On structure of graded restricted simple Lie algebras of Cartan type as modules over the Witt algebra
aa r X i v : . [ m a t h . R T ] F e b ON STRUCTURE OF GRADED RESTRICTED SIMPLE LIE ALGEBRAS OFCARTAN TYPE AS MODULES OVER THE WITT ALGEBRA
KE OU AND YU-FENG YAO
Abstract.
Any graded restricted simple Lie algebra of Cartan type contains a subalgebra isomor-phic to the Witt algebra over a field of prime characteristic. As some analogue of study on branchingrules for restricted non-classical Lie algebras, it is shown that each graded restricted simple Lie al-gebra of Cartan type can be decomposed into a direct sum of restricted baby Verma modules andsimple modules as an adjoint module over the Witt algebra. In particular, the composition factorsare precisely determined. Introduction
It is well known that in the late 1930’s E. Witt firstly found a non-classical simple Lie algebraover prime characteristic field which is called the Witt algebra W (1) . This contributed to thestudy of non-classical simple Lie algebras which were called Cartan-type Lie algebras later. Awell-known classification result on simple modular Lie algebras asserts that each finite dimensional(restricted) simple Lie algebra over a field of prime characteristic p > is of either classical type orCartan type (cf. [1, 5]). There are four families of simple Lie algebras X ( n ) of Cartan type X for X ∈ { W, S, H, K } . They are subalgebras of derivation algebra of truncated polynomial algebras.Each simple Lie algebra of Cartan type contains a subalgebra isomorphic to the Witt algebra, whichplays a similar role as the three dimensional simple Lie algebra sl of type A in classical simpleLie algebras. In view of this point, the Witt algebra is a "fundamental" non-classical simple Liealgebra.The representation theory of the Witt algebra W (1) was firstly studied by Ho-Jui Chang in theearly 1940’s (cf. [2]). Its irreducible representations were completely determined. Based on thisresult, in the present paper we precisely determine the structure of graded restricted simple Liealgebras of Cartan type as adjoint modules over the Witt algebra. It is shown that any gradedrestricted simple Lie algebra of Cartan type can be decomposed as a direct sum of restricted babyVerma modules and simple modules over the Witt algebra. As a consequence, the compositionfactors are precisely determined. We hope the study on the decomposition of X ( n ) as a W (1) -modules will provide some useful and interesting intrinsic observation on the structure of irreduciblerestricted X ( n ) -modules and branching rules in resticted representation category for X ( n ) , where X ∈ { W, S, H, K } .This paper is organized as follows. In section 2, we introduce some basic concepts on restrictedLie algebras and their (restricted) representations, and the algebra structure on graded Lie algebrasof Cartan type. In particular, we present precisely the embedding of the Witt algebra to the fourfamilies of Lie algebras of Cartan type. Moreover, the restricted representation theory of the Wittalgebra is recalled. Section 3 is devoted to studying the decomposition of the Jacobson-Witt algebra Mathematics Subject Classification.
Key words and phrases.
Lie algebras of Cartan type, Witt algebra, restricted modules, restricted baby Vermamodules.This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11771279and 12071136), the Fundamental Research Funds of Yunnan Province (Grant No. 2020J0375) and the FundamentalResearch Funds of YNUFE (Grant No. 80059900196). as a module over the Witt algebra into a direct sum of submodules. In section 4, we first preciselygive a basis for the special algebra. By using this basis, we decompose the special algebra into adirect sum of restricted baby Verma modules and simple modules over the Witt algebra. Sections5 and 6 are devoted to determining the decomposition of the Hamiltonian algebra and the contactalgebra as direct sums of restricted baby Verma modules and simple modules over the Witt algebra,respectively. 2.
Preliminaries
Throughout this paper, F is assumed to be an algebraically closed field of prime characteristic p > . All modules (vector spaces) are over F and finite-dimensional. Set I = { , , · · · , p − } . For a finite set S , let | S | denote the number of elements in S . For a Lie algebra g , let U ( g ) be itsuniversal enveloping algebra, and Z ( g ) be the center of U ( g ) . For a g -module M , let [ M ] be theformal sum of all composition factors of M .2.1. Restricted Lie algebras and their irreducible representations.
Recall that a restrictedLie algebra g over F is a Lie algebra with a so-called restricted mapping [ p ] : g → g sending x x [ p ] satisfying that ad ( x [ p ] ) = ( ad x ) p and that ξ : g → Z ( g ) sending x x p − x [ p ] is semi-linear.For a restricted Lie algebra ( g , [ p ]) and a simple g -module M , since x p − x [ p ] ∈ Z ( g ) for any x ∈ g , the element x p − x [ p ] must act as a scalar, denoted by χ ( x ) p . The semilinearity of ξ impliesthat χ ∈ g ∗ . In general, a g -module M is said to be χ -reduced if x p · v − x [ p ] · v = χ ( x ) p v for all x ∈ g , v ∈ M . In particular, it is called a restricted module if χ = 0 . Let U χ ( g ) be the quotient ofthe universal enveloping algebra U ( g ) by the ideal generated by { x p − x [ p ] − χ ( x ) p | x ∈ g } which iscalled a χ -reduced enveloping algebra of g , i.e., U χ ( g ) = U ( g ) / ( x p − x [ p ] − χ ( x ) p | x ∈ g ¯0 ) . If χ = 0 ,the algebra U ( g ) is called the restricted enveloping algebra and denoted by u ( g ) for brevity. All the χ -reduced (resp. restricted) g -modules constitute a full subcategory of the g -module category, whichcoincides with the U χ ( g ) (resp. u ( g ) )-module category. Each simple g -module is a U χ ( g ) -modulefor a unique χ ∈ g ∗ .2.2. Graded Lie algebras of Cartan type.
Fix a positive integer n . Denote by A ( n ) the indexset { α = ( α , · · · , α n ) | ≤ α i ≤ p − , i = 1 , , · · · , n } , and denote ( p − , · · · , p − by τ for brevity. We have a truncated polynomial algebra A ( n ) which is by definition a commutativeassociative algebra with a basis { x α | α ∈ A ( n ) } , and multiplication subject to the following rule x α x β = x α + β , ∀ α, β ∈ A ( n ) , additionally with x α = 0 if α / ∈ A ( n ); x i := x ǫ i for ǫ i = ( δ , i , · · · , δ n, i ) . There is a natural graded structure on A ( n ) , and consequently a filtered structure there. Thegradation and filtration of A ( n ) induce the corresponding ones on the so-called Jacobson-Wittalgebra W ( n ) , which is the derivation algebra of A ( n ) . Then W ( n ) is free A ( n ) -module with a basis { ∂ , · · · , ∂ n } , where ∂ i ( x j ) = δ ij , ≤ i, j ≤ n . For more details, the readers are referred to thereference [6, 7].We can get other three series of subalgebras in W ( n ) , which are called (graded) Cartan type Liealgebras of series S, H , and K respectively, arising from the three exterior differentials ω S , ω H , ω K .Below, we recall the definitions, and cite some basic notations and facts we need later. The defini-tions here will be given by using some operators (cf. [6, Chapter 4]), instead of using the originaldifferential forms (cf. [4]).Set e S ( n ) = { D ∈ W ( n ) | div ( D ) = 0 } , where div ( P f i ∂ i ) = P ∂ i ( f i ) for any P f i ∂ i ∈ W ( n ) .Then by definition, the derived algebra of e S ( n ) is called the special algebra S ( n ) , i.e. S ( n ) =[ e S ( n ) , e S ( n )] . N STRUCTURE OF GRADED RESTRICTED SIMPLE LIE ALGEBRAS OF CARTAN TYPE 3
The Hamiltonian algebra is by definition H (2 r ) = F -span { D H ( x α ) | ≺ α ≺ τ } . Here D H is theHamiltonian operator from A (2 r ) to W (2 r ) defined as follows: D H : A (2 r ) −→ W (2 r ) f D H ( f ) = r X i =1 σ ( i ) ∂ i ( f ) ∂ i ′ where σ ( i ) := (cid:26) , if ≤ i ≤ r, − , if r + 1 ≤ i ≤ r, and i ′ := (cid:26) i + r, if ≤ i ≤ r,i − r, if r + 1 ≤ i ≤ r. Set e K (2 r + 1) = F -span { D K ( x α ) | α ∈ A (2 r + 1) } , where the contact operator D K from A (2 r + 1) to W (2 r + 1) is defined as follows: D K : A (2 r + 1) −→ W (2 r + 1) f D K ( f ) = r +1 X i =1 f i ∂ i where f j = x j ∂ r +1 ( f ) + σ ( j ′ ) ∂ j ′ ( f ) , j ≤ r,f r +1 = 2 f − r X i =1 σ ( j ) x j f j ′ . The contact algebra K (2 r + 1) is the derived algebra of e K (2 r + 1) , i.e. K (2 r + 1) = [ e K (2 r +1) , e K (2 r + 1)] .Both subalgebras S ( n ) and H ( n ) naturally inherit the graded and filtered structure of W ( n ) .But the contact algebra K (2 r + 1) is not a graded subalgebra of W (2 r + 1) . One can define a newgradation on K (2 r + 1) which is not inherited from the gradation of W (2 r + 1) . For that, define || α || = P r +1 i =1 α i + α r +1 − for α ∈ A (2 r + 1) and K (2 r + 1) [ i ] = F -span { D K ( x α ) | || α || = i } . Then K (2 r + 1) = L i ≥− K (2 r + 1) [ i ] is a gradation of K (2 r + 1) . Associated with this gradation, onecan also obtain the corresponding filtration K (2 r + 1) = K (2 r + 1) − ⊃ K (2 r + 1) − ⊃ · · · where K (2 r + 1) i = L j ≥ i K (2 r + 1) [ j ] . Let L = X ( n ) , X ∈ { W, S, H, K } . Then L is a restricted Lie algebra with the restricted [ p ] -mapping given by taking the p -th power of derivations. Moreover, L has a Z -grading L = ⊕ i ≥− − δ XK L [ i ] , associated with which there is a natural filtration L = L − − δ XK ⊃ L − δ XK ⊃ · · · with L j = ⊕ i ≥ j L [ i ] for j ≥ − − δ XK . KE OU AND YU-FENG YAO
Embeddings of the Witt algebra to restricted Lie algebras of Cartan type.
Definethe linear mappings from the Witt algebra to restricted Lie algebras of Cartan type as follows. Θ W : W (1) −→ W ( n ) x i ∂ x i ∂ , ∀ ≤ i ≤ p − , Θ S : W (1) −→ S ( n ) x i ∂ D ( x i x ) = x i ∂ − ix i − x ∂ , ∀ ≤ i ≤ p − , Θ H : W (1) −→ H (2 r ) x i ∂ D H ( x i x r +1 ) = x i ∂ − ix i − x r +1 ∂ r +1 , ∀ ≤ i ≤ p − , Θ K : W (1) −→ K (2 r + 1) x i ∂ D K ( x i r +1 ) , ∀ ≤ i ≤ p − . The following result asserts that the Witt algebra is a restricted subalgebra of any restricted Liealgebra of Cartan type. The proof follows from straightforward computation, we omit the details.
Lemma 2.1.
Keep notations as above. Then Θ X is an injective restricted Lie algebra homo-morphism from the Witt algebra W (1) to the restricted Lie algebra X ( n ) of Cartan type X for X ∈ { W, S, H, K } .Remark . Thanks to Lemma 2.1, any Lie algebra of Cartan type is a restricted module overthe Witt algebra under the adjoint action. We will determine this module structure in the sequelsections.2.4.
Restricted representations of the Witt algebra W (1) . In this subsection, we alwaysassume that g = W (1) is the Witt algebra over F . We will recall the known classification resultson simple g -modules given by Chang in [2]. Recall that g has a natural Z -gradation g = ⊕ p − i = − g [ i ] ,associated with which there is a filtration g = g − ⊃ g · · · ⊃ g p − ⊃ , where g i = ⊕ j ≥ i g [ j ] for − ≤ i ≤ p − . For any λ ∈ I , let F λ be the one dimensional restricted g [0] -module given bymultiplication by the scalar λ . Then we can regard F λ as a restricted g -module with trivial actionby g . Let V ( λ ) = u ( g ) ⊗ u ( g ) F λ which is called a restricted baby Verma g -module. Each restrictedbaby Verma g -module V ( λ ) has a unique simple quotient denoted by L ( λ ) for λ ∈ I . Then the set { L ( λ ) | λ ∈ I } exhausts all non-isomorphic irreducible restricted g -modules. Moreover, L ( λ ) = V ( λ ) if and only if < λ < p − . While both V (0) and V ( p − have two composition factors L (0) and L ( p − . The natural module A (1) is isomorphic to V ( p − , while the adjoint module W (1) isisomorphic to L ( p − .3. Structure of the Jacobson-Witt algebras as modules over the Witt algebra
In this section, we study the structure of the Jacobson-Witt algebra W ( n ) as a module over theWitt algebra W (1) . For that, denote x i = x i · · · x i n n for any i = ( i , · · · , i n ) ∈ I n − . Then(3.1) W ( n ) = n M j =1 M i ∈ I n − A (1) x i ∂ j . N STRUCTURE OF GRADED RESTRICTED SIMPLE LIE ALGEBRAS OF CARTAN TYPE 5
Moreover, each summand in (3.1) is a W (1) -module. More precisely, for each i ∈ I n − , we have thefollowing isomorphism as modules over the Witt algebra W (1) , A (1) x i ∂ j ∼ = ( W (1) , if j = 1 ,A (1) , if ≤ j ≤ n. Consequently, we have
Theorem 3.1.
As a module over the Witt algebra W (1) , we have (3.2) W ( n ) ∼ = V ( p − ⊕ ( n − p n − ⊕ L ( p − ⊕ p n − . Hence, (3.3) [ W ( n )] = ( n − p n − ([ L (0)] + [ L ( p − p n − [ L ( p − . Proof.
Since A (1) ∼ = V ( p − and W (1) ∼ = L ( p − as W (1) -modules, (3.2) follows. Furthermore,note that the restricted baby Verma module V ( p − has two composition factors L (0) and L ( p − .This together with (3.2) yields (3.3). (cid:3) Structure of the special algebras as modules over the Witt algebra
In this section, we study the structure of the special algebra S ( n ) as a module over the Wittalgebra W (1) . For that, for each a = ( a , · · · , a n ) ∈ I n , we define x a = x a · · · x a n n , Ω( a ) = { i | a i = p − } and ℓ ( a ) = | Ω( a ) | . In the following, if Ω( a ) = { i , · · · , i s } , we always assume that ≤ i < · · · < i s ≤ n. Basis of S ( n ) . In this subsection, we give a basis of S ( n ) . This may be known for experts.However, we do not find it in literature.Set B = { x a ∂ i | a ∈ I n , ≤ i ≤ n, and a i = 0 , a j = p − for some j = i } . and B = (cid:8) D i j i j +1 ( x a x i j x i j +1 ) | a ∈ I n with Ω( a ) = { i , · · · , i s } , ≤ j ≤ s − (cid:9) . Let V be the linear subspace of S ( n ) spanned by elements in B , and let V be the linearsubspace of S ( n ) spanned by D ij ( x a ) for a ∈ I n with a i = 0 and a j = 0 , ≤ i < j ≤ n . Note that D ij ( x b ) ∈ V if and only if x b = x a x i x j for some a ∈ I n such that a i , a j = p − . We have the following basic observation.
Lemma 4.1.
Keep notations as above. Then the following statements hold. (1) S ( n ) = V ⊕ V . (2) B is a basis of V . (3) dim( V ) = n ( p n − − , dim( V ) = np n − ( p −
1) + 1 .Proof. (1) It’s obvious that V ∩ V = { } . Moreover, if a ∈ I n with a i = 0 and a j = 0 , then D ij ( x a ) = a j x a − ǫ j ∂ i ∈ V . Hence, (1) follows.(2) is obvious.(3) The first assertion for dim( V ) follows from (2). Furthermore, it follows from (1) and [7,Theorem 3.7, Chapter 4] that dim( V ) = dim S ( n ) − dim( V ) = ( n − p n − − n ( p n − −
1) = np n − ( p −
1) + 1 . (cid:3) The following result is crucial to our final determination of a basis of S ( n ) . KE OU AND YU-FENG YAO
Lemma 4.2.
Suppose ℓ ( a ) = s and Ω( a ) = { i , · · · , i s } . Then for any k, l ∈ Ω( a ) ,D kl ( x a x k x l ) ∈ span F { D i j i j +1 ( x a x i j x i j +1 ) | ≤ j ≤ s − } . Proof.
Without loss of generality, we can suppose that k = i , l = i s . It is readily shown that { D i j i j +1 ( x a x i j x i j +1 ) | ≤ j ≤ s − } are linear independent. Recall that D kl ( x a x k x l ) = x a (( a l +1) x k ∂ k − ( a k + 1) x l ∂ l ) . Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i + 1 0 · · · a i s + 1 − ( a i + 1) . . . . . . ... . . . . . . . . . ... · · · a i s + 1 00 · · · − ( a i s − + 1) − ( a i + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 ,D kl ( x a x k x l ) ∈ span F { D i j i j +1 ( x a x i j x i j +1 ) | ≤ j ≤ s − } . (cid:3) We need the following combination formulas for later use.
Lemma 4.3.
The following equalities hold. n X s =0 sC sn ( p − s = n ( p − p n − . n X s =2 C sn ( p − s − ( s −
1) = n − X i =1 ip i − . Proof.
Take derivations on both sides of the following equations respectively, n X s =0 C sn ( x − s = x n , n X s =2 C sn ( x − s − = 1 x − x n − n ( x − −
1) = n − X i =0 x i − n, and put x = p. Then the desired equalities follows. (cid:3)
As a consequence of Lemma 4.2 and Lemma 4.3, we have
Corollary 4.4.
The subspace V has a basis B . Consequently, S ( n ) has a basis B ∪ B .Proof. By Lemma 4.2, V = X D ∈ B F D = n X s =2 X a ∈ In Ω( a )= { i , ··· ,i s } s − X j =1 F D i j i j +1 ( x a x i j x i j +1 ) . Moreover, it follows from Lemma 4.1(3) and Lemma 4.3 that | B | = n X s =2 C sn ( p − s ( s −
1) = ( p − n X s =2 C sn ( p − s − ( s −
1) = ( p − n − X i =1 ip i − = dim( V ) . Hence, B is a basis of V . This together with Lemma 4.1(1)(2) yields the second assertion. (cid:3) N STRUCTURE OF GRADED RESTRICTED SIMPLE LIE ALGEBRAS OF CARTAN TYPE 7
The structure of S ( n ) as a module over the Witt algebra. Recall the Lie algebraembedding Θ S : W (1) ֒ → S ( n ) given by Θ S ( x i ∂ ) = D ( x i x ) = x i ∂ − ix i − x ∂ , i ∈ I. In thissubsection, we study the structure of the special algebra S ( n ) as a module over the Witt algebra W (1) . For that, for each l = ( l , · · · l n ) ∈ I n − , we denote x l = x l · · · x l n n . For any ≤ i ≤ n and l ∈ I n − with l i = 0 , let N i := span F { x t x p − · · · x p − i − x p − i +1 · · · x p − n ∂ i | ≤ t ≤ p − } and N l,i := span F { x t x l ∂ i | t ∈ I } if l j = p − for some j = i . Then both N i and N l,i are submodulesover the Witt algebra. More precisely, we have Lemma 4.5.
Keep notations as above. Then as modules over the Witt algebra W (1) , we have (1) For each ≤ i ≤ n, N i ∼ = L ( p − . (2) Suppose l ∈ I n − with l = 0 and l j = p − for some j = 2 . Then N l, ∼ = V (0) . (3) For each ≤ i ≤ n, suppose l ∈ I n − with l i = 0 and l j = p − for some j = i. Then N l,i ∼ = V ( p − − l ) . Proof. (1) For each ≤ i ≤ n , let b = ( p − , · · · , p − − ( p − ǫ i ∈ I n − . Then [Θ S ( x r ∂ ) , x t x b ∂ i ] = ( t + r ) x r + t − x b ∂ i , ∀ r ∈ I, ≤ t ≤ p − . Therefore, N i is a simple W (1) -module with a maximal vector x p − x b ∂ i of weight p − . Hence, N i ∼ = L ( p − . (2) Since l ∈ I n − with l = 0 , l i = p − for some i = 2 , we have [Θ S ( x r ∂ ) , x t x l ∂ ] = ( t + r ) x r + t − x l ∂ , ∀ r, t ∈ I. This implies that N l, , as a W (1) -module, has a maximal vector x p − x l ∂ of weight . Consequently, N l, ∼ = V (0) . (3) For each i ≥ , if l ∈ I n − with l i = 0 , l j = p − for some j = i, then [Θ S ( x r ∂ ) , x t x l ∂ i ] = ( t − rl ) x r + t − x l ∂ i , ∀ r, t ∈ I. This implies that N l,i as a W (1) -module, has a maximal vector x p − x l ∂ i of weight p − − l .Consequently, N l,i ∼ = V ( p − − l ) . (cid:3) For each ≤ i < j ≤ n and a ∈ I n − , let M a,ij := span F { D ij ( x t x a ) | t ∈ I } . Lemma 4.6.
Suppose l = ( l , · · · , l n ) ∈ I n − with ℓ ( l ) = s and Ω( l ) = { i , · · · , i s } ⊆ { , · · · , n } . Then as modules over the Witt algebra W (1) , we have (1) For each ≤ j ≤ s − , M l + ǫ ij + ǫ ij +1 ,i j i j +1 ∼ = V ( p − − l ) . (2) If i = 2 , then M l + ǫ , ∼ = V ( p − − l ) . (3) If i > , then M l + ǫ i , i ∼ = V ( p − . Proof. (1) For each ≤ j ≤ s − and r, t ∈ I, [Θ S ( x r ∂ ) , D i j i j +1 ( x t x l x i j x i j +1 )] = ( t − rl ) D i j i j +1 ( x r + t − x l x i j x i j +1 ) . This implies that M l + ǫ ij + ǫ ij +1 ,i j i j +1 , as a W (1) -module, has a maximal vector D i j i j +1 ( x p − x l x i j x i j +1 ) of weight p − − l . Hence, M l + ǫ ij + ǫ ij +1 ,i j i j +1 ∼ = V ( p − − l ) . (2) For each r, t ∈ I, we have [Θ S ( x r ∂ ) , D ( x t x l + ǫ )] = ( t − rl − r ) D ij ( x r + t − x l + ǫ ) . This implies that M l + ǫ , , as a W (1) -module, has a maximal vector D ( x p − x l + ǫ ) of weight p − − l . Hence, M l + ǫ , ∼ = V ( p − − l ) . KE OU AND YU-FENG YAO (3) Since i > , l = p − . For each r, t ∈ I, we have [Θ S ( x r ∂ ) , D i ( x t x l + ǫ i )] = tD i ( x r + t − x l + ǫ i ) . This implies that M l + ǫ i , i , as a W (1) -module, has a maximal vector D ( x p − x l + ǫ i ) of weight p − . Hence, M l + ǫ i , i ∼ = V ( p − . (cid:3) We are now in the position to present the following main result on the decomposition of S ( n ) asa direct sum of W (1) -modules. Theorem 4.7.
As a W (1) -module, we have S ( n ) ∼ = V (0) ⊕ (cid:0) ( n − p n − − (cid:1) ⊕ V ( p − ⊕ (cid:0) ( n − p n − − (cid:1) ⊕ (cid:16) p − M i =1 L ( i ) ⊕ ( n − p n − (cid:17) ⊕ L ( p − ⊕ ( n − . Proof.
Set V := span F { x a ∂ | a ∈ I n with a = 0 and a j = p − for some j = 1 } ,V := span F { x a ∂ i | ≤ i ≤ n, a ∈ I n with a i = 0 and a j = p − for some j = i } . Then V = V ⊕ V as a vector space. Moreover, both V and V + V are W (1) -modules, and S ( n ) = V ⊕ ( V + V ) . It follows from Lemma 4.5 that V = U ⊕ U ⊕ U ⊕ U , where U = n M i =2 N i ∼ = L ( p − ⊕ ( n − , U = M l ∈ I n − , l =0 l =(0 ,p − , ··· ,p − N l, ∼ = V (0) ⊕ ( p n − − ,U = n M i =3 p − M j =0 M l ∈ I n − l i =0 , l = j N l,i ∼ = n M i =3 p − M j =0 M l ∈ I n − l i =0 , l = j V ( p − − j ) ∼ = p − M i =1 V ( i ) ⊕ ( n − p n − ,U = n M i =3 M l ∈ I n − l i =0 , l = p − l = τ − ( p − ǫ i N l,i ∼ = n M i =3 M l ∈ I n − l i =0 , l = p − l = τ − ( p − ǫ i V (0) ∼ = V (0) ⊕ ( n − p n − − . Since V ( i ) = L ( i ) for ≤ i ≤ p − (see §2.4), it follows that V ∼ = V (0) ⊕ (( n − p n − − p n − − ⊕ (cid:16) p − M i =1 L ( i ) ⊕ ( n − p n − (cid:17) (4.1) ⊕ L ( p − ⊕ ( n − ⊕ V ( p − ⊕ ( n − p n − . It follows from Corollary 4.4 and Lemma 4.6 that V + V = W ⊕ W ⊕ W ⊕ W , where W = n − M s =2 M b ∈ I n − , i =2 E ( b )= { i , ··· ,i s } s − M j =1 M b + ǫ ij + ǫ ij +1 ,i j i j +1 , W = n − M s =2 M b ∈ I n − , i > E ( b )= { i , ··· ,i s } s − M j =1 M b + ǫ ij + ǫ ij +1 ,i j i j +1 ,W = n − M s =1 M b ∈ I n − , i =2 E ( b )= { i , ··· ,i s } M b + ǫ , , W = n − M s =1 M b ∈ I n − , i > E ( b )= { i , ··· ,i s } M b + ǫ i , i . N STRUCTURE OF GRADED RESTRICTED SIMPLE LIE ALGEBRAS OF CARTAN TYPE 9
Thanks to Lemma 4.3, we have n − X s =2 ( s − C s − n − ( p − s − = ( n − p − p n − , and n − X s =2 ( s − C sn − ( p − s = ( n − p − p n − − ( p n − − . Moreover, It follows from Lemma 4.6 that W ∼ = p − M i =1 n − M s =2 V ( i ) ⊕ ( s − C s − n − ( p − s − = p − M i =1 V ( i ) ⊕ ( n − p − p n − ,W ∼ = n − M s =2 V (0) ⊕ ( s − C sn − ( p − s = V (0) ⊕ (( n − p − p n − − p n − +1) ,W ∼ = p − M i =0 V ( i ) ⊕ ( L n − s =1 C s − n − ( p − s − ) = p − M i =0 V ( i ) ⊕ p n − ,W ∼ = n − M s =1 V ( p − ⊕ C sn − ( p − s = V ( p − ⊕ ( p n − − . Since V ( i ) = L ( i ) for ≤ i ≤ p − (see §2.4), it follows that V + V ∼ = V (0) ⊕ (( n − p − p n − +1) ⊕ (cid:16) p − M i =1 L ( i ) ⊕ (( n − p − p n − + p n − ) (cid:17) ⊕ V ( p − ⊕ (( n − p − p n − + p n − − . (4.2)Now the desired assertion follows from (4.1) and (4.2). (cid:3) As a consequence of Theorem 4.7, we further have
Corollary 4.8.
As a module over the Witt algebra W (1) , [ S ( n )] = (2( n − p n − − n + 1)[ L (0)] + p − X i =1 ( n − p n − [ L ( i )] + 2( n − p n − [ L ( p − . Proof.
Since [ V (0)] = [ V ( p − L (0)] + [ L ( p − , the assertion follows directly from Theorem4.7. (cid:3) Structure of the Hamiltonian algebras as modules over the Witt algebra
Recall the Lie algebra embedding Θ H : W (1) ֒ → H (2 r ) given by Θ H ( x i ∂ ) = − D H ( x i x r +1 ) = x i ∂ − ix i − x r +1 ∂ r +1 , i ∈ I. In this section, we study the structure of the Hamiltonian algebra H (2 r ) as a module over the Witt algebra W (1) .We first investigate the case H (2) . Set H j = span F { D H ( x i x j ) | ≤ i ≤ p − } , if j = 0 , span F { D H ( x i x j ) | i ∈ I } , if 1 ≤ j ≤ p − , span F { D H ( x i x j ) | ≤ i ≤ p − } , if j = p − . Since(5.1) [Θ H ( x s ∂ ) , D H ( x i x j )] = ( i − sj ) D H ( x s + i − x j ) , ∀ i, j, s ∈ I, each H j is a W (1) -module for any j ∈ I . Moreover, we have the following decomposition of H (2) as a W (1) -module. Lemma 5.1.
As a W (1) -module, H (2) is completely reducible and H (2) ∼ = (cid:16) p − M i =1 L ( i ) (cid:17) ⊕ L ( p − ⊕ . In particular, (5.2) [ H (2)] = p − X j =1 [ L ( j )] + 2[ L ( p − . Proof.
For any ≤ j ≤ p − , it follows from (5.1) that H j has a maximal vector x p − x j ofweight p − − j . Since H j is p -dimensional, H j ∼ = V ( p − − j ) ∼ = L ( p − − j ) as W (1) -modules.Furthermore, it again follows from (5.1) that H has a maximal vector x p − of weight p − , and H p − has a maximal vector x p − x p − of weight p − . And dim H = dim H p − = p − , it follows that H ∼ = H p − ∼ = L ( p − as W (1) -modules. Since H (2) = ⊕ p − j =0 H j , the desired assertion follows. (cid:3) Remark . (5.2) was obtained in [3, Lemma 2.6].In general, for r > , j ∈ I , l = ( l , · · · , l r , l r +2 , · · · , l r ) ∈ I r − , let x l := x l · · · x l r r x l r +2 r +2 · · · x l r r , H j,l := span F { D H ( x i x jr +1 x l ) | i ∈ I such that ( i, j, l ) = (0 , · · · , , ( p − , · · · , p − } , and H l := span F { D H ( x i x kr +1 x l ) | i, k ∈ I such that ( i, k, l ) = (0 , · · · , , ( p − , · · · , p − } . Since(5.3) [Θ H ( x s ∂ ) , D H ( x i x jr +1 x l )] = ( i − sj ) D H ( x s + i − x jr +1 x l ) , ∀ i, j, s ∈ I, l ∈ I n − , both H t,l and H l are W (1) -modules for any t ∈ I, l ∈ I r − . The following result describes the W (1) -module structure on H l . Lemma 5.3.
Keep notations as above, then the following decompositions hold as W (1) -modules. (1) If l = (0 , · · · , , then H l ∼ = V (0) ⊕ (cid:16) p − M i =1 L ( i ) (cid:17) . (2) If l = ( p − , · · · , p − , then H l ∼ = (cid:16) p − M i =1 L ( i ) (cid:17) ⊕ V ( p − . (3) If l = (0 , · · · , and ( p − , · · · , p − , then H l ∼ = V (0) ⊕ (cid:16) p − M i =1 L ( i ) (cid:17) ⊕ V ( p − . Proof.
Note that H l = L j ∈ I H j,l . We need to determine the W (1) -module structure of H t,l for any t ∈ I . We just show the assertion for the case l = (0 , · · · , . Similar arguments yield the assertionfor the other cases.In the following, we always assume that l = (0 , · · · , . For each ≤ j ≤ p − , it follows from(5.3) that H j,l , as a W (1) -module, contains a maximal vector D H ( x p − x jr +1 x l ) of weight p − − j . N STRUCTURE OF GRADED RESTRICTED SIMPLE LIE ALGEBRAS OF CARTAN TYPE 11
Since dim H j,l = p for ≤ j ≤ p − , H j,l ∼ = V ( p − − j ) as a W (1) -module. While dim H ,l = p − ,we have H ,l ∼ = L ( p − . Hence, H l = M j ∈ I H j,l ∼ = L ( p − ⊕ (cid:16) p − M j =1 V ( p − − j ) (cid:17) ∼ = V (0) ⊕ (cid:16) p − M i =1 L ( i ) (cid:17) . (cid:3) We are now in the position to present the following main result on the structure of the Hamiltonianalgebra H (2 r ) as a module over the Witt algebra W (1) . Theorem 5.4.
As a module over the Witt algebra W (1) , we have H (2 r ) ∼ = V (0) ⊕ ( p r − − ⊕ (cid:16) p − M i =1 L ( i ) ⊕ p r − (cid:17) ⊕ V ( p − ⊕ ( p r − − ⊕ L ( p − ⊕ . In particular, [ H (2 r )] = (2 p r − − L (0)] + p r − p − X j =1 [ L ( j )] + 2 p r − [ L ( p − . Proof.
For r = 1 , the assertion follows from Lemma 5.1. While for r > , H (2 r ) = L l ∈ I r − H l ,then the desired assertion follows directly from Lemma 5.3. (cid:3) Structure of the contact algebras as modules over the Witt algebra
Recall the Lie algebra embedding Θ K : W (1) ֒ → K (2 r + 1) given by Θ K ( x i ∂ ) = D K ( x i r +1 ) = P rj =1 i x i − r +1 x j ∂ j + x i r +1 ∂ r +1 , i ∈ I. In this section, we study the structure of the contact algebra K (2 r + 1) as a module over the Witt algebra W (1) .For i ∈ I , set Γ i := l = ( l , · · · l r ) ∈ I r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) r X j =1 l j − (cid:17) ≡ i ( mod p ) . For arbitrary i ∈ I and ( l , · · · , l r − ) ∈ I r − , there is a unique l r ∈ I such that l r ≡ i + 4 − r − X j =1 l j (mod p ) . Hence, | Γ i | = p r − for any i ∈ I .For any l = ( l , · · · l r ) ∈ I r , we denote x l := x l · · · x l r r and K l := ( span F { x l x t r +1 | ≤ t ≤ p − } , if r + 4 ≡ mod p ) and l = τ − ( p − ǫ r +1 , span F { x l x t r +1 | ≤ t ≤ p − } , otherwise.Since(6.1) [Θ K ( x i ∂ ) , x l x t r +1 ] = 12 (cid:16) i (cid:16) r X j =1 l j − (cid:17) + 2 t (cid:17) x l x t + i − r +1 , ∀ i, t ∈ I, l ∈ I r ,K l is a W (1) -module for any l ∈ I r . More precisely, we have Lemma 6.1.
Keep notations as above, then for any l = ( l , · · · l r ) ∈ I r , as a W (1) -module K l ∼ = ( L ( p − , if r + 4 ≡ mod p ) and l = τ − ( p − ǫ r +1 ,V (cid:0) (cid:0) P rj =1 l j (cid:1) − (cid:1) , otherwise. Proof. If r + 4 ≡ mod p ) and l = τ − ( p − ǫ r +1 , it follows from (6.1) that x l x p − r +1 is a maximalvector of weight p − in K l , so that K l , as a W (1) -module, is a homomorphic image of V ( p − .Furthermore, since dim K l = p − , it follows that K l ∼ = L ( p − .If r + 4 mod p ) or l = τ − ( p − ǫ r +1 , it follows from (6.1) that x l x p − r +1 is a maximalvector of weight (cid:0) P rj =1 l j (cid:1) − in K l . This together with the dimension of K l yields that K l ∼ = V (cid:0) (cid:0) P rj =1 l j (cid:1) − (cid:1) . (cid:3) As a consequence of Lemma 6.1, we have the following main result on the decomposition of thecontact algebra K (2 r + 1) as a module over the Witt algebra W (1) . Theorem 6.2.
As a W (1) -module, we have K (2 r +1) ∼ = V (0) ⊕ ( p r − − ⊕ V ( p − ⊕ p r − ⊕ (cid:16) p − L i =1 L ( i ) ⊕ p r − (cid:17) ⊕ L ( p − , if r + 4 ≡ mod p ) ,V (0) ⊕ p r − ⊕ V ( p − ⊕ p r − ⊕ (cid:16) p − L i =1 L ( i ) ⊕ p r − (cid:17) , if r + 4 mod p ) . Consequently, [ K (2 r + 1)] = (2 p r − − L (0)] + p − P i =1 p r − [ L ( i )] + 2 p r − [ L ( p − , if r + 4 ≡ mod p ) , p r − [ L (0)] + p − P i =1 p r − [ L ( i )] + 2 p r − [ L ( p − , if r + 4 mod p ) . Proof.
Note that K (2 r + 1) = L l ∈ I r K l , It follows from Lemma 6.1 that K (2 r +1) ∼ = V (0) ⊕ ( | Γ |− ⊕ V ( p − ⊕| Γ p − | ⊕ (cid:16) p − L i =1 L ( i ) ⊕| Γ i | (cid:17) ⊕ L ( p − , if r + 4 ≡ mod p ) ,V (0) ⊕| Γ | ⊕ V ( p − ⊕| Γ p − | ⊕ (cid:16) p − L i =1 L ( i ) ⊕| Γ i | (cid:17) , if r + 4 mod p ) . Since | Γ i | = p r − for any i ∈ I , the first assertion holds. Moreover, since [ V (0)] = [ V ( p − L (0)] + [ L ( p − , the second assertion follows. (cid:3) Acknowledgment
We would like to thank Prof. A. Premet for helpful discussion on the basisof S ( n ) . References [1] R. E. Block and R. L. Wilson,
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School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming,650221, China.
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