Open Math. 2015; 13: 518–527
Open Mathematics
Open Access
Research Article Bing Sun and Liangyun Chen*
Restricted and quasi-toral restricted Lie-Rinehart algebras DOI 10.1515/math-2015-0049 Received February 4, 2015; accepted August 20, 2015.
Abstract: In this paper, we introduce the definition of restrictable Lie-Rinehart algebras, the concept of restrictability
is by far more tractable than that of a restricted Lie-Rinehart algebra. Moreover, we obtain some properties of p-mappings and restrictable Lie-Rinehart algebras. Finally, we give some sufficient conditions for the commutativity of quasi-toral restricted Lie-Rinehart algebras and study how a quasi-toral restricted Lie-Rinehart algebra with zero center and of minimal dimension should be. Keywords: Restricted Lie-Rinehart algebras, Restrictable Lie-Rinehart algebras, Quasi-toral restricted Lie-Rinehart
algebras MSC: 17B30, 17B50
1 Introduction The concept of a restricted Lie algebra is attributable to N. Jacobson in 1943. It is well known that the Lie algebras associated with algebraic groups over a field of characteristic p are restricted Lie algebras [17]. Now, restricted Lie algebras attract more and more attentions. For example: restricted Lie superalgebras [9], restricted Leibniz algebras [7], restricted Lie triple system [12] and restricted Lie-Rinehart algebras [6] were studied, respectively. Lie-Rinehart algebras were introduced and studied by Herz [11], Palais [15] and Rinehart [16]. A Lie-Rinehart algebra is a Lie K-algebra, which is also an A-module and these two structures are related in an appropriate way [13]. A precise definition will be reproduced in Section 2 below. The leading example of Lie-Rinehart algebras is the set Der.A/ of all K-derivations of A. In recent years the study of restricted Lie-Rinehart obtained some important results. In [6], I. Dokas introduced the notion of restricted Lie-Rinehart algebras and constructed its restricted enveloping algebra. As a natural generalization of a restricted Lie algebra, it seems desirable to investigate the possibility of establishing a parallel theory for restricted Lie-Rinehart algebras. n.x/ Jacobson [14] conjectured that every restricted Lie algebra .L; Œp/ satisfying the requirement x Œp D x, for any x 2 L, is abelian, where n.x/ 2 N, N is a positive integer. Though his conjecture remains an open problem, the study of analogous conditions with this conjecture has been developed in [3, 4, 8, 10]. The presence of these results motivates an analogous discussion to restricted Lie-Rinehart algebras in this paper. The paper is organized as follows. In Section 2, we recall some basic definitions of restricted Lie-Rinehart algebras. In Section 3, we introduce the definition of restrictable Lie-Rinehart algebras, which is by far more tractable than that of a restricted Lie-Rinehart algebras in [6]. Moreover, we obtain some properties of p-mappings and restrictable Lie-Rinehart algebras. In Section 4, we give some sufficient conditions for the commutativity of quasi-
Bing Sun: School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China *Corresponding Author: Liangyun Chen: School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China, E-mail:
[email protected] © 2015 Bing Sun and Liangyun Chen, licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
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toral restricted Lie-Rinehart algebras and study how a quasi-toral restricted Lie-Rinehart algebra with zero center and of minimal dimension should be. In this paper, K is a field and A is a commutative algebra over K.
2 Preliminaries Definition 2.1 ([1]). A Lie-Rinehart algebra over A consists of a Lie K-algebra L together with an A-module structure on L and a mapping called anchor ˛ W L ! DerK .A/ which is simultaneously an A-module and a Lie algebra homomorphism such that the following relation holds ŒX; aY D aŒX; Y C ˛.X /.a/Y; for any X; Y 2 L, a 2 A. Example 2.2 ([1]). It is clear that the Lie-Rinehart algebras with ˛ D 0 are exactly the Lie A-algebras. On the other hand, any commutative K-algebra A defines a Lie-Rinehart algebra with L D Der.A/. If A D K, then Der.A/ D 0, and in this case, there is no difference between Lie algebras and Lie-Rinehart algebras. Definition 2.3 ([1]). Lie-Rinehart algebra L is called abelian if ŒL; L D 0. Definition 2.4 ([2]). If L1 and L2 are Lie-Rinehart algebras over A, then a Lie-Rinehart homomorphism f W L1 ! L2 is a mapping, which is simultaneously a Lie K-algebra homomorphism and an A-module homomorphism. Further, one requires that the diagram ˛1 / Der.A/ L1 ; f
˛2
L2 commutes.
We denote by LR.A/ the category of Lie-Rinehart algebras over A. As we mentioned above, one has the full inclusion L.A/ LR.A/, where L.A/ denotes the category of Lie A-algebras. Observe that the kernel of any Lie-Rinehart algebra homomorphism is a Lie A-algebra. Definition 2.5 ([2]). Let L be a Lie-Rinehart algebra over A. A Lie-Rinehart subalgebra N of L consists of a K-Lie subalgebra N which is an A-module and N acts on A via the composition ˛
N ,! L ! Der.A/: Definition 2.6 ([2]). Let L be a Lie-Rinehart algebra over A. A subalgebra N of a Lie-Rinehart algebra L is said to be an ideal if N is an ideal of the Lie K-algebra L and the composition ˛
N ,! L ! Der.A/ is trivial. Note that an ideal only has the structure of a Lie A-algebra. It should be noted that a Lie-Rinehart algebra L is an ideal of L itself if and only if ˛ D 0, i.e., if and only if L is a Lie A-algebra. In particular, this holds if L is a Lie algebra .A D K/. Let M and N be ideals of the Lie-Rinehart algebra L. The commutator ideal of M and N , denoted by ŒM; N , is the ideal of L spanned by the brackets ŒX; Y for X 2 M and Y 2 N . Obviously, ŒM; N M \ N . Note that in the situation of Lie A-algebras, L is an ideal of L, so the derived algebra ŒL; L makes sense.
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Lemma 2.7 ([2]). Let L be a Lie-Rinehart algebra over A. The center of L is the ideal C.L/ D fX 2 L j ŒX; Y D 0; 8Y 2 L and ˛.X / D 0g: It is clear that L is an abelian Lie-Rinehart algebra if and only if C.L/ D L. Definition 2.8. Suppose that L is a Lie-Rinehart algebra over A. The sequence fL.n/ gn2N defined by means of L.0/ WD L; L.nC1/ D ŒL.n/ ; L.n/ is called the derived series of L. L is called solvable if there is an n 2 N such that L.n/ D 0 and the anchor mapping ˛ satisfies ˛ D 0. Definition 2.9. [5] Let L be a Lie-Rinehart algebra over A. A derivation of L is a pair .D; ı/, where D W L ! L is a linear mapping, ı 2 Der.A/ and such that .1/ DŒx; y D ŒD.x/; y C Œx; D.y/; .2/ D.ax/ D aD.x/ C ı.a/x; for all x; y 2 L and a 2 A. Obviously, .adx; ı/ is a derivation. Derivations of this form are called inner. If L is a Lie A-algebra, then ı D 0 in the definition above. Definition 2.10. Let L be a Lie-Rinehart algebra and .Œx; z; y/ D .x; Œz; y/.
a symmetric bilinear form on L.
is called associative, if
Definition 2.11. Let L be a Lie-Rinehart algebra and a symmetric bilinear form on L. Set L? D fx 2 ? Lj .x; y/ D 0; 8y 2 Lg. L is called nondegenerate, if L D 0. Definition 2.12. Let L be a Lie-Rinehart algebra. L is called semisimple if it contains no non-zero soluble ideal. If L has no ideals except itself and 0 and if moreover ŒL; L ¤ 0, we call L simple. Definition 2.13 ([6]). A restricted Lie-Rinehart algebra .L; Œp/ over A, is a Lie-Rinehart algebra over A such that .L; Œp/ is a restricted Lie algebra over K, the anchor mapping ˛ is a restricted Lie homomorphism, and the following relation holds: .aX /Œp D ap X Œp C .˛.aX //p 1 .a/X; for all a 2 A and X 2 L. Definition 2.14. Let L be a Lie-Rinehart algebra over A. A mapping Œp W L ! L is called a p-mapping if .1/ adx Œp D .adx/p ; 8x 2 L. .2/ .kx/Œp D k p x Œp ; 8x 2 L; k 2 K. .3/ .ax/Œp D ap x Œp C .˛.ax//p 1 .a/x; 8x; y 2 L; a 2 A. P 1 .4/ .x C y/Œp D x Œp C y Œp C p i D1Psi .x; y/, 1 where .ad.x ˝ X C y ˝ 1//p 1 .x ˝ 1/ D p i D1 i si .x; y/ ˝K KŒX ; 8x; y 2 L, X is an indeterminate over K. By Definition 2.13, the pair .L; Œp/ is a restricted Lie-Rinehart algebra. Example 2.15. It is clear that the restricted Lie-Rinehart algebras with ˛ D 0 are exactly the restricted Lie Aalgebras. On the other hand, any commutative K-algebra A defines a restricted Lie-Rinehart algebra with L D Der.A/. If A D K, then Der.A/ D 0, and in this case, there is no difference between restricted Lie algebras and restricted Lie-Rinehart algebras. Definition 2.16. Let .L1 ; Œp1 / and .L2 ; Œp2 / be restricted Lie-Rinehart algebras. A mapping f W L1 ! L2 is called restricted morphism if .1/ f is a Lie-Rinehart morphism, .2/ f .x Œp1 / D f .x/Œp2 ; 8x 2 L1 : A representation W L ! gl.V / is called restricted if .x Œp / D .x/p ; 8x 2 L.
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Definition 2.17. Let .L; Œp/ be a restricted Lie-Rinehart algebra over A. A subalgebra H L (ideal I G L) is called a p-subalgebra (p-ideal) if x Œp 2 H; 8x 2 H .x Œp 2 I; 8x 2 I /.
3 Restrictable Lie-Rinehart algebras Proposition 3.1. Let .L; Œp/ be a restricted Lie-Rinehart algebra and G be a Lie-Rinehart algebra over A. Suppose that f W L ! G is a Lie-Rinehart homomorphism of L into G such that ker.f / is a p-ideal of L. Then there exists exactly one p-mapping on f .L/ such that f W L ! f .L/ is a restricted morphism. 0
Proof. We put y Œp WD f .x Œp / if y D f .x/. If f .x1 / D f .x2 /, then x1 x1Œp D ..x1
x2 /Œp C x2Œp C
x2 / C x2 /Œp D .x1
x2 2 kerf and p X1
si .x1
x2 ; x2 /:
i D1
Note that kerf is a p-ideal, which ensures that the first and the last summand of the righthand side of the equation 0 above are contained in kerf . Consequently, f .x1Œp / D f .x2Œp / and Œp W f .L/ ! f .L/ is well defined. We claim 0 that the definition of Œp on f .L/ entails the properties of a p-mapping. In fact, for all x 2 L, 0
0
.af .x//Œp D .f .ax//Œp
D f ..ax/Œp / D f .ap x Œp C .˛1 .ax//p p
D a f .x
Œp
p
Œp
D a f .x p
D a f .x/
/ C .˛1 .ax//
1
.a/x/
p 1
/ C .˛2 ı f .ax//
Œp0
.a/f .x/
p 1
p 1
C .˛2 .af .x///
.a/f .x/
.a/f .x/:
It is also clear that this is the only p-mapping on f .L/, making f into a restricted homomorphism. Proposition 3.2. Let L be an abelian Lie-Rinehart algebra over A and suppose that f W L ! L is a mapping. Then .L; f / is restricted if and only if f .ax C y/ D ap f .x/ C f .y/ (1) and f .kx C y/ D k p f .x/ C f .y/; for all x; y 2 L, a 2 A and k 2 K. Mappings with this property are referred to as p-semilinear. Proof. .)/ For all x; y 2 L, a 2 A and k 2 K, we have f .ax C y/ D f .ax/ C f .y/ C
p X1
si .ax; y/ D ap f .x/ C f .y/
iD1
and f .kx C y/ D f .kx/ C f .y/ C
p X1
si .kx; y/ D k p f .x/ C f .y/:
iD1
.(/ Since L is an abelian Lie-Rinehart algebra over A, we have adf .x/.y/ D Œf .x/; y D 0 D .adx/p .y/; 8y 2 L: Hence adf .x/ D .adx/p : Set y D 0; k D 1 in Equations (3.2) and (3.2) respectively, we have f .ax/ D ap f .x/; f .kx/ D k p f .x/ C f .y/
(2)
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and f .x C y/ D f .x/ C f .y/ D f .x/ C f .y/ C
p X1
si .x; y/;
iD1
for all x; y 2 L; a 2 A; k 2 K. Consequently, we prove the proposition. Proposition 3.3. Let L be a subalgebra of a restricted Lie-Rinehart algebra .G; Œp/ and Œp1 W L ! L be a mapping. Then the following statements are equivalent: .1/ Then Œp1 is a p-mapping on L and ˛ ı .Œp1 Œp/ D 0. .2/ There exists a p-semilinear mapping f W L ! CG .L/ such that Œp1 D Œp C f , where CG .L/ D fx 2 GjŒx; L D 0 and ˛.x/ D 0g. Proof. .1/ ) .2/ Consider f W L ! G; f .x/ WD x Œp1 x Œp . Since adf .x/.y/ D 0 and ˛.f .x// D ˛ ı .Œp1 Œp/.x/ D 0; 8x; y 2 L; f actually maps L into CG .L/. For x; y 2 L; a 2 A and k 2 K, we obtain f .ax C y/ D ap x Œp1 C y Œp1 C
p X1
si .ax; y/
ap x Œp
y Œp
p X1
si .ax; y/ D ap f .x/ C f .y/;
i D1
iD1
and f .kx C y/ D k p x Œp1 C y Œp1 C
p X1
si .kx; y/
k p x Œp
y Œp
p X1
si .kx; y/ D k p f .x/ C f .y/;
i D1
iD1
which proves that f is p-semilinear. .2/ ) .1/ For x; y 2 L; a 2 A; k 2 F, we obtain .kx/Œp1 D .kx/Œp C f .kx/ D k p x Œp1 ; adx Œp1 .y/ D Œx Œp C f .x/; y D Œx Œp ; y D .adx/p .y/; .x C y/Œp1 D x Œp C y Œp C
p X1
si .x; y/ C f .x/ C f .y/ D x Œp1 C y Œp1 C
si .x; y/;
iD1
iD1
.ax/Œp1 D .ax/Œp C f .ax/ D ap x Œp C .˛.ax//p
p X1
1
.a/x C ap f .x/ D ap x Œp1 C .˛.ax//p
1
.a/x;
which proves the proposition. Note that L is not necessarily a p-subalgebra of .G; Œp/. Corollary 3.4. The following statements hold. .1/ If C.L/ D 0, then L admits at most one p-mapping. .2/ If L is free as an A-module and two p-mapping coincide on a basis, then they are equal. 0 .3) If .L; Œp/ is a restricted Lie-Rinehart algebra over A, then there exists a p-mapping Œp of L such that 0 x Œp D 0 8x 2 C.L/. Proof. (1) Since for G D L we have CG .L/ D C.L/, the only p-semilinear occurring in proposition 3.3 is the zero mapping. (2) Let Œp; Œp1 be two p-mappings of L and .ej /j 2J a basis of L satisfying ejŒp D ejŒp1 . By proposition 3.3, there exists a p-semilinear mapping f W L ! C.L/ such that Œp1 D Œp C f . Hence f .ej / D ejŒp1 ejŒp D 0, i.e., 0 D f D Œp1 Œp. (3) ŒpjC.L/ defines a p-mapping on C.L/. Since C.L/ is abelian, it is p-semilinear by proposition 3.1. Extend 0 this to a p-semilinear mapping f W L ! C.L/. Then Œp WD Œp f is a p-mapping of L by proposition 3.3, vanishing on C.L/. Lemma 3.5 ([6]). Let L be a Lie-Rinehart algebra over A such that L is free as an A-module. Let fui ; I g be an ordered A-basis of L. If there is a mapping ui ! uŒp such that .adui /p D aduŒp for all i 2 I , then L can be i i equipped with the structure of restricted Lie-Rinehart algebra with a p-mapping which extends the mapping Œp.
Restricted and quasi-toral restricted Lie-Rinehart algebras
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Definition 3.6. Let L be a Lie-Rinehart algebra over A such that L is free as an A-module. L is called restrictable if .adx/p 2 ad.L/ 8x 2 L. Proposition 3.7. Let L be a Lie-Rinehart algebra over A such that L is free as an A-module. Then L is restrictable if and only if there is a mapping Œp W L ! L which makes L a restricted Lie-Rinehart algebra. Proof. .)/ By virtue of Lemma 3.5, there exists a p-mapping Œp such that .L; Œp/ is a restricted Lie-Rinehart algebra. .(/ Since L is a restricted Lie-Rinehart algebra, we have .adx/p D adx Œp 2 ad.L/; 8x 2 L: Hence L is restrictable. Example 3.8. .1/ Every abelian Lie-Rinehart algebra is restrictable. .2/ Every nilpotent Lie-Rinehart algebra L such that LpC1 D 0 and ˛.L/ D 0 is restrictable since then .ady/p D 0; 8y 2 L. Theorem 3.9. Let f W L1 ! L2 be a surjective homomorphism of Lie-Rinehart algebras over A. L2 is free as an A-module. If L1 is restrictable, so is L2 . Proof. Since f is a surjective homomorphism, for all y; z 2 L2 there are x; x1 2 L1 which makes f .x/ D y; f .x1 / D z. Since L1 is restrictable, there is x2 2 L1 such that .adx/p D adx2 . Then .ady/p .z/ D .adf .x//p .f .x1 // D f ..adx/p .x1 // D f .adx2 .x1 // D adf .x2 /.z/: Hence .ady/p D adf .x2 / 2 ad.L2 /. Definition 3.10. Let .L; Œp/ be a restricted Lie-Rinehart algebra over A. A derivation .D; ı/ 2 Der.L/ is called a restricted Lie-Rinehart derivation if D.x Œp / D .adx/p 1 .D.x// holds and .D; ı/ is a derivation of a Lie-Rinehart algebra. Obviously, every inner derivation .adx; ı/; 8x 2 L, is a restricted derivation. Definition 3.11. Let L1 be a Lie-Rinehart algebra over A and L2 be a Lie A-algebra and ' W L1 ! Der.L2 / be a homomorphism of Lie K-algebras and module. On the vector space L1 ˚ L2 , define a Lie bracket by means of 0
0
0
0
Œl1 C l2 ; l1 C l2 D Œl1 ; l1 C '.l1 /.l2 /
0
0
'.l1 /.l2 / C Œl2 ; l2 :
This algebra, which is denoted by L1 ˚' L2 , is called the semidirect product of L1 and L2 . Theorem 3.12. Notions such as Definition 3.11, then L1 ˚' L2 is a Lie-Rinehart algebra. Proof. In [17], the authors had proved that L1 ˚' L2 is a Lie algebra over K. It is clear that L1 ˚' L2 is an A-module. So we need to check that Equation (3) is true in Definition 2.14. For all x1 ; y1 2 L1 , x2 ; y2 2 L1 , a 2 A, and ı 2 Der.A/, we have Œx1 C x2 ; a.y1 C y2 / D Œx1 C x2 ; ay1 C ay2 D Œx1 ; ay1 C '.x1 /.ay2 /
'.ay1 /.x2 / C Œx2 ; ay2
D aŒx1 ; y1 C ˛1 .x1 /.a/y1 C a'.x1 /.y2 / D a.Œx1 ; y1 C '.x1 /.y2 /
a'.y1 /.x2 / C aŒx2 ; y2
'.y1 /.x2 / C Œx2 ; y2 / C ˛1 .x1 /.a/y1
D aŒx1 C x2 ; y1 C y2 C ˛1 .x1 /.a/y1 : Let ˛.x1 C x2 /.a/.y1 C y2 / D ˛1 .x1 /.a/y1 . Then ˛ W L1 ˚' L2 ! DerK .A/ is an A-module and a Lie algebra homomorphism. Hence L1 ˚' L2 is a Lie-Rinehart algebra. Theorem 3.13. Let L1 be a Lie-Rinehart algebra over A and L2 be a Lie A-algebra. If ' W L1 ! Der.L2 / is a restricted Lie-Rinehart homomorphism such that '.x/ is restricted for every x 2 L1 and L1 ; L2 are free as an A-module. Then L1 ˚' L2 is restrictable.
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B. Sun, L. Chen
Proof. Let x 2 L1 . Then .adx/p
adx Œp jL1 D 0
and .adx/p .y/ D .adx/p
1
.'.x/.y// D .'.x//p .y/ D '.x Œp /.y/ D adx Œp .y/ 8y 2 L2
holds, hence .adx/p 2 ad.L1 ˚' L2 / 8x 2 L1 . If x 2 L2 , then .adx/p
0
adx Œp jL2 D 0;
and for y 2 L1 we obtain 0
..adx/p
adx Œp /.y/ D
.adx/p
1
0
.'.y/.x// C '.y/.x Œp / D 0; 8x 2 L2 ;
since '.y/ is a restricted derivation. Hence .adx/p 2 ad.L1 ˚' L2 / 8x 2 L2 . Therefore, L1 ˚' L2 is restrictable by Lemma 3.5. Corollary 3.14. Let M; N be ideals of a Lie-Rinehart algebra L such that L D M ˚ N . Then L is restrictable if and only if M; N are. Proof. .(/ If M; N are restrictable, setting ' D 0 we conclude that L D M ˚ N is restrictable. .)/ Since M ˚ N=N Š M=M \ N , we have L=N Š M; L=M Š N and M; N are free as an A-module. If .A; L/ is restrictable, so are M; N by Theorem 3.9. Corollary 3.15. Let M; N be restrictable ideals of a Lie-Rinehart algebra L such that L D M C N , and ŒM; N D 0. Then L is restrictable. Proof. Since M; N be restrictable, L D M C N is free as an A-module. Define a mapping f W M ˚ N ! L D M C N; x C y 7! x C y. Clearly, f is a surjective homomorphism. For x1 C y1 ; x2 C y2 2 M ˚ N , by ŒM; N D 0, one gets Œx1 ; y2 D Œy1 ; x2 D 0. We have f .Œx1 C y1 ; x2 C y2 / D f .Œx1 ; x2 C Œy1 ; y2 / D Œx1 ; x2 C Œy1 ; y2 D Œx1 ; x2 C Œx1 ; y2 C Œy1 ; x2 C Œy1 ; y2 D Œx1 C y1 ; x2 C y2 D Œf .x1 C y1 /; f .x2 C y2 /: By Corollary 3.14, we have M ˚ N is restrictable. By Theorem 3.9, one gets L is restrictable. Theorem 3.16. Let L be a finite dimensional subalgebra of the restricted Lie-Rinehart algebra .G; Œp/ over A such that L is free as an A-module. Assume W G G ! K to be an associative symmetric bilinear form, which is nondegenerate on L L. Then L is restrictable. Proof. Since is nondegenerate on L L, every linear form f on L is determined by a suitably chosen element y 2 L W f .z/ D .y; z/; 8z 2 L. Let x 2 L. Then there exists y 2 L such that .x Œp ; z/ D .y; z/; 8z 2 L: This implies that 0 D .x Œp
y; L.1/ / D .Œx Œp
y; L; L/ and Œx Œp
y; L D 0. Therefore, we have
.adL x/p D adL x Œp D adL y 2 adL .L/; proving that L is restrictable.
4 Quasi-toral restricted Lie-Rinehart algebras Definition 4.1. Let .L; Œp/ be a restricted Lie-Rinehart algebra over A. A restricted Lie-Rinehart algebra is called n.x/ quasi-toral if there exists a positive integer n.x/ such that x Œp D x for any element x 2 L; n.x/ 2 N.
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Restricted and quasi-toral restricted Lie-Rinehart algebras
Definition 4.2. Let .L; Œp/ be a restricted Lie-Rinehart algebra over A. A p-ideal I Gp L is referred to as p-solvable n n if there is n 2 N such that I.n/ D ŒI Œp ; I Œp D 0. n
n
n
Theorem 4.3. If x D x Œp C.ady/p 1 .x/; .adx/p .y/ D 0 and x is a quasi-toral element, then y is a quasi-toral element if and only if x C y is a quasi-toral element for x; y 2 L. m
Proof. Since x is quasi-toral, there is m n.m 2 N/ such that x Œp D x. Then m
m
Œx; y D Œx Œp ; y D .adx/p .y/ D .adx/p
m
pn
n
.adx/p .y/ D 0; 8y 2 L:
k
.)/ Since y is quasi-toral, there is k 2 N such that y Œp D y. By virtue of routine computation, we obtain mk Œpkm x D x; y Œp D y. Then km
.x C y/Œp
D x Œp
km
C y Œp
km
DxCy
by means of Œx; y D 0. So x C y is a quasi-toral element. u .(/ Since x C y is quasi-toral, there is u 2 N such that .x C y/Œp D x C y. Then u
u
u
u
.x C y/Œp D x Œp C y Œp D x Œp C y Œp by virtue of Œx; y D 0. We obtain x C y D .x C y/Œp um Hence y Œp D y, i.e., y is a quasi-toral element.
um
um
D x Œp
C y Œp
um
u
and x Œp
um
m
D x since x Œp D x.
Theorem 4.4. Let .L; Œp/ be a quasi-toral restricted Lie-Rinehart algebra over A. If L is p-solvable, then L is abelian. m
m
n.x/
Proof. Since L is p-solvable, there is m 2 N such that L.m/ D ŒLŒp ; LŒp D f0g. If x Œp D x for all t n.x/ n.x/ n.x/ x 2 L, then x Œp D .:::.x Œp /:::/Œp D x for all x 2 L. As there is k 2 N such that 1 n.x/ k, kŠ x Œp D x for all x 2 L. kŠ kŠ kŠ (i) If m kŠ, then ŒLŒp ; LŒp D f0g. Since 1 n.x/ k and x Œp D x for all x 2 L, ŒL; L D f0g, i.e., L is abelian. kŠ (ii) If m > kŠ, then there exist u; v such that m D ukŠ C v; 0 v < kŠ. Since 1 n.x/ k and x Œp D x kŠ m m ukŠCv ukŠCv v v for all x 2 L, LŒp D L, f0g D L.m/ D ŒLŒp ; LŒp D ŒLŒp ; LŒp D ŒLŒp ; LŒp . Since 0 v < kŠ, ŒL; L D f0g by virtue of (i), i.e., L is abelian. Lemma 4.5. Let .L; Œp/ be a quasi-toral restricted Lie-Rinehart algebra over A. Then adx is semisimple for every x 2 L. n.x/
n.x/
Proof. Since L is quasi-toral, there is n.x/ 2 N such that x Œp D x for all x 2 L. Then (adx/p D adx: Let mx .X/ 2 KŒX be the minimum polynomial of adx: So there is f .X / 2 KŒX such that f .X / mx .X / D n.x/ X . Taking the derivative we obtain f .X /0 mx .X / C f .X /mx .X /0 D 1, which means adx is semisimple Xp for every x 2 L. Theorem 4.6. Let .L; Œp/ be a quasi-toral restricted Lie-Rinehart algebra over A. Then L is solvable if and only if L is abelian. Proof. .(/ It is obvious. .)/ Let J be an ideal of L such that ŒJ; J C.L/. For any x 2 J , we have (adx/3 D 0. By virtue of Lemma 4.5 have adx D 0, then J C.L/, i.e., if J is an ideal of L such that ŒJ; J C.L/, then J C.L/. Since L is solvable, there is k 2 N such that L.k/ D ŒL.k 1/ ; L.k 1/ D f0g. Obviously, for any 1 m k 1, L.m/ is an ideal of L. Since ŒL.k 1/ ; L.k 1/ D f0g, L.k 1/ D ŒL.k 2/ ; L.k 2/ C.L/. Then L.k 2/ C.L/, i.e., L.k 2/ D ŒL.k 3/ ; L.k 3/ C.L/. So L.k 2/ C.L/. Using the same method, we have L.0/ C.L/, i.e., L is abelian. Corollary 4.7. Let .L; Œp/ be a finite-dimensional solvable restricted Lie-Rinehart algebra over A. If L is not abelian, then there is x 2 L such that x is not a quasi-toral element of L.
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Proof. If every element x 2 L is a quasi-toral element, then L is abelian since L is solvable by virtue of Theorem 4.6. We have arrived at a contradiction since L is not abelian. Corollary 4.8. Let .L; Œp/ be a nonsolvable quasi-toral restricted Lie-Rinehart algebra over A. Then C.L/ is a maximal solvable ideal of L and L=C.L/ is semisimple. Proof. Let J be an abelian ideal of L. For any x 2 J , y 2 L, we have Œx Œp ; y D .adx/p .y/ D Œx; ; Œx; Œx; y; . Since J is an abelian ideal of L, Œx; y 2 J and Œx; Œx; y D 0. Then Œx Œp ; y D 0. n.x/ n.x/ Since x Œp D x for any x 2 L, we have Œx; y D Œx Œp ; y D 0. So J C.L/. If J is a maximal abelian ideal of L, then C.L/ J . Hence J D C.L/. Obviously, C.L/ is restricted. By virtue of Theorem 4.6, C.L/ is a maximal solvable ideal of L, then L=C.L/ is semisimple. Theorem 4.9. Let .L; Œp/ be a minimal-dimensional restricted Lie-Rinehart algebra over A such that .i/ L is quasi-toral, .ii/ C.L/ D f0g, .iii/ L is not simple. Then the following statements hold: .1/ If J is a proper ideal of L and ˛.L/ D 0, then L=J is abelian. .2/ If J is a proper ideal of L, then J is semisimple. .3/ If J is a maximal ideal of L, then the codimension of J is 1. .4/ ŒL; L is simple. Proof. (1) Assume that H is a proper restricted ideal of L. Since L is a minimal-dimensional restricted Lie algebra over A such that (i) and (ii) hold, C.H / ¤ f0g. By virtue of Corollary 4.8, we have H WD H=C.H / is semisimple, n.x/ i.e., the center of H is zero. Let x 2 H D H=C.H /: Since x Œp D x for any x 2 L, where n.x/ 2 N, n.x/ n.x/ Œp Œp .x C C.H // D x C C.H /, i.e., x D x for any x 2 L. Then H D H=C.H / ¤ f0g such that (i) and (ii) hold. Since dimH < dimL, this contradicts the choice of dimL. Consequently, L has not any proper restricted ideal. Let J be a proper ideal of L. Since J is not restricted, there is x1 2 J such that x1Œp … J . Since ŒJ u Œp Ax1 ; L J J u Ax1Œp , J u Ax1Œp is an ideal of L. If J u Ax1Œp ¤ L, then J u Ax1Œp is not restricted, i.e., there is x2 2 J u Ax1Œp such that x2Œp … J u Ax1Œp . Then J u Ax1Œp u Ax2Œp is an ideal of L. Using the same method, there are x1 2 J; x2 2 J; ; xn 2 J such that x1Œp … J; x2Œp … J; ; xnŒp … J and L D J uAx1Œp uAx2Œp u uAxnŒp since L is finite dimensional. By routine computation, we obtain ŒL; L J . Hence L=J is abelian. Let J be a proper ideal of L and I be an abelian ideal of J . Then there are x1 2 J; x2 2 J; ; xn 2 J such that x1Œp … J; x2Œp … J; ; xnŒp … J and L D J u Ax1Œp u Ax2Œp u u AxnŒp : Then ŒI; L D ŒI; J u Ax1Œp u Ax2Œp u u AxnŒp I , i.e., I is an abelian ideal of L. By virtue of the proof of Corollary 4.8, C.L/ is a maximal abelian ideal of L, then I D f0g by C.L/ D f0g. Hence J is semisimple. (2) Let J be maximal ideal of L. Since J is not a proper restricted ideal of L, there is x 2 J such that x Œp … J . According to ad.ax/Œp .J / .ad.ax//p .J / D Œax; ; Œax; Œax; J ; 2 J , we obtain ŒAx Œp u J; Ax Œp u J J Ax Œp u J . Then Ax Œp u J is a subalgebra of L. Since ŒAx Œp u J; L J , Ax Œp u J is a nontrivial ideal of L. If J is a maximal ideal of L, then L D Ax Œp u J . So dimJ D dimL 1 D n 1, i.e., the codimension of J is 1. (3) Due to the proof of (1), ŒL; L is a minimal proper non-restricted ideal of L. Let I be a proper ideal of ŒL; L. Then there are x1 2 ŒL; L; x2 2 ŒL; L; ; xn 2 ŒL; L such that x1Œp … ŒL; L; x1Œp … ŒL; L; ; x1Œp … ŒL; L and L D ŒL; L u Ax1Œp u Ax2Œp u u AxnŒp : So ŒI; L D ŒI; ŒL; L u Ax1Œp u Ax2Œp u u AxnŒp I , i.e., I is an ideal of L. According to the proof of (1), we obtain I ŒL; L and I D ŒL; L. As a result, ŒL; L is simple. Acknowledgement: Supported by NNSF of China (Nos. 11171055 and 11471090), NSF of Jilin province
(No.201115006). The authors would like to thank the referee for valuable comments and suggestions on this article.
Restricted and quasi-toral restricted Lie-Rinehart algebras
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