.
We consider
On the other hand,
Comparing the coefficients at the basis elements and z, we have
Now consider
On the other hand,
Comparing the coefficients at the basis elements and z, we get
Similarly, considering
On the other hand,
Comparing the coefficients at the basis elements p0 and z, we obtain □
Lemma 3.5.
Proof.
We consider
On the other hand,
Comparing the coefficients at the basis elements and z, we get
Now consider,
On the other hand,
Comparing the coefficients at the basis elements and z, we have
Similarly, considering
On the other hand,
Comparing the coefficients at the basis elements and z, we get
We considering,
On the other hand,
Comparing the coefficients at the basis elements p1 and z, we have □
Lemma 3.6.
Let Then where
Proof.
We consider
On the other hand,
Comparing the coefficients at the basis elements and h, we get
We consider
On the other hand,
Comparing the coefficients at the basis elements and we have
We consider
On the other hand,
Comparing the coefficients at the basis elements and for we have and for we get only
We consider
On the other hand,
Comparing the coefficients at the basis elements f and we have
We consider
On the other hand,
Comparing the coefficients at the basis elements h and we get
We consider
On the other hand,
Comparing the coefficients at the basis elements p0 and we have
We consider
On the other hand,
Comparing the coefficients at the basis elements p0 and for which implies
We consider
On the other hand,
Comparing the coefficients at the basis elements p0 and we have
We consider
On the other hand,
Comparing the coefficients at the basis elements and for we have and for we get only
Therefore, for case take an element
On the other hand,
Comparing the coefficients at the basis elements and we have
Therefore, for case take an element
On the other hand,
Comparing the coefficients at the basis elements , and we have □
Lemma 3.7.
Proof.
We considering, take an element (3.4) (3.4)
On the other hand, (3.5) (3.5)
Comparing the coefficients at the basis elements and z, (3.4) and (3.5) we obtain that (3.6) (3.6)
Similarly, considering take an element (3.7) (3.7)
On the other hand, (3.8) (3.8)
Comparing the coefficients at the basis elements and z, by EquationEquations (3.7)(3.7) (3.7) and Equation(3.8)(3.8) (3.8) we obtain that (3.9) (3.9)
Similarly, considering take an element (3.10) (3.10)
On the other hand, (3.11) (3.11)
Comparing the coefficients at the basis elements and z, (3.10) and (3.11) we obtain that (3.12) (3.12)
Comparing (3.6), (3.9), and (3.12) we obtain that
□
Lemma 3.8.
Proof.
We consider
On the other hand,
Comparing the coefficients at the basis elements and z, we get
Similarly, considering
On the other hand,
Comparing the coefficients at the basis elements and z, we get □
Lemma 3.9.
Let Then
Proof.
We consider take an element (3.13) (3.13)
On the other hand, (3.14) (3.14)
Comparing the coefficients at the basis elements , and z, we get which implies (3.15) (3.15)
We consider take an element
On the other hand,
Comparing the coefficients at the basis elements , and z, we get (3.16) (3.16) which implies (3.17) (3.17) and i = m we obtain that
Comparing (3.15) and (3.17) we obtain that (3.18) (3.18) on recurrent equation instead of i, gradually substituting we obtain the following (3.19) (3.19)
Comparing (3.15) and (3.19) we obtain that
□
Lemma 3.10.
Let Then
Proof.
We consider
On the other hand,
Comparing the coefficients at the basis elements e and f, we get
We consider take an element
On the other hand,
Comparing the coefficients at the basis elements , and z, we get (3.20) (3.20) which implies
We consider take an element
On the other hand,
Comparing the coefficients at the basis elements and z, we get (3.21) (3.21) which implies □
Lemma 3.11.
Δ is an inner derivation on with
Proof.
For any We consider
Then Δ is an inner derivation. □
Now we are in position to prove Theorem 3.1.
Proof of Theorem 3.1.
Let Δ be a local derivation of Take a derivation Dx such that
Set Then is a local derivation such that By Lemma 3.11, Thus Then Δ is a derivation. The proof is complete. □
Acknowledgments
We thank professor K. K. Kudaybergenov for the helpful comments and suggestions that contributed to improving this paper.
Data availability statement
No data was used for the research described in the article.
References
- Adashev, J. Q., Yusupov, B. B. (2022). Local derivations and automorphisms of direct sum null-filiform Leibniz algebras. Lobachevskii J. Math. 43(12):1–7.
- Alauadinov, A. K., Yusupov, B. B. (2023). Local derivations of the Schrödinger algebras. Algebra Colloq.
- Andrzejewski, K., Gonera, J., Maslanka, P. (2012). Nonrelativistic conformal groups and their dynamical realizations. Phys. Rev. D 86:065009. DOI: 10.1103/PhysRevD.86.065009.
- Aizawa, N., Isaac, P. S. (2011). On irreducible representations of the exotic conformal Galilei algebra. J. Phys. A 44:035401. DOI: 10.1088/1751-8113/44/3/035401.
- Ayupov, Sh. A., Elduque, A., Kudaybergenov, K. K. (2022). Local derivations and automorphisms of Cayley algebras. J. Pure Appl. Algebra. 227(5):107277. DOI: 10.1016/j.jpaa.2022.107277.
- Ayupov, Sh. A., Khudoyberdiyev, A. Kh., Yusupov, B. B. (2020). Local and 2-local derivations of solvable Leibniz algebras. Int. J. Algebra Comput. 30(6):1185–1197. DOI: 10.1142/S021819672050037X.
- Ayupov, Sh. A., Kudaybergenov, K. K. (2016). Local derivations on finite-dimensional Lie algebras. Linear Algebra Appl. 493:381–398. DOI: 10.1016/j.laa.2015.11.034.
- Ayupov, Sh. A., Kudaybergenov, K. K., Allambergenov, A. (2023). Local and 2-local derivations on octonion algebras. J. Algebra Appl. 22(7):2350147. DOI: 10.1142/S0219498823501475.
- Ayupov, Sh. A., Kudaybergenov, K. K., Yusupov, B. B. (2020). Local and 2-local derivations of p-filiform Leibniz algebras. J. Math. Sci. 245(3):359–367.
- Ayupov, Sh. A., Kudaybergenov, K. K., Yusupov, B. B. (2022). Local and 2-local derivations of locally simple Lie algebras. Contemp. Math. Fundam. Dir. 68(1):59–69. DOI: 10.22363/2413-3639-2022-68-1-59-69.
- Cai, Y., Cheng, Y., Shen, R. (2014). Quasi-Whittaker modules for Schrodinger algebra. Linear Algebra Appl. 463: 16–32. DOI: 10.1016/j.laa.2014.09.001.
- Chen, Y., Zhao, K., Zhao, Y. (2022). Local derivations on Witt algebras. Linear Multilinear Algebra 70(6):1159–1172. DOI: 10.1080/03081087.2020.1754750.
- Elisova, A. P., Zotov, I. N., Levchuk, V. M., Suleymanova, G. S. (2011). Local automorphisms and local derivations of nilpotent matrix algebras. Izv. Irkutsk Gos. Univ. 4(1):9–19.
- Ferreira, B., Kaygorodov, I., Kudaybergenov, K. K. (2021). Local and 2-local derivations of simple n-ary algebras. Ricerchedi Matematica. DOI: 10.1007/s11587-021-00602-3.
- Johnson, B. E. (2001). Local derivations on C*-algebras are derivations. Trans. Amer. Math. Soc. 353:313–325.
- Galajinsky, A., Masterov, I. (2013). Dynamical realization of l-conformal Galilei algebra and oscillators. Nuclear Phys. B, 866(2):212–227. DOI: 10.1016/j.nuclphysb.2012.09.004.
- Kadison, R. V. (1990). Local derivations. J. Algebra 130:494–509. DOI: 10.1016/0021-8693(90)90095-6.
- Larson, D. R., Sourour, A. R. (1990). Local derivations and local automorphisms of B(X). Proc. Sympos. Pure Math. 51:187–194.
- Lü, R., Mazorchuk, V., Zhao, K. (2014). On simple modules over conformal Galilei algebras. J. Pure Appl. Algebra 140:1885–1899.
- Kudaybergenov, K. K., Omirov, B. A., Kurbanbaev, T. K. (2022). Local derivations on solvable Lie algebras of maximal rank. Commun. Algebra 50(9):1–11.
- Kudaybergenov, K. K., Kaygorodov, I., Yuldashev, I. (2022). Local derivations of semisimple Leibniz algebras. Commun. Math. 30(2):1–12.
- Yu, Y., Chen, Zh. (2020). Local derivations on Borel subalgebras of finite-dimensional simple Lie algebras. Commun. Algebra 48(1):1–10. DOI: 10.1080/00927872.2018.1541465.
- Yao, Y. F. (2022). Local derivations on the Witt algebra in prime characteristic. Linear Multilinear Algebra 70: 2919–2933. DOI: 10.1080/03081087.2020.1819189.
- Zhao, Yu., Cheng, Y. (2021). 2-local derivation on the conformal Galilei algebra, arXiv:2103.04237.
- Zhang, X., Cheng, Y. (2015). Simple Schrodinger modules which are locally finite over the positive part. J. Pure Appl. Algebra 219:2799–2815. DOI: 10.1016/j.jpaa.2014.09.029.