Local Automorphisms and Local Superderivations of Model Filiform Lie Superalgebras

Sheng, Yuqiu; Liu, Wende; Liu, Yang

doi:https://doi.org/10.1155/2024/6650997

Journal of Mathematics

On this page

Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 6650997 | https://doi.org/10.1155/2024/6650997

Local Automorphisms and Local Superderivations of Model Filiform Lie Superalgebras

Yuqiu Sheng,¹Wende Liu,²and Yang Liu³

Academic Editor: Ljubisa Kocinac

Received04 Jun 2023

Revised16 Feb 2024

Accepted13 Mar 2024

Published27 Mar 2024

Abstract

In this paper, we give the forms of local automorphisms (resp. superderivations) of model filiform Lie superalgebra in the matrix version. Linear 2-local automorphisms (resp. superderivations) of are also characterized. We prove that each linear 2-local automorphism of is an automorphism.

1. Introduction and Basics

As a significant class of nilpotent Lie algebras, filiform Lie algebras were introduced by Vergne [1] and have been studied extensively, see [2–6] and references in them. Model filiform Lie algebra is the simplest filiform Lie algebra. Vergne proved that each filiform Lie algebra can be obtained by deformations of model filiform Lie algebra (see [1]). Similarly, model filiform Lie superalgebra is the simplest filiform Lie superalgebra.

Automorphisms and superderivations are also important in the study of the structure of Lie superalgebras. In recent years, some new generalized derivations of finite-dimensional Lie algebras and Lie superalgebras were proposed and studied (see [7–9]). Local automorphisms and local derivations were introduced by Kadison in [10] and Larson and Sourour in [11]. The idea of local came from [12, 13]. The idea of 2-local was introduced by emrl in [14]. Later, more and more results of such problem on various algebras were obtained by many scholars (see [15–21] and references in them). In particular, local and 2-local automorphisms (resp. derivations) on some Lie algebras were proved to be automorphisms (resp. derivations) (see [22–26]). For Lie superalgebras, such problem were studied in [27–30] and some other papers.

In this paper, we will use matrices to study local automorphisms (resp. superderivations) of model filiform Lie superalgebra . We will give concrete forms of local automorphisms (resp. superderivations) of . For finite-dimensional nilpotent Lie algebra with . In [23], Ayupov and Kudaybergenov proved that there is a 2-local automorphism of which is not an automorphism. Then, it is impossible that every 2-local automorphism of Lie superalgebra is an automorphism. But if a 2-local automorphism is linear, then we can prove that it must be an automorphism. So, we add an additional linear condition in the definition of 2-local automorphism, we call it linear 2-local automorphism. We will prove that all linear 2-local automorphisms of are automorphisms. But for 2-local superderivation of , the situation is different. We also add an additional linear condition in the definition of 2-local superderivation, and we call it linear 2-local superderivation. In this paper, we will show that not all linear 2-local superderivations of are superderivations, but they is very close to a superderivations. The same situation also occurs in 2-local automorphisms (resp. derivations) of model filiform Lie algebra . We find that not all linear 2-local automorphisms (resp. derivations) of Lie algebra are automorphisms (resp. derivations), and the linear 2-local automorphisms (resp. derivations) which are not automorphisms (resp. derivations) are very close to automorphisms (resp. derivations).

Model filiform Lie superalgebra is a superalgebra with multiplicationwhere is the homogeneous basis and the other brackets vanished. If we only consider the Lie algebra with a basis, their multiplication are same to (1), then it is the model filiform Lie algebra .

For a Lie superalgebra , a linear bijective map is called an automorphism of Lie superalgebra if

Denote the group consisting of all automorphisms of by . Suppose is a linear map of degree , we call a superderivation of degree if

Denote all superderivations of degree by . The elements of are called superderivations of . Linear map is called a local automorphism (resp. superderivation), if for any , there exists (resp. ) such that . A linear map is called a linear 2-local automorphism (resp. superderivation), if for any , there exists (resp. ) such that and . Denote the group consisting of all local automorphisms of by and the superalgebra consisting of all local superderivations of by , respectively.

Throughout the paper, we assume that . All mappings mentioned in this paper are linear. The matrices of mappings of are all with respect to the homogeneous basis , and the matrices of mappings of are all with respect to the basis . stands for an arbitrary field of characteristic zero, is the set of all nonzero elements of , and is the -dimensional column vector space over . and represent the matrix unit and unit vector, respectively.

Denote block matricesby and , respectively.

2. Local Automorphism and Linear 2-Local Automorphism of

Suppose and . Denote

Lemma 1. (1)Let be an invertible lower triangular matrix. Then, for any , there exists such that , where is of the form , with ;(2)Let be an invertible lower triangular matrix. Then, for any and , there exists such that , where is of the form , with .

Proof. (1)Denote where and is an invertible lower triangular matrix. For any , put . We will prove that there exist and with such that Case 1. . Put . Then, it is easy to see that there exists such that (7) holds. Case 2. . Assume that the first nonzero component of vector is the th. Put . Then, it is easy to prove that there exist and such that (7) holds.(2)In a similarly way to the proof of (1), one can come to the conclusion.

Theorem 2. Let be a linear mapping of . Then, if and only if the matrix of is of the formwith .

Proof. If , we can assume that the matrix of is , where is an matrix. DenoteFor any and , using , and successively, we obtainwhere .
By (12), we have .
If , then by (11), we have . Note that in (10), they contradict the invertibility of . Therefore, . Consequently, we have since is invertible. For any , according to (10) and (11), we have and . Denote . Thus, .
Conversely, if the matrix of is with , then is a Lie automorphism of by verification.

Theorem 3. Let be a linear mapping of . Then, if and only if the matrix of is of the form , where and are and invertible lower triangular matrices, respectively.

Proof. Assume that the matrix of iswhere is an matrix.
If , by Theorem 2, for any , there exist and such thatwhere is of the form (8).
We let for all . Since each (where ) in (14) is arbitrary, we see that .
Similarly, we let for all . Since each (where ) in (14) is arbitrary, we see that .
If is not invertible, then there exists a nonzero vector such that . Substituting into (14), we have , which contradicts the invertibility of . Thus, is invertible. Similarly, is also invertible.
Substituting into (14) in turn, by the arbitrariness of , we obtain that is a lower triangular matrix. Similarly, is also a lower triangular matrix.
Conversely, if the matrix of is , where and are and invertible lower triangular matrices respectively, then by Lemma 1, is a local automorphism of .

Theorem 4. Every linear 2-local automorphism of is an automorphism.

Proof. If is a linear 2-local Lie superalgebra automorphism of , then . By Theorem 3, the matrix of is of the form , where and are and invertible lower triangular matrices, respectively. Denotewhere .
By Theorem 2, for any , there exist and such thatwhere . In fact, are all related to and . But for the sake of simplicity, we still denote them in this way without causing confusion.
Substituting into (16), we haveThen, substituting into (16), we have .
Similarly, we can conclude that . Next, substituting into (16), we have . Finally, substituting into (16), we obtain . Thus, by Theorem 2, we have .
From the proof of the above theorems, we get the following conclusions immediately.

Corollary 5. where

Corollary 6. where .

Corollary 7. Let be a linear mapping of .(1)The automorphism group of is where ;(2)The local automorphism group of is where ;(3)The linear mapping is a linear 2-local automorphism of if and only if there exist and such thatwhere is a linear mapping of whose matrix is , , and with such that the matrix of is .

Proof. From the proof of the above theorems, (1), (2), and the necessity of (3) hold immediately. Next, we only need to prove the sufficiency of (3).
Assume the matrix of is . For any , we will find appropriate such thatwhere with , and are all related to and .

Case 8. If and are linear dependent, then by Theorem 3, the existence of is obvious.

Case 9. If and are linear independent, then without loss of generality, we only need to consider the case ofwhere .

Subcase 10. . Put , and, therefore, (24) holds.

Subcase 11. . Denote . We will find such that (24) holds, i.e.,First, we find . Then, it is easy to find to satisfy (27) and . Put . Finally, it is easy to find to satisfy (26).

3. Local Superderivations and Linear 2-Local Superderivations of

Suppose . Denote

As early as 1996, Goze and Khakimdjanov had characterized derivations of in [31]. The following lemma comes from [31].

Lemma 12. where

From this lemma, we can easily get the following conclusion.

Corollary 13. Let be a linear mapping of . Then, if and only if there exist and such that the matrix of is .

Next, we will characterize the matrix form of the superderivation of .

Theorem 14. Let be a linear mapping of , then is a superderivation if and only if its matrix is of the form , where , and are in the forms of (28)–(31), respectively.

Proof. Clearly, a direct verification can prove the sufficiency, and so we only need to prove the necessity of the theorem.
If , we can assume that the matrix of is , whereFirst, we will deduce the form of the matrix of even derivation.
By Corollary 13,where .
Using , we can conclude thatThat is,Next, we will deduce the form of the matrix of odd derivation. Similar to the above process, substituting into the next equation successively,then we can get the following equations in turn:Then,where refers the degree of , .
By (35), (37), and (40), we complete the proof of the necessity of the theorem.

Theorem 15. Let be a linear mapping of whose matrix is , where is an matrix. Then, if and only if and are both lower triangular matrices, and and are of the formrespectively.

Proof. First, we prove the necessity of the theorem. If , then for any , there exist , and such thatwhere , and are of the forms , and in Theorem 14, respectively.
If we let , then by the arbitrariness of and (43) and in a similar way to the proof of Theorem 3, we obtain that and have the required forms.
Similarly, if we let , then by the arbitrariness of in (43) and in a similar way to the proof of Theorem 3, we deduce that and have the required forms.
Next, we will prove the sufficiency of the theorem.
For any , in a similar way to prove Lemma 1, we have and such that and , where and are of the forms and in Theorem 14, respectively.
Similarly, for any and which is (1, 1)-entry of , there exist and such that and , where and are of the forms and in Theorem 14, respectively.
Thus, for any , we haveHence, .

Corollary 16. where

Corollary 17. where

Theorem 18. Let be a linear mapping of . Then, is a linear 2-local superderivation of if and only if there exist and such thatwhere is a linear mapping of whose matrix is .

Proof. If is a linear 2-local superderivation of , then by Theorem 18, we can assume that the matrix of is , where and are both lower triangular matrices, and and are of the forms (41) and (42), respectively. Thus, for any , there exist , and such thatwhere , and are of the forms of (28)–(31), respectively.
Denote and with . For any , substituting into (50), we have . Then, for any , substituting into (50), we have . Denote . Thus,Similarly, we conclude thatIf , substituting into (50), we have . Else if , substituting into (50), we have . Thus, is desired.
Next, we will prove the sufficiency of the theorem. Assume the matrix of is , where , and are the same as in (51) and (52), respectively, and . For any , we want to find and such that (50) holds, where , and are of the forms (29)–(31), respectively.

Case 19. If and are linear dependent, then by Theorem 18, the existence of , , and is obvious.

Case 20. If and are linear independent, then without loss of generality, we only need to consider the case ofwhere .

Subcase 21. . Put and , and, therefore, (50) holds.

Subcase 22. and . LetWe will choose appropriate such that (50) holds.
Put . For any , it is easy to choose appropriate such thatthen we can choose such that (50) holds.

Subcase 23. and . Similar to the proof in Subcase 22, we can achieve the goal.
From the proof of the above theorems, we get the following conclusion immediately.

Corollary 24. Let be a linear mapping of . Then,(1) if and only if there exist and such that the matrix of is ;(2) if and only if the matrix of is a lower triangular matrix;(3) is a linear 2-local automorphism of if and only if there exist and such that the matrix of is .

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the NSF of Hainan Province of China (nos. 121MS0784 and 120RC587), the NSF of Heilongjiang Province of China (no. LH2020A020)}, and the NSF of China (no. 12061029).

References

M. Vergne, “Cohomologie des algébres de Lie nilpotentes. Application álétude de la variétédes algébres de Lie nilpotentes,” The Bulletin de la Société Mathématique de France, vol. 98, pp. 81–116, 1970.
View at: Google Scholar
M. Gilg, “On deformations of the filiform Lie Superalgebra L_n,m,” Communications in Algebra, vol. 32, no. 6, pp. 2099–2115, 2004.
View at: Publisher Site | Google Scholar
Y. Khakimdjanov and R. M. Navarro, “A complete description of all the infinitesimal deformations of the Lie superalgebra,” Journal of Geometry and Physics, vol. 60, no. 1, pp. 131–141, 2010.
View at: Publisher Site | Google Scholar
Q. Wang, H. J. Chen, and W. D. Liu, “On representations of the filiform Lie superalgebra L_n,m,” Journal of Geometry and Physics, vol. 97, pp. 93–104, 2015.
View at: Publisher Site | Google Scholar
Y. Yang and W. D. Liu, “On cohomology of filiform Lie superalgebras,” Journal of Geometry and Physics, vol. 134, pp. 212–234, 2018.
View at: Publisher Site | Google Scholar
W. D. Liu and Y. Yang, “Cohomology of model filiform Lie superalgebras,” Journal of Algebra and Its Applications, vol. 17, no. 04, Article ID 1850074, 2018.
View at: Publisher Site | Google Scholar
G. F. Leger and E. M. Luks, “Generalized derivations of Lie algebras,” Journal of Algebra, vol. 228, no. 1, pp. 165–203, 2000.
View at: Publisher Site | Google Scholar
R. X. Zhang and Y. Z. Zhang, “Generalized derivations of Lie superalgebras,” Communications in Algebra, vol. 38, no. 10, pp. 3737–3751, 2010.
View at: Publisher Site | Google Scholar
H. L. Chang, Y. Chen, and R. X. Zhang, “A generalization on derivations of Lie algebras,” Electronic Research Archive, vol. 29, no. 3, pp. 2457–2473, 2021.
View at: Publisher Site | Google Scholar
R. V. Kadison, “Local derivations,” Journal of Algebra, vol. 130, no. 2, pp. 494–509, 1990.
View at: Publisher Site | Google Scholar
D. R. Larson and A. R. Sourour, “Local derivations and local automorphisms of B(X),” Proceedings of Symposia in Pure Mathematics, vol. 51, pp. 187–194, 1990.
View at: Google Scholar
A. I. Loginov and V. S. Šulman, “Hereditary and intermediate reflexivity of W^∗-algebras,” Mathematics of the USSR-Izvestiya, vol. 9, no. 6, pp. 1189–1201, 1975.
View at: Publisher Site | Google Scholar
D. Hadwin, “Algebraically reflexive linear transformations,” Linear and Multilinear Algebra, vol. 14, no. 3, pp. 225–233, 1983.
View at: Publisher Site | Google Scholar
P. Šemrl, “Local automorphisms and derivations on B(H),” Proceedings of the American Mathematical Society, vol. 125, no. 9, pp. 2677–2680, 1997.
View at: Publisher Site | Google Scholar
L. Moln ár, “Local automorphisms of operator algebras on Banach spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 6, pp. 1867–1874, 2003.
View at: Google Scholar
S. Ayupov and K. Kudaybergenov, “2-local derivations and automorphisms on B(H),” Journal of Mathematical Analysis and Applications, vol. 395, no. 1, pp. 15–18, 2012.
View at: Publisher Site | Google Scholar
S. O. Kim and J. S. Kim, “Local automorphisms and derivations on M_n,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1389–1392, 2003.
View at: Publisher Site | Google Scholar
Z. X. Chen and D. Y. Wang, “2-Local automorphisms of finite-dimensional simple Lie algebras,” Linear Algebra and Its Applications, vol. 486, pp. 335–344, 2015.
View at: Publisher Site | Google Scholar
X. M. Tang, “2-Local derivations on the W-algebra W(2,2),” Journal of Algebra and Its Applications, vol. 20, no. 12, Article ID 2150237, 2021.
View at: Publisher Site | Google Scholar
S. Ayupov, A. Khudoyberdiyev, and B. Yusupov, “Local and 2-local derivations of solvable Leibniz algebras,” International Journal of Algebra and Computation, vol. 30, no. 06, pp. 1185–1197, 2020.
View at: Publisher Site | Google Scholar
S. Ayupov, K. Kudaybergenov, and B. Omirov, “Local and 2-local derivations and automorphisms on simple Leibniz algebras,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 43, no. 3, pp. 2199–2234, 2020.
View at: Publisher Site | Google Scholar
S. Ayupov, K. Kudaybergenov, and I. Rakhimov, “2-Local derivations on finite-dimensional Lie algebras,” Linear Algebra and Its Applications, vol. 474, pp. 1–11, 2015.
View at: Publisher Site | Google Scholar
S. Ayupov and K. Kudaybergenov, “2-Local automorphisms on finite-dimensional Lie algebras,” Linear Algebra and Its Applications, vol. 507, pp. 121–131, 2016.
View at: Publisher Site | Google Scholar
S. Ayupov and K. Kudaybergenov, “Local derivations on finite-dimensional Lie algebras,” Linear Algebra and Its Applications, vol. 493, pp. 381–398, 2016.
View at: Publisher Site | Google Scholar
M. Costantini, “Local automorphisms of finite dimensional simple Lie algebras,” Linear Algebra and Its Applications, vol. 562, pp. 123–134, 2019.
View at: Publisher Site | Google Scholar
S. Ayupov and A. K. Khudoyberdiyev, “Local derivations on solvable Lie algebras,” Linear and Multilinear Algebra, vol. 69, no. 7, pp. 1286–1301, 2021.
View at: Publisher Site | Google Scholar
L. Yu, Y. Wang, H. X. Chen, and J. Z. Nan, “2-Local automorphisms on basic classical Lie superalgebras,” Acta Mathematica Sinica, English Series, vol. 35, no. 3, pp. 427–437, 2019.
View at: Publisher Site | Google Scholar
H. X. Chen, Y. Wang, and J. Z. Nan, “Local superderivations on basic classical Lie superalgebras,” Algebra Colloquium, vol. 24, no. 04, pp. 673–684, 2017.
View at: Publisher Site | Google Scholar
Y. C. Su, X. Q. Yue, and X. Y. Zhu, “2-Local superderivations on the Lie superalgebra of Block type,” Linear and Multilinear Algebra, vol. 70, no. 21, pp. 6425–6440, 2021.
View at: Publisher Site | Google Scholar
J. X. Yuan, L. Y. Chen, and Y. Cao, “Local superderivations on Cartan type Lie superalgebras,” Revista de la Unión Matemática Argentina, vol. 62, no. 2, pp. 433–442, 2021.
View at: Publisher Site | Google Scholar
M. Goze and Y. Khakimdjanov, Nilpotent Lie Algebras, Kluwers Academics Publishers, Dordrecht, Netherlands, 1996.

Copyright

Copyright © 2024 Yuqiu Sheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

118

Downloads

85

Citations