Irreducible modules over finite simple Lie pseudoalgebras IV. Non-primitive pseudoalgebras
aa r X i v : . [ m a t h . QA ] M a y IRREDUCIBLE MODULES OVER FINITE SIMPLE LIEPSEUDOALGEBRAS IV. NON-PRIMITIVE PSEUDOALGEBRAS
ALESSANDRO D’ANDREA
Abstract.
Let d ⊂ d ′ be finite-dimensional Lie algebras, H = U ( d ) , H ′ = U ( d ′ ) the corresponding universal enveloping algebras endowed with the canon-ical commutative Hopf algebra structure. We show that if L is a primitive Liepseudoalgebra over H then all finite irreducible L ′ = Cur H ′ H L -modules are ofthe form Cur H ′ H V , where V is an irreducible L -module, with a single class of ex-ceptions. Indeed, when L ≃ H ( d , χ, ω ), we introduce non current L ′ -modules V H χ,ω,t, d ′ ( R ) that are obtained by modifying the current pseudoaction withan extra term depending on an element t ∈ d ′ \ d , which must satisfy sometechnical conditions. This, along with results from [BDK1, BDK2, BDK3],completes the classification of finite irreducible modules of finite simple Liepseudoalgebras over the universal enveloping algebra of a finite-dimensionalLie algebra. Introduction
This is the last issue in a series of papers addressing the structure and represen-tations of simple Lie pseudoalgebras over a cocommutative Hopf algebra H = U ( d ),where d is a finite-dimensional Lie algebra. A classification of finite irreducible mod-ules over all primitive simple Lie pseudoalgebras [BDK] has already been achievedin [BDK1, BDK2, BDK3]; in this paper, I give a complete description of finiteirreducible representations of non-primitive ones.The Lie pseudoalgebra language provides a common generalization to both Liealgebras and Lie conformal algebras [DK], that are strictly related to algebraicproperties of the Operator Product Expansion in vertex algebras [K]. Finite simpleLie pseudoalgebras over H = U ( d ) have been classified in [BDK]: they all arisefrom the so-called primitive simple Lie pseudoalgebra by means of a current con-struction. The list of primitive Lie H -pseudoalgebras, up to isomorphism, is givenin Section 4. Apart from simple finite-dimensional Lie algebras, which occur when d = (0), they all arise as subalgebras of W ( d ), see Example 4.2, and are denoted by S ( d , χ ) , H ( d , χ, ω ) , K ( d , θ ). While W ( d ) and S ( d , χ ) exist for all choices of the Liealgebra d , and non isomorphic examples of S ( d , χ ) are parametrized by 1-cocycles χ ∈ d ∗ , instances of H ( d , χ, ω ), K ( d , θ ) depend on more elusive properties that d must satisfy, see [BDK].Representation theory of finite-dimensional simple Lie algebras is certainly wellknown. Finite irreducible representations of the Lie pseudoalgebras W ( d ) and S ( d , χ ), H ( d , χ, ω ), K ( d , θ ) have been classified in [BDK1, BDK2, BDK3] respec-tively: the main result is that each irreducible L -module, where L is one of the above Date : May 18, 2020.
Lie pseudoalgebras, arises as a quotient of special representations called tensor mod-ules . These are parametrized by finite-dimensional irreducible representation of afinite-dimensional Lie algebra associated with L ; only finitely many of the abovetensor modules fail irreducibility, and then fit nicely in complexes that providepseudoalgebraic translations of differential-geometric constructions such as the deRham complex, along with its generalizations by Rumin [Ru] and Eastwood [E] inthe context of contact and conformally symplectic geometry. The above results areclosely connected with the study of finite height representations of Cartan type Liealgebras undertaken by Rudakov [R1, R2] and Kostrikhin [Ko].It is not difficult to show, as in Corollary 6.1, that if V is an irreducible rep-resentation of any Lie pseudoalgebra L , then the current construction yields anirreducible representation Cur H ′ H V of the current Lie pseudoalgebra Cur H ′ H L . Theconverse does not hold in general, as one may verify by choosing L to be an abelianLie pseudoalgebra L . However, when L is simple, the only case that has been in-vestigated, apart from the trivial one H = H ′ , is H = k , H ′ = k [ ∂ ] from [CK],where it is shown that all finite irreducible representations of the current Lie con-formal algebra Cur g = Cur k [ ∂ ] k g arise as currents of irreducible g -modules. It isthus tempting to conjecture that finite irreducible modules of Cur H ′ H L are alwayscurrent representations whenever L is a simple primitive Lie pseudoalgebra L .The present paper shows that this expectation holds, with a single class of excep-tions: indeed, the simple Lie pseudoalgebra Cur H ′ H H ( d , χ, ω ) may have noncurrentirreducible representations, that are thoroughly classified. The main result is thefollowing Theorem.
Let d ⊂ d ′ be finite-dimensional Lie algebras, H ⊂ H ′ their universalenveloping algebras endowed with the canonical cocommutative Hopf algebra struc-ture. The following is a complete list of finite irreducible representations of thecurrent Lie pseudoalgebra L ′ = Cur H ′ H L , where L is a primitive Lie pseudoalgebra: — Cur H ′ H V , where V is a finite irreducible L -module; — V H χ,ω,t, d ′ ( R ) , where L = H ( d , χ, ω ) , R is a finite-dimensional irreduciblerepresentation of d + ⊕ sp ( d , ω ) , and t ∈ d ′ \ d satisfies(i) ad χ t preserves d and lies in sp ( d , ω ) ;(ii) [ s, t ] = 0 , where s satisfies χ = ι s ω . A description of all nontrivial isomorphism between the above irreducible mod-ules is also provided. Notice that here d + denotes the extension0 → k χ → d + → d → d by the one-dimensional abelian ideal of 1-cocycle χ corresponding to the 2-cocycle ω and ad χ t : d → d ′ is defined as (ad χ t )( ∂ ) := [ t, ∂ ] + χ ( ∂ ) t . Noncurrentrepresentations V H χ,ω,t, d ′ ( R ) are introduced in Section 9. They are obtained byadding an extra term ( t ⊗ ⊗ H ′ (1 ⊗ u ) to the expression providing the pseudoaction e ∗ (1 ⊗ u ) in a current tensor module Cur H ′ H V H χ,ω ( R ).The special behaviour of H ( d , χ, ω ) depends on the presence of nontrivial centralelements in the corresponding annihilation Lie algebra. This fact also plays a majorrole in the less standard description from [BDK3] of finite irreducible representa-tions of H ( d , χ, ω ) when compared to the more unified treatment of other primitivetypes. RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 3
Unfortunately, many conflicting notations from [BDK1, BDK2, BDK3] have tobe resolved in this paper. Explanations are provided in footnotes whenever needed.2.
Hopf algebra preliminaries
Hopf notation.
In this paper all vector spaces, algebras and tensor productsare, unless otherwise specified, over an algebraically closed field k of characteristiczero. We will deal with pairs d ⊂ d ′ of finite dimensional Lie algebras and denoteby H , respectively H ′ , the universal enveloping algebra U ( d ), resp. U ( d ′ ). No otherHopf algebras will be considered with the exception of such universal envelopingalgebras.Notice that both H and H ′ are Hopf algebras with respect to the coproduct ∆,antipode S , and counit ε given by:∆( ∂ ) = ∂ ⊗ ⊗ ∂ , S ( ∂ ) = − ∂ , ε ( ∂ ) = 0 , ∂ ∈ d ′ . (2.1)More precisely, H ⊂ H ′ is a Hopf subalgebra, so that the inclusion ι : H → H ′ is aHopf-algebra homomorphism. We will employ the notation:∆( h ) = h (1) ⊗ h (2) = h (2) ⊗ h (1) , (2.2)(∆ ⊗ id)∆( h ) = (id ⊗ ∆)∆( h ) = h (1) ⊗ h (2) ⊗ h (3) , (2.3)( S ⊗ id)∆( h ) = h ( − ⊗ h (2) , h ∈ H . (2.4)Then the antipode and counit axioms can be written as follows: h ( − h (2) = h (1) h ( − = ε ( h ) , (2.5) ε ( h (1) ) h (2) = h (1) ε ( h (2) ) = h, (2.6)while the fact that ∆ is a homomorphism of algebras translates as:( f g ) (1) ⊗ ( f g ) (2) = f (1) g (1) ⊗ f (2) g (2) , f, g ∈ H. (2.7)Eqs. (2.5), (2.6) imply the following useful relations: h ( − h (2) ⊗ h (3) = 1 ⊗ h = h (1) h ( − ⊗ h (3) . (2.8)Set dim d = N, dim d ′ = N + r . If { ∂ , . . . , ∂ N + r } is a basis of d ′ , we denote by { x , . . . , x N + r } the corresponding dual basis of d ′∗ . The structure constant c kij of d ′ , are defined by [ ∂ i , ∂ j ] = N + r X k =1 c kij ∂ k , i, j = 1 , . . . , N + r . (2.9)We have a corresponding (reduced) Poincar´e-Birkhoff-Witt basis ∂ ( K ) = ∂ k · · · ∂ k N + r N + r /k ! · · · k N + r ! , K = ( k , . . . , k N + r ) ∈ Z N + r + (2.10)of H ′ . Remark . PBW bases may be used to show that H ′ is free both as a left and asa right H -module. The embedding ι : H ֒ → H ′ is thus a pure homomorphism. A. D’ANDREA
Straightening.
Let M be a left H -module. The coproduct ∆ : H → H ⊗ H makes H ⊗ H into a right H -module, so that it makes sense to consider the tensorproduct ( H ⊗ H ) ⊗ H M , which is a left H ⊗ H -module by left multiplication onthe first ⊗ H factor. Elements lying in ( H ⊗ H ) ⊗ H M may be expressed in verymany ways, and this can make it difficult to verify whether two distinct expressionsactually describe the same quantity. This problem is solved by the left- and right-straightening techniques, introduced in [BDK], that we recall here briefly. Proposition 2.1.
Let M , resp. N , be a right, resp. left, H -module. Then theassignments M ⊗ N ∋ m ⊗ n ( m ⊗ ⊗ H n ∈ ( M ⊗ H ) ⊗ H N (2.11) M ⊗ N ∋ m ⊗ n (1 ⊗ m ) ⊗ H n ∈ ( H ⊗ M ) ⊗ H N (2.12) extend to linear isomorphisms.Proof. One checks easily that the maps( M ⊗ H ) ⊗ H N ∋ ( m ⊗ h ) ⊗ H n mh ( − ⊗ h (2) n (2.13)( H ⊗ M ) ⊗ H N ∋ ( h ⊗ m ) ⊗ H n mh ( − ⊗ h (1) n (2.14)extend to explicit inverses to (2.11), (2.12). (cid:3) Corollary 2.1.
Let N be a left H -module. Every element α ∈ ( H ⊗ H ) ⊗ H N canbe expressed either in the form α = X i ( h i ⊗ ⊗ H m i , h i ∈ H, m i ∈ N (2.15) or in the form α = X i (1 ⊗ k i ) ⊗ H n i , k i ∈ H, n i ∈ N. (2.16) Similarly, every element in ( H ⊗ H ⊗ H ) ⊗ H N has a unique representative both in ( H ⊗ H ⊗ k ) ⊗ H N and in ( k ⊗ H ⊗ H ) ⊗ H N .Proof. Use M = H , resp. H ⊗ H , in Proposition 2.1. (cid:3) Corollary 2.2.
Let N ⊂ M be left H -modules, and α = X i ( h i ⊗ ⊗ H m i ∈ ( H ⊗ H ) ⊗ H M = X i (1 ⊗ k i ) ⊗ H n i , where h i , resp. k i , are linearly independent elements in H . Then α lies in ( H ⊗ H ) ⊗ H N if and only if m i ∈ N , or equivalently n i ∈ N , for every i .Remark . The above corollary shows that there always exists a smallest N ⊂ M such that α ∈ ( H ⊗ H ) ⊗ H N , which may be computed by taking the left- or right-straightened expression for α , and considering the H -linear span of all elements onthe right of ⊗ H .Corollary 2.2 also allows one to check equalities in ( H ⊗ H ) ⊗ H N and ( H ⊗ H ⊗ H ) ⊗ H N . It is enough to straighten everything, say to the left, and then verify theequality in the vector spaces H ⊗ N , H ⊗ H ⊗ N respectively, which is a trivial job.We end this section with a technical statement which can be thought of as apartial straightening. We split the set N n of PBW indices for H = U ( d ) as follows:elements of R := N k × { } ⊂ N n are supported on the first k indices, whereasthose lying in S := { } × N n − k vanish on the first k indices, so that they are RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 5 supported on the subsequent n − k ones. Clearly, if K i ∈ R, L i ∈ S, i = 1 , , then K + L = K + L if and only if K = K , L = L . Moreover, the sum of twoelements in N n lies in R (resp. S ) if and only if both summands lie in R (resp. S ). Lemma 2.1.
Let U ⊂ V be left H -modules. Then the element α = X K ∈ R,L ∈ S ( ∂ ( K ) ⊗ ∂ ( L ) ) ⊗ H v K + L ∈ ( H ⊗ H ) ⊗ H V (2.17) lies in ( H ⊗ H ) ⊗ H U precisely when all v J , J ∈ N n , lie in U .Proof. Clearly, if all v K,L lie in U , then α ∈ ( H ⊗ H ) ⊗ H U . As for the converse,assume α ∈ ( H ⊗ H ) ⊗ H U and left-straighten (2.17) to obtain α = X K ∈ R X L ,L ∈ S ( ∂ ( K ) S ( ∂ ( L ) ) ⊗ ⊗ H ∂ ( L ) v K + L + L (2.18)= X K ∈ R X L ,L ∈ S ( ∂ ( K + L ) ) ⊗ ⊗ H ( − | L | ∂ ( L ) v K + L + L (2.19)= X J ∈ Z n ( ∂ ( J ) ⊗ ⊗ H X L ∈ S ± ∂ ( L ) v J + L ∈ ( H ⊗ H ) ⊗ H U, (2.20)where we set J = K + L . Then Corollary 2.2 shows that X L ∈ S ± ∂ ( L ) v J + L (2.21)lies in U for each choice of J ∈ Z n , L ∈ S .Now, proceed by contradiction and assume that J ∈ Z n is (lexicographically)maximal such that v J / ∈ U . Then (2.21) rewrites as the sum ± v J + X = L ∈ S ± ∂ ( L ) v J + L ∈ U. However, the summation on the right lies in the H -submodule U ⊂ V by maximalityof J , whence v J ∈ U , yielding a contradiction. (cid:3) Remark . It is important to stress here that the above Lemma 2.1 may be appliedto any totally ordered basis of a Lie algebra. In particular, it may be applied to anybasis of d ′ , and not just those where the last elements constitute a basis of d ⊂ d ′ as we will consider below; even in this case, k does not need to equal the differencedim d ′ − dim d but may be any number. Corollary 2.3.
Let V be a left H -module. Then equality X K ∈ R,L ∈ S ( ∂ ( K ) ⊗ ∂ ( L ) ) ⊗ H u K + L = X K ∈ R,L ∈ S ( ∂ ( K ) ⊗ ∂ ( L ) ) ⊗ H v K + L holds in ( H ⊗ H ) ⊗ H V if and only if u N = v N for every N ∈ N n .Proof. Use U = 0 in Lemma 2.1. (cid:3) Filtrations.
The PBW basis may be used to set up a canonical increasingfiltration of H ′ given byF p H ′ = span k { ∂ ( K ) | | K | ≤ p } , where | K | = k + · · · + k N + r . (2.22)This filtration does not depend on the choice of basis of d ′ , and is compatiblewith the Hopf algebra structure of H . We have: F − H ′ = { } , F H ′ = k , andF H ′ = k ⊕ d ′ . A. D’ANDREA
The dual X ′ := Hom k ( H ′ , k ) is a commutative associative algebra and inheritsan induced decreasing filtration F i X ′ = (F i H ′ ) ⊥ making it into a linearly compactvector space. We will identify d ′∗ as a subspace of X ′ by letting h x i , ∂ i i = 1 and h x i , ∂ ( I ) i = 0 for all other basis vectors (2.10). Mapping x i t i gives rise to anisomorphism from X to the algebra k [[ t , . . . , t N + r ]] of formal power series in N + r indeterminates. Notice that the above filtration satisfies F − X ′ = X ′ , whereasF X ′ is the (maximal) ideal generated by x , . . . , x N + r . Similarly, F p X ′ coincideswith (F X ′ ) p +1 . Remark . The choice of indices in the filtration for X ′ is natural, but somewhatclumsy, and one has (F p X ′ )(F q X ′ ) = F p + q +1 X ′ so that indices do not add upproperly. This will later have consequences in choosing the right indexing of thefiltration of annihilation algebras. Remark . If H = U ( d ), then H ⊗ H = U ( d ) ⊗ U ( d ) ≃ U ( d ⊕ d ). We willoccasionally denote by F p ( H ⊗ H ) the corresponding filtration, which satisfiesF p ( H ⊗ H ) = P i + j = p F i H ⊗ F j H .The Lie algebra d ′ has left and right actions on X ′ by derivations, given by h ∂x, h i = −h x, ∂h i , (2.23) h x∂, h i = −h x, h∂ i , ∂ ∈ d ′ , x ∈ X ′ , h ∈ H ′ , (2.24)where ∂h and h∂ are the products in H ′ . These two actions coincide only when d ′ is abelian, the difference ∂x − x∂ giving the coadjoint action of ∂ ∈ d ′ on x ∈ X ′ : h ∂x − x∂, h i = −h x, [ ∂, h ] i . Splitting a projection.
Throughout the rest of the paper, we will chose abasis { ∂ , . . . , ∂ N + r } of d ′ so that the last N elements ∂ r +1 , . . . , ∂ N + r are a basisof d ⊂ d ′ . Then the PBW basis of H gets identified to a subset of the PBW basisof H ′ ; correspondingly, the subalgebra k [[ x r +1 , . . . , x N + r ]] ⊂ X ′ may be identifiedwith X . Notice that the inclusion ι : H → H ′ induces a surjective commutativealgebra homomorphism ι ∗ : X ′ → X whose kernel we denote by I = ker ι ∗ , and theabove identification provides a splitting ς : X → X ′ to ι ∗ , so that X ′ = X + I isa direct sum decomposition (as vector spaces). More specifically, ι ∗ is the uniquehomomorphism mapping x , . . . , x r to 0 and fixing each x r +1 , . . . , x N + r ; thus, I ⊂ X ′ is the ideal generated by x , . . . , x r . Remark . Denote by H + the augmentation ideal of H . It is easy to show that thesubalgebra k [[ x , . . . , x r ]] ⊂ X ′ equals ( H ′ H + ) ⊥ and is thus canonically determined,and independent of the choice of the basis of d ′ .Henceforth, we will denote this canonical subalgebra of X ′ by O , and its uniquemaximal ideal by m = h x , . . . , x r i . Notice that one has an isomorphism X ′ ≃O b ⊗ X of linearly compact vector spaces, which identifies I with m b ⊗ X .It is important to highlight the fact that we are provided with two distinct pairsof left and right actions of d on X : one is as above in the case where r = 0; the otheris obtained by restricting to the subalgebra d the action of d ′ on X ≃ ς ( X ) ⊂ X ′ .In general, these actions will differ, but they are still nicely compatible. RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 7
Proposition 2.2. (i) The right action of d ⊂ d ′ on X ′ preserves X ⊂ X ′ and coincides with the natural right action of d on X . Moreover, the rightaction of d on x , . . . , x r is trivial.(ii) The right action of d on X ′ is O -linear.(iii) The restriction to X ⊂ X ′ of the left action of d ⊂ d ′ on X ′ coincides, upto elements in I , with the natural left action of d on X .(iv) I is stabilized by both the left- and right- action of d ⊂ d ′ on X ′ .Proof.(i) Let ∂ ∈ d . Then by definition h x k ∂, h i = −h x k , h∂ i . When h ∈ U ( d ) ⊂ U ( d ′ ), then h∂ lies in the augmentation ideal of U ( d ); consequently, when h is a member of the PBW basis (2.10) of U ( d ′ ), the only contributionto h x k , ∂ ( K ) · ∂ i , where r + 1 ≤ k ≤ N + r , will arise from those K thatare supported in indices r + 1 through N + r . Thus, the right action of ∂ ∈ d , viewed as an element of d ′ , on x k ∈ X , viewed as an element of X ′ , will coincide with the right action of ∂ on x k . The first claim nowfollows by observing that the right action od d ′ on X ′ is by (continuous)derivations, and that x k , r + 1 ≤ k ≤ N + r, are (topological) algebragenerators of X ⊂ X ′ , so that the right action of d ⊂ d ′ on X ⊂ X ′ isuniquely determined by its action on elements x k , r + 1 ≤ k ≤ N + r .As for the second claim, we similarly argue that if k ≤ r and ∂ ∈ d , then h x k ∂, ∂ ( K ) i = −h x k , ∂ ( K ) ∂ i vanishes, as ∂ ( K ) ∂ lies in the left ideal of U ( d ′ )generated by the augmentation ideal of U ( d ) ⊂ U ( d ′ ), which lies in ( x k ) ⊥ . (ii) Follows immediately from part (i) and continuity of the action of d ′ on X ′ . (iii) It is enough to compute h ∂x k , ∂ ( K ) i = −h x k , ∂∂ ( K ) i when r +1 ≤ k ≤ N + r and K is supported over the same indices, as other choices of K will yieldcontributions lying in the ideal I . However, only the structure of d isinvolved in the computation of the above expressions h x k , ∂∂ ( K ) i . (iv) The ideal I = ker ι ∗ ⊂ X ′ coincides with H ⊥ . The claim follows fromthe fact that both left- and right- multiplication by elements of d stabilize H ⊂ H ′ . (cid:3) Lie pseudoalgebra preliminaries
Pseudoalgebraic definitions.
Let H = U ( d ). An H -pseudoalgebra is a left H -module L endowed with an H ⊗ H -linear pseudoproduct L ⊗ L → ( H ⊗ H ) ⊗ H L ,where ( H ⊗ H ) ⊗ H L is defined as in Section 2.2.A pseudoproduct is usually denoted by a ⊗ b a ∗ b , and one may make sense of( a ∗ b ) ∗ c, a ∗ ( b ∗ c ) as elements in ( H ⊗ H ⊗ H ) ⊗ H L , as in [BDK, (3.15)-(3.19)]. Thena Lie pseudoalgebra is a pseudoalgebra whose pseudoproduct satisfies a pseudo-version of the skew-symmetry and Jacobi identity axioms for a Lie algebra. In thiscontext, the pseudoproduct is called Lie pseudobracket and the most usual notationfor it is [ a ∗ b ]. The correct pseudoalgebraic translation of the axioms is[ a ∗ b ] = − ( σ ⊗ H id L )[ b ∗ a ]; (3.1)[ a ∗ [ b ∗ c ]] = [[ a ∗ b ] ∗ c ] + (( σ ⊗ id H ) ⊗ H id L )[ b ∗ [ a ∗ c ]] , (3.2)where a, b, c ∈ L and σ : H ⊗ H → H ⊗ H is the flip map switching the twotensor factors. If A, B ⊂ L are H -submodules, then it is convenient to define [ A, B ] A. D’ANDREA as the smallest H -submodule S ⊂ L such that [ a ∗ b ] ∈ ( H ⊗ H ) ⊗ H S for all a ∈ A, b ∈ B . Then A ⊂ L is a subalgebra if [ A, A ] ⊂ A , and an ideal if [ L, I ] ⊂ I .A Lie pseudoalgebra L is abelian if [ L, L ] = 0 and simple if it is not abelian and itsonly ideals are the trivial ones (0) , L .One may similarly define representations of pseudoalgebras. In our setting, if L isan H -Lie pseudoalgebra and M a left H -module, we may consider a pseudoaction to be an H ⊗ H -linear map L ⊗ M → ( H ⊗ H ) ⊗ H M that we will denote by a ⊗ m a ∗ m . This defines a Lie pseudoalgebra representation when[ a ∗ b ] ∗ m = a ∗ ( b ∗ m ) − (( σ ⊗ id H ) ⊗ H id M ) b ∗ ( a ∗ m ) , (3.3)for all a, b ∈ L, m ∈ M, and equality is understood to hold inside ( H ⊗ H ⊗ H ) ⊗ H M as before. We will also say that M is an L -module. Remark . We stress the fact that a pseudoaction of a Lie pseudoalgebra L on theleft H -module M is nothing else than a H ⊗ H -linear maps L ⊗ M → ( H ⊗ H ) ⊗ H M .This only makes M into a Lie pseudoalgebra representation of L when (3.3) issatisfied.Once again, if A ⊂ L, N ⊂ M , one may define A · N to be the smallest H -submodule S ⊂ M such that [ a ∗ n ] ∈ ( H ⊗ H ) ⊗ H S for all a ∈ L, n ∈ N . Then N ⊂ M is an L -submodule if L · N ⊂ N and M has a trivial action of L if L · M = (0);notice that when the action of L on M is trivial, then any H -submodule of M isautomatically an L -submodule.An L -module M is irreducible if its only L -submodules are (0) , M and it doesnot have a trivial action of L . An L -submodule N ( M is maximal if the onlysubmodules of M containing it are N and M ; then M/N is either an irreducible L -module or it is a simple (nonzero) H -module with a trivial action of L . Inparticular, if M/N is an irreducible L -module, then L · M = M . Remark . When
A, B ⊂ L , then [ A, B ] denotes an H -submodules of L , whereaswe reserve the notation [ A ∗ B ] for the subset of ( H ⊗ H ) ⊗ H L containing all[ a ∗ b ] , a ∈ A, b ∈ B . The same applies to A · B, A ∗ B , where A ⊂ L, B ⊂ M and M is a Lie pseudoalgebra representation of L .3.2. Annihilation algebras.
Let L be a Lie pseudoalgebra over H , and denoteas usual by X = H ∗ the commutative algebra dual to H . Then L = X ⊗ H L maybe endowed with a bilinear product defined by[ x ⊗ H a, y ⊗ H b ] = X i ( xh i )( yk i ) ⊗ H c i , (3.4)as soon as the Lie pseudobracket on L satisfies [ a ∗ b ] = P i ( h i ⊗ k i ) ⊗ H c i . Then H -bilinearity along with the pseudoalgebra analogue of skew-symmetry and Jacobiidentity for [ ∗ ] ensure that (3.4) is a Lie bracket on L .Let now d be a finite-dimensional Lie algebra, H = U ( d ) the correspondinguniversal enveloping algebra, and assume that the finitely generated H -module L admits a Lie pseudoalgebra structure over H . Then we may use the filtration on X = H ∗ so as to build up a corresponding linearly compact topology on L .More explicitly, choose a finite dimensional vector subspace S ⊂ L such that L = HS . If { s i } is a basis of S , then[ s i ∗ s j ] = X i ( h kij ⊗ k kij ) ⊗ H s k RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 9 for some choice of h kij , k kij ∈ H , and one may find ℓ ∈ N so that h kij ⊗ k kij ∈ F ℓ ( H ⊗ H )for all choices of i, j, k .Setting L i = (F i + ℓ − X ) ⊗ H S then provides L with a decreasing filtration L = L − ℓ ⊃ L − ℓ +1 ⊃ · · · ⊃ L ⊃ L ⊃ . . . (3.5)which makes it into a linearly compact vector space, and satisfies [ L i , L j ] ⊂ L i + j for all i, j . Thus, the Lie bracket is continuous with respect to the topology and L is a linearly compact topological Lie algebra. Notice that the filtration dependson the choice of the generating subspace S , but the topology it induces does not.Some choices of S allow for more convenient, i.e., lower, values of ℓ . In next section,we exhibit some convenient choices of S when L is a primitive Lie pseudoalgebra. Proposition 3.1.
The action of H on X is continuous. In particular, both theright- and the left-action of d on X is by continuous derivations.Proof. It follows from (F p H ) . (F n L ) ⊂ F n − p L and the fact that elements in d ⊂ H are primitive in H . The right-action case is done in the same way. (cid:3) When dealing with representations of the Lie pseudoalgebra L , it is also con-venient to introduce the so-called extended annihilation algebra e L . This is thesemi-direct product e L := d ⋉ L , where the adjoint action of d on L is defined as[ ∂, x ⊗ H a ] := ( ∂x ) ⊗ H a. Example 3.1.
As a trivial example, let us consider the case d = (0), so that H = k .If L is a finite H -Lie pseudoalgebra, then ( H ⊗ H ) ⊗ H L = ( k ⊗ k ) ⊗ k L can becanonically identified with L . Thus the Lie pseudobracket is simply a (bilinear) Liebracket, and L is a finite-dimensional Lie algebra.The corresponding annihilation algebra L = X ⊗ H L = ( k ) ∗ ⊗ k L is canonicallyisomorphic to the Lie algebra L . The filtration on both H and X is trivial, andthe corresponding topology on L is discrete; however, the concepts of discrete andlinearly compact topologies coincide for finite-dimensional vectors spaces. As d =(0), we also get e L = L ≃ L .The relevance of the extended annihilation algebra lies in the following fact: Theorem 3.1 ([BDK, Proposition 9.4]) . The notions of Lie pseudoalgebra actionof L on the H -module V is equivalent to that of continuous action of the Lie algebra e L on V endowed with the discrete topology. More specifically, if [ a ∗ v ] = P i ( f i ⊗ g i ) ⊗ H v i , then ( x ⊗ h a ) .v = X i h S ( x ) , f i g i ( − i g i (2) v i . Conversely, if V is a discrete module over e L , one may use the d ⊂ e L -actionon V in order to endow it with an H = U ( d ) -module structure, and recover thepseudoaction by a ∗ v = X i ( S ( h i ) ⊗ ⊗ H ( x i ⊗ H a ) .v, where { h i } and { x i } are dual bases of H and X . It is known that the above equivalence preserves the natural notions of irre-ducibility, so that the study of irreducible representations of a Lie pseudoalgebra L translates into that of irreducible representations of the corresponding extendedannihilation algebra e L . We end this section by recalling a fact stated in [BDK, Lemma 14.4] that we aregoing to use multiple times. If V is a Lie pseudoalgebra representation of L , set • ker V := { v ∈ V | L ∗ v = 0 } ; • ker n V := { v ∈ V | L n .v = 0 } ,whereas L n is defined as in (3.5), according to some generating subspace S ⊂ L .Then Proposition 3.2. If V is a finite L -module, then the quotient ker n V / ker V is afinite-dimensional vector space for every choice of n ≥ − ℓ , and it is nonzero when n is sufficiently large. Primitive simple Lie pseudoalgebras
Examples of primitive simple Lie pseudoalgebras.
Let d be a finite di-mensional Lie algebra. In this paper, by primitive Lie pseudoalgebra , we mean asimple Lie pseudoalgebra over the cocommutative Hopf algebra H = U ( d ) whichcannot be obtained by means of a non-trivial current construction, see [BDK, Sec-tion 4.2] and Section 6 below. More specifically, they are either finite-dimensionalsimple Lie algebras over H = k = U ( { } ), or one of the primitive pseudoalgebrasof vector fields from [BDK, Section 8]. Let us review them briefly. Example 4.1 (Simple Lie algebras) . Let g be a finite dimensional simple Liealgebra over H = k . As in Example 3.1 its Lie bracket may be rewritten in apseudoalgebraic fashion as follows:[ a ∗ b ] = (1 ⊗ ⊗ k [ a, b ] , a, b ∈ g . Here we like to stress the trivial fact that S = g is a finite-dimensional vector spacewith the property that [ S ∗ S ] ∈ (1 ⊗ ⊗ k S. We have already seen that the (extended) annihilation algebra of g is g itself. Example 4.2 ( W ( d )) . Let d be a (nonzero) finite dimensional Lie algebra over k , H = U ( d ). Then L = W ( d ) = H ⊗ d is a simple Lie pseudoalgebra when endowedwith the unique pseudo-Lie bracket H -bilinearly extending[1 ⊗ a ∗ ⊗ b ] = (1 ⊗ ⊗ H (1 ⊗ [ a, b ]) + ( b ⊗ ⊗ H (1 ⊗ a ) − (1 ⊗ a ) ⊗ H (1 ⊗ b ) , (4.1)where a, b, ∈ d . Then S = k ⊗ d ⊂ H ⊗ d is a finite dimensional vector subspace of L which generates it as an H -module and direct inspection of (4.1) shows that[ S ∗ S ] ∈ ( d ⊗ k + k ⊗ d + k ⊗ k ) ⊗ H S. The annihilation algebra W = X ⊗ H W ( d ) is isomorphic, as a Lie algebra, tothe Lie algebra W N from Cartan’s classification [C]. We denote by E ∈ W theelement corresponding to the Euler vector field E = P Ni =1 x i ∂/∂x i ∈ W N undersuch isomorphism. Example 4.3 ( S ( d , χ )) . Let d be a finite dimensional Lie algebra over k , χ ∈ d ∗ atrace form, and set S ( d , χ ) := (cid:8)P i h i ⊗ ∂ i ∈ W ( d ) | P i h i ( ∂ i + χ ( ∂ i )) = 0 (cid:9) . Then S ( d , χ ) ⊂ W ( d ) is the H -submodule generated by elements s ab = ( a + χ ( a )) ⊗ b − ( b + χ ( b )) ⊗ a − ⊗ [ a, b ] , a, b ∈ d , and is a simple subalgebra of W ( d ). Lie pseudobrackets between the above elementsmay be read off [BDK, Proposition 8.4]. If we denote by S the k -linear span of RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 11 elements s ab , a, b ∈ d , then S is a finite-dimensional vector subspace of S ( d , χ )generating it as an H -module, and satisfying[ S ∗ S ] ∈ (( d + k ) ⊗ ( d + k )) ⊗ H S. The annihilation algebra S = X ⊗ H S ( d , χ ) is isomorphic, as a Lie algebra, to theLie algebra S N from Cartan’s list. Example 4.4 ( H ( d , χ, ω ) and K ( d , θ )) . Let d be a finite dimensional Lie algebraover k , and let the rank one free H -module L = He support a Lie pseudoalgebrastructure over H = U ( d ). Then one may see [BDK, Section 4.3] that[ e ∗ e ] = ( r + s ⊗ − ⊗ s ) ⊗ H e (4.2)where 0 = r ∈ V d , s ∈ d satisfy opportune conditions, under which e
7→ − r + 1 ⊗ s ∈ H ⊗ d ≃ W ( d ) (4.3)extends to an injective homomorphism of Lie pseudoalgebras L ֒ → W ( d ).We obtain the primitive Lie pseudoalgebras H ( d , χ, ω ), respectively K ( d , θ ),when d is even dimensional and r is non-degenerate, resp. when d is odd dimen-sional and it is linearly generated by s , along with the support of r . Once again,choosing S = k e provides a finite-dimensional vector subspace of L satisfying[ S ∗ S ] ∈ (( d + k ) ⊗ ( d + k )) ⊗ H S. The annihilation Lie algebra K = X ⊗ H K ( d , θ ) is isomorphic to the Cartantype Lie algebra K N , and we denote by E ′ ∈ K the element corresponding to theEuler vector field E ′ = 2 x ∂/∂x + P Ni =2 x i ∂/∂x i ∈ K N under such isomorphism.The annihilation algebra H = X ⊗ H H ( d , χ, ω ) is instead isomorphic to the uniqueirreducible central extension P N of the Cartan Lie algebra H N .We may sum up the above examples in the following Proposition 4.1.
Let d be a finite dimensional Lie algebra and L be a simpleprimitive Lie pseudoalgebra over the cocommutative Hopf algebra H = U ( d ) as inExamples 4.1-4.4. Then there exists a finite dimensional subspace S ⊂ L and ℓ ∈ N such that L = HS and [ S ∗ S ] ∈ (( d + k ) ⊗ ( d + k ) ∩ F ℓ ( H ⊗ H )) ⊗ H S, where — ℓ = 0 if L = g is a finite-dimensional simple Lie algebra; — ℓ = 1 if L = W ( d ) ; — ℓ = 2 if L = S ( d , χ ) , H ( d , χ, ω ) , K ( d , θ ) . Annihilation algebra of primitive Lie pseudoalgebras of vector fields.
In this section, we recall a few facts from [BDK, Section 6] on infinite-dimensionalsimple linearly compact Lie algebras and their irreducible central extensions. Here L is a primitive simple finite Lie pseudoalgebra over H = U ( d ), where d is a Liealgebra of finite dimension N > W N , S N , K N , P N ,where P N is the unique irreducible central extension of H N . All such Lie algebrasadmit a Z -grading:— The annihilation algebra W ≃ W N of W ( d ) is graded in indices ≥ − E ∈ W . — The inclusion S ( d , χ ) ֒ → W ( d ) induces an embedding of annihilation alge-bras S ֒ → W . Then S ≃ S N is graded in indices ≥ − E , where E ∈ W is as above.— The inclusion K ( d , θ ) ֒ → W ( d ) from (4.3) induces an embedding of annihi-lation algebras K ֒ → W . Then K ≃ K N is graded in indices ≥ − E ′ ∈ K .— The inclusion H ( d , χ, ω ) ֒ → W ( d ) from (4.3) induces a non-injective ho-momorphism of annihilation algebras P ֒ → W , whose kernel equals thecenter Z ( P ). Then P ≃ P N is graded in indices ≥ − E ∈ ad W acting on P /Z ( P ) =: H ≃ H N ⊂ W N , where E ∈ W is as above.Denote by L the annihilation algebra of our primitive Lie pseudoalgebra L , and set L i to be the graded part of degree i in L , with respect to the above defined grading.Then L = Q i ≥− L i and • [ L i , L j ] ⊂ L i + j ; • L ⊂ L is a reductive subalgebra, isomorphic to gl N , sl N , sp N , csp N − = sp N − ⊕ k when L = W ( d ) , S ( d , χ ) , H ( d , χ, ω ) , K ( d , θ ) respectively; • each L i is a finite-dimensional completely reducible representation of L .Central elements from P N all lie in the degree − L >n = Y j>n L j , L ≥ n = Y j ≥ n L j , L =0 = Y j =0 L j . We will also denote by L ⊥ the sum of all isotypical L -components relative tonontrivial L -actions. Then L ⊥ ⊂ L is a complement to the trivial isotypical L -component, and L > ⊂ L ⊥ . In all cases L equals its derived subalgebra, so that L = [ L ⊥ , L ⊥ ]. Notice that {L ≥ n } n ∈ Z is a decreasing filtration of L which coincideswith {L n } n ∈ Z when L = W ( d ) , S ( d , χ ) , H ( d , χ, ω ), see [BDK1, BDK3].In the remaining case L = K ( d , θ ), the two filtrations are distinct but equivalent,see [BDK2], and they induce the same linearly compact topology. More precisely, K n ⊂ K ≥ n , K ≥ n ⊂ K ⌊ n − ⌋ , so that K n certainly contains K ≥ n +1 . Also notice that K ⊂ K ⊥ .4.3. H ( d , χ, ω ) , d + and sp ( d , ω ) . Choose r ∈ V d , s ∈ d . Setting[ e ∗ e ] = ( r + s ⊗ − ⊗ r ) ⊗ H e (4.4)defines a Lie pseudobracket on the free H -module L = He of rank 1 precisely whenidentities [ r, ∆( s )] = 0 (4.5)([ r , r ] + r s ) + (cyclic permutations) = 0 (4.6)hold, see [BDK, Equations (4.3)-(4.4)].If r = P ij r ij ∂ i ⊗ ∂ j is nondegenerate, then ( r ij ) is an invertible skew-symmetricmatrix, whose inverse we denote by ( ω ij ). Setting ω ( ∂ i ∧ ∂ j ) then defines a non-degenerate skew-symmetric 2-form on d , that we may use to set χ = ι s ω . Then RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 13 L ≃ H ( d , χ, ω ) and identities (4.5) are equivalent tod ω + χ ∧ ω = 0; (4.7)d χ = 0 . (4.8)This means that the 1-cocycle, or traceform , χ may be used to define a 1-dimensional d -module k χ , and that ω is a 2-cocycle with values in k χ . In other words, we mayset up an abelian extension → k χ → d + π → d → , whose representations prove useful in the description of H ( d , χ, ω )-tensor modules.Notice that d + ≃ d ⊕ k χ as vector spaces: we shall denote the linear generator of k χ by c and abuse the notation by denoting with ∂ both elements from d and thosefrom d + . The Lie bracket in d + then satisfies[ ∂, ∂ ′ ] + = [ ∂, ∂ ′ ] + ω ( ∂ ∧ ∂ ′ ) c. If we set ∂ = ∂ − χ ( ∂ ), and ∂ i = P j r ij ∂ j , then ω ( ∂ i ∧ ∂ j ) = δ ij and ∂ i = P j ω ij ∂ j .Also, (4.4) rewrites as [ e ∗ e ] = X k ( ∂ k ⊗ ∂ k ) ⊗ H e. (4.9)Notice that skew-symmetry of r implies P k ∂ k ⊗ ∂ k = − P k ∂ k ⊗ ∂ k , so that X k ∂ k ∂ k = − X k ∂ k ∂ k = 12 X k [ ∂ k , ∂ k ]belongs to [ d , d ]. This implies that every traceform on d vanishes on the aboveelement.One may use the symplectic form ω on d to define the symplectic subalgebra sp ( d , ω ) ⊂ gl d . More precisely, φ ∈ gl d lies in sp ( d , ω ) if and only if ω ( φ ( ∂ ) ∧ ∂ ′ ) + ω ( ∂ ∧ φ ( ∂ ′ )) = 0 . Now define ad χ ∂ : d → d as(ad χ ∂ )( ∂ ′ ) := [ ∂, ∂ ′ ] + χ ( ∂ ′ ) ∂. We will later need the following technical fact:
Lemma 4.1.
Let d be an even dimensional Lie algebra, χ ∈ d ∗ a traceform on d ,and assume that -form ω ∈ V d ∗ is nondegenerate and satisfies d ω + χ ∧ ω = 0 .Choose s ∈ d so that χ = ι s ω . If δ ∈ d , then the following are equivalent: (1) [ s, δ ] = 0 and ad χ δ ∈ sp ( d , ω ) ; (2) ι δ ω ∈ d ∗ is a traceform and χ ( δ ) = 0 .Proof. We have s = P k χ ( ∂ k ) ∂ k and χ ( s ) = ω ( s ∧ s ) = 0. Identity d ω + χ ∧ ω = 0translates into ω ([ a, b ] ∧ c )+ ω ([ b, c ] ∧ a )+ ω ([ c, a ] ∧ b ) − χ ( a ) ω ( b ∧ c ) − χ ( b ) ω ( c ∧ a ) − χ ( c ) ω ( a ∧ b ) = 0 , (4.10)for all choices of a, b, c ∈ d . Substituting a = s into (4.10) then yields ω ([ s, b ] ∧ c ) + ω ( b ∧ [ s, c ]) = 0 , (4.11) The Lie algebra d + is denoted d ′ in [BDK3]. as ι s ω = χ vanishes on [ d , d ] and χ ( b ) ω ( c ∧ s ) + χ ( c ) ω ( s ∧ b ) = − χ ( b ) χ ( c ) + χ ( c ) χ ( b ) = 0 . (1) = ⇒ (2). If we plug b = δ into (4.11), we obtain ω ( δ ∧ [ x, c ]) = 0 for all c ∈ d , thus showing that the linear functional ι δ ω ∈ d ∗ vanishes on Im ad s . Setting a = δ in (4.10) and using ad χ δ ∈ sp ( d , ω ) yields ω ([ δ, b ] ∧ c ) + ω ([ b, c ] ∧ δ ) + ω ([ c, δ ] ∧ b ) − χ ( δ ) ω ( b ∧ c ) − χ ( b ) ω ( c ∧ δ ) − χ ( c ) ω ( δ ∧ b ) = 0 ,ω ([ δ, b ] ∧ c ) + ω ( b ∧ [ δ, c ]) + χ ( b ) ω ( δ ∧ c ) + χ ( c ) ω ( b ∧ δ ) = 0 , which together give ω ( δ ∧ [ b, c ]) + χ ( δ ) ω ( b ∧ c ) = 0 . (4.12)Using b = s in (4.12) then shows that χ ( δ ) ω ( s ∧ c ) = 0 for all c ∈ d . As ω is nondegenerate, this forces χ ( δ ) = 0, whence ω ( δ ∧ [ b, c ]) = 0 for all b, c ∈ d ,implying ι δ ω is a trace-form on d .(2) = ⇒ (1). Substitute b = δ in (4.11) to obtain ω ([ x, δ ] ∧ c )+ ω ( δ ∧ [ x, c ]) = 0. As ι δ ω is a traceform, then ω ([ s, δ ] ∧ c ) = 0 for all c ∈ d . However, ω is nondegenerate,hence [ δ, s ] = 0. Now set a = δ in (4.10), and use χ ( δ ) = 0 along with the fact that ι δ ω is a traceform. This gives ω ([ δ, b ] ∧ c ) + ω ([ c, δ ] ∧ b ) − χ ( b ) ω ( c ∧ δ ) − χ ( c ) ω ( δ ∧ b ) = 0 . (4.13)The left hand side now rewrites as ω (([ δ, b ] + χ ( b )) δ ∧ c ) + ω ( b ∧ ([ δ, c ] + χ ( c ) δ )) = 0 , (4.14)thus showing that ad χ δ ∈ sp ( d , ω ). (cid:3) We end this section by recalling notation that we will later employ. If V is arepresentation of a Lie algebra g , and χ ∈ g ∗ is a traceform, then we will denoteby V χ the tensor product V ⊗ k χ , where k χ is the one-dimensional representationcorresponding to χ : g → k viewed as a Lie algebra homomorphism. Notice thatthis notation is compatible with denoting by k the trivial representation of g . Also,if π : g → h is a Lie algebra homomorphism, and χ is a traceform of h , we willdenote by π ∗ χ = χ ◦ π the pullback traceform on g .Finally, if V, W are Lie algebra representations of the Lie algebras g , h respec-tively, then V ⊠ W is the corresponding tensor product representation of the directsum Lie algebra g ⊕ h . Notice that every irreducible g ⊕ h -module is isomorphic to V ⊠ W for a suitable choice of irreducible g -, resp. h -, modules V, W .5.
Tensor modules of primitive Lie pseudoalgebras
Each finite irreducible representations of a primitive Lie pseudoalgebra L as inExamples 4.2-4.4 arises as a quotient of an opportune L -module from a specialclass of representations called tensor modules , that are parametrized by (finite-dimensional irreducible) Lie algebra representations of the direct sum of L and aLie algebra isomorphic either to d in types W, S, K, or d + in type H. this section,we recall their definition from [BDK1, BDK2, BDK3]. As usual, dim d = N and { ∂ , . . . , ∂ N } is a basis of d . RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 15
Tensor modules for W ( d ) . When L = W ( d ), the degree zero component L of the annihilation algebra is isomorphic to gl d ≃ gl N . Let { ∂ i | ≤ i ≤ N } bea basis of the Lie algebra d and R = (Π ⊠ U, ρ ) be an irreducible representationof d ⊕ gl d . Then there is a Lie pseudoalgebra action of W ( d ) = H ⊗ d on H ⊗ R defines by(1 ⊗ ∂ i ) ∗ (1 ⊗ v ) = N X j =1 ( ∂ j ⊗ ⊗ H (1 ⊗ ρ ( e ji ) v ) (5.1)+ (1 ⊗ ⊗ H (1 ⊗ ρ ( ∂ i + ad ∂ i ) v − ∂ i ⊗ v ) , where v ∈ R and e ji ∈ gl d denotes the elementary matrix satisfying e ji ( ∂ j ) = ∂ i .The notation for this W ( d )-module from [BDK1, Definition 6.2] is V ( R ) = V (Π , U ),but we will add a W superscript to distinguish it from tensor modules over primitiveLie pseudoalgebras of other types.Notice that V W (Π , U ) is irreducible unless U is isomorphic to V n d for some0 ≤ n < N , in which case it has a unique nontrivial maximal W ( d )-submodule.The corresponding quotient is thus a finite irreducible W ( d )-module, with the singleexception of the case n = 0, when it has a trivial W ( d )-action.5.2. Tensor modules for S ( d , χ ) . In this case, L ≃ sl d ≃ sl N . Recall that S ( d , χ ) has a unique Lie pseudoalgebra embedding in W ( d ), so that every W ( d )-tensor modules as in (5.1) becomes an S ( d , χ )-module by restriction. Then a rep-resentation of S ( d , χ ) is a tensor module if it is the restriction of a W ( d )-tensormodule; notice that each S ( d , χ )-tensor module arises as such a restriction in morethan one way, and that a tensor module is reducible if and only if it is the restrictionof at least one reducible W ( d )-tensor module.If R = (Π ⊠ U, ρ ) is an irreducible representation of d ⊕ sl d , we denote by V S χ ( R ) = V S χ (Π , U ) the tensor module obtained by restricting the W ( d )-tensormodule V W (Π , U ), where U is the gl d -module obtained by extending the sl d -action on U so that id ∈ gl d = sl d ⊕ k id acts trivially.Once again, the S ( d , χ )-tensor module V S χ (Π , U ) is irreducible unless U is eithertrivial or isomorphic to one of the fundamental representations V n d , < n < N .In the latter cases, there is a unique maximal submodule which yields an irreduciblequotient, whereas when U is the trivial sl d -module, the corresponding quotient hasa trivial S ( d , χ )-action.5.3. Tensor modules for K ( d , θ ) . If d = ker θ , then d θ is a symplectic formon d , and L ≃ csp ( d , d θ ) = sp ( d , d θ ) ⊕ k c ≃ sp N − ⊕ k . Here, elements ∂ , . . . , ∂ N − form a basis of d and ∂ N = s as in (4.2). Elements ∂ k are defined asin Section 4.3. We similarly raise indices by introducing elements e ij ∈ gld thatsatisfy e ij ∂ k = δ jk ∂ i . The Lie subalgebra sp ( d , d θ ) is then generated by elements f ij = −
12 ( e ij + e ji ) , ≤ i ≤ j ≤ N − . Notice that differences e ij − e ji span a (skew-symplectic) complement to sp ( d , d θ )in gl d , so that one may consider the projection π sp to the symplectic summand.Then we denote by ad sp ( ∂ ) the expression π sp (ad ∂ − ∂ N · ι ∂ ω ). This is denoted V χ ( R ) = V χ (Π , U ) in [BDK1, Definition 7.2]. If R = (Π ⊠ U, ρ ) is a finite-dimensional irreducible representation of d ⊕ sp ( d , d θ )then e ∗ (1 ⊗ v ) = n X i,j =1 ( ∂ i ∂ j ⊗ ⊗ H (1 ⊗ ρ ( f ij ) v ) (5.2) − N − X k =1 ( ∂ k ⊗ ⊗ H (1 ⊗ ρ ( ∂ k + ad sp ( ∂ k )) v − ∂ k ⊗ v )+ 12 ( ∂ N ⊗ ⊗ H (1 ⊗ ρ ( c ) v )+ (1 ⊗ ⊗ H (1 ⊗ ρ ( ∂ N + ad ∂ N ) v − ∂ N ⊗ v ) , where v ∈ R and c denotes the central element in csp ( d , d θ ) = sp ( d , d θ ) ⊕ k c ,defines a Lie pseudoalgebra action of L = K ( d , θ ) = He on H ⊗ R , that we willdenote by V K θ ( R ) = V K θ (Π , U ). The K ( d , θ )-tensor module V K θ (Π , U ) is irreducibleunless either the csp ( d , d θ )-action on U is trivial; or U is the p -th fundamental rep-resentation of sp ( d , d θ ) and c acts via scalar multiplication by either p or N + 1 − p .Whenever V K θ (Π , U ) is reducible, it has a unique maximal K ( d , θ )-submodule yield-ing an irreducible K ( d , θ )-module, unless when U is the trivial sp ( d , d θ )-module,in which case the K ( d , θ )-action on the quotient is trivial.5.4. Tensor modules for H ( d , χ, ω ) . As χ is a 1-cocycle on d , we set ∂ = ∂ − χ ( ∂ ). Then ∂ ∂ extends to an algebra automorphism of the universal envelopingalgebra H = U ( d ).Here d + = d + k c is the abelian extension of d corresponding to the 2-cocycle ω with values in k χ , and L ≃ sp ( d , ω ) ≃ sp N . For each choice of ∂ ∈ d , we denoteby ad χ ∂ ∈ gl d the map x [ ∂, x ] + χ ( x ) ∂ ; we also set ad sp χ ( ∂ ) = π sp (ad χ ∂ ). If R = (Π + ⊠ U, ρ ) is a finite-dimensional irreducible representation of d + ⊕ sp ( d , ω ),then e ∗ (1 ⊗ v ) = N X i,j =1 ( ∂ i ∂ j ⊗ ⊗ H (1 ⊗ ρ ( f ij ) v ) (5.3) − N X k =1 ( ∂ k ⊗ ⊗ H (1 ⊗ ρ ( ∂ k + ad sp χ ( ∂ k )) v − ∂ k ⊗ v )+(1 ⊗ ⊗ H (1 ⊗ ρ ( c ) v )defines a Lie pseudoalgebra representation of H ( d , χ, ω ) = He on H ⊗ R , that wedenote by V H χ,ω ( R ) = V H χ,ω (Π + , U ). The tensor module V H χ,ω (Π + , U ) is irreducibleunless:— either U is trivial and ρ ( c ) = 0;— or U is one of the fundamental representations of sp ( d , ω ).When V H χ,ω (Π + , U ) is reducible, it has a unique maximal submodule if ρ ( c ) = 0,when the corresponding quotient has a nontrivial H ( d , χ, ω )-action unless U is thetrivial sp ( d , ω )-module. When instead ρ ( c ) = 0, each reducible tensor modulesdecomposes into the direct sum of its two irreducible maximal submodules. This is denoted V ( R ) = V (Π , U ) in [BDK2, Definition 5.3]. This is denoted ad sp ∂ in [BDK3]. This is denoted V ( R ) = V (Π + , U ) in [BDK3, Definition 6.2]. RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 17 The current functor
Let H ⊂ H ′ be our usual Hopf algebras. One may use the inclusion homomor-phism ι : H → H ′ to define a scalar extension functor H ′ ⊗ H which associates witheach left H -module M the corresponding left H ′ -module H ′ ⊗ H M . As H ′ is freeas a right H -module, the above functor is exact.When L is an H -Lie pseudoalgebra, the associated left H ′ -module H ′ ⊗ H L maybe endowed with a corresponding H ′ -Lie pseudoalgebra structure, which is uniquelydetermined by setting[1 ⊗ H a ∗ ⊗ H b ] = X i ( f i ⊗ g i ) ⊗ H ′ (1 ⊗ H c i ) , whenever a, b ∈ L and [ a ∗ b ] = P i ( f i ⊗ g i ) ⊗ H c i . This Lie H ′ -pseudoalgebra isusually denoted by Cur H ′ H L and called current Lie pseudoalgebra of L . It followsfrom the structure theory developed in [BDK] that Cur H ′ H L is simple whenever L issimple, and that all finite simple Lie pseudoalgebras are obtained in this way fromone of the primitive Lie pseudoalgebras listed above in Section 4.The current functor H ′ ⊗ H may also be applied to L -modules, and similarlyyields Lie pseudoalgebra representations of Cur H ′ H L , as it will be made explicit inSection 6.2 below. We will later see that irreducibility of modules behaves well withrespect to the current functor. Remark . Recall that the injection ι : H → H ′ is pure, so that M ∋ m ⊗ H m ∈ Cur H ′ H M is always injective. In other words, every left H -module M embeds H -linearly intoits current module Cur H ′ H M Current simple Lie pseudoalgebras.
Let L be a Lie pseudoalgebra over H ,and L ′ = Cur H ′ H L := H ′ ⊗ H L its current pseudoalgebra over H ′ . Then L = X ⊗ H L is the annihilation algebra of L and L ′ = X ′ ⊗ H ′ ( H ′ ⊗ H L ) ≃ X ′ ⊗ H L is the annihilation algebra of L ′ ; one has therefore a natural surjection ι ∗ ⊗ H id L : L ′ ≃ X ′ ⊗ H L → X ⊗ H L ≃ L . Recall that the Lie bracket (3.4) of a Lie pseudoalgebra over H is obtained byrephrasing the Lie pseudobracket in terms of the right action of H on its dual X .As the right d -action on X ⊂ X ′ coincides with the natural right d -action on X ,then the map L ≃ X ⊗ H L → X ′ ⊗ H L ≃ L ′ is a continuous Lie algebra homomorphism and provides a(n injective) splitting ς ⊗ H id L to ι ∗ ⊗ H id L . Notice that X ′ = O ⊗ X and we may use Proposition 2.2 (ii) in order to conclude that the Lie bracket on L ′ extends O -linearly the Lie bracketon the subalgebra L ⊂ L ′ . Lemma 6.1.
Let L be a primitive simple Lie pseudoalgebra of vector fields, anddenote by L denote the reductive Lie subalgebra of the annihilation algebra L con-sisting of degree element according to the grading recalled in Section 4.2. Thenthe only trivial summands for the adjoint action of L on L are — k E when L = W ( d ) ; — the center Z ( L ) when L = H ( d , χ, ω ) ; — k E ′ when L = K ( d , θ ) ,whereas there is no such trivial summand when L = S ( d , χ ) .Proof. When L = W ( d ), resp. K ( d , θ ), then the grading on L is given by theeigenspace decomposition of the inner derivation ad E , resp. ad E ′ . Consequently,all trivial summands for the adjoint action of L must lie in degree 0. As L ≃ gl N = sl N ⊕ k , resp. csp N − = sp N +1 ⊕ k , the adjoint action of L is trivial on Z ( L ), that is on k E , resp. k E ′ .When L = H ( d , χ, ω ), [BDK, Lemma 6.7 (iii)] shows that each graded summand L n ⊂ L is an irreducible representation of L , which is only trivial when n = − L − coincides with the center Z ( L ), as the L action on Z ( L ) is trivialand dim Z ( L ) = dim L − .Absence of trivial L -summand in the L = S ( d , χ ) case follows from [BDK,Lemma 6.7 (ii)]. (cid:3) As we have identified L with a subalgebra of L ′ , L ⊂ L also arises as a subalge-bra of the annihilation algebra L ′ of the current Lie pseudoalgebra L ′ = Cur H ′ H L .In this setting Proposition 6.1.
The trivial isotypical component corresponding to the adjointaction of L ⊂ L ⊂ L ′ on L ′ = O b ⊗ L is — O ⊗ k E , where E ∈ W is the Euler vector field, when L = W ( d ) ; moreover m ⊗ k E ⊂ [ O b ⊗ W > , m b ⊗ W ⊥ ] . — (0) when L = S ( d , χ ) . — O ⊗ k E ′ , where E ′ ∈ K is the contact Euler vector field, when L = K ( d , θ ) ;moreover m ⊗ k E ′ ⊂ [ O b ⊗ K > , m b ⊗ K ⊥ ] . — O ⊗ Z ( P ) when L = H ( d , χ, ω ) ; moreover m ⊗ Z ( P ) ⊂ [ m b ⊗ P ⊥ , m b ⊗ P ⊥ ] .Proof. The Lie bracket on L ′ = O b ⊗ L is O -bilinear, so that if X ⊂ L is a non-trivialirreducible L -summand, then O b ⊗ X is X -isotypical with respect to the action of L ⊂ L ⊂ L ′ . Then the description of the trivial isotypical component follow fromLemma 6.1.As for m ⊗ k E ⊂ [ O b ⊗ L > , m b ⊗ L ⊥ ] when L = W ( d ), resp. K ( d , θ ), it suffices toprove that E , resp. E ′ , lies in [ L > , L ⊥ ]. This follows from the last claim in each of[BDK, Lemmas 6.5, 6.7], as the semisimple Lie subalgebra of L denoted there by p equals its derived subalgebra.Lastly, when L = H ( d , χ, ω ), L = P ≃ P N is an irreducible central extensionof P /Z ( P ) = H ≃ H N , so that P equals its derived subalgebra. However [BDK,Lemma 6.7] states that P ⊥ = P ≥− is a complement, as vector spaces, to the center Z ( P ), so that [ P ⊥ , P ⊥ ] contains Z ( P ). The claim m ⊗ Z ( P ) ⊂ [ m b ⊗ P ⊥ , m b ⊗ P ⊥ ]now follows by O -bilinearity of the Lie bracket of L ′ ≃ O b ⊗ P . (cid:3) A further description of L ′ , which is valid for all current Lie pseudoalgebraand not only for simple ones, is in order. We have set up an explicit splitting ς : X → X ′ to ι ∗ : X ′ → X . As I := ker ι ∗ , we obtain the direct sum decomposition X ′ = ς ( X ) ⊕ I as right H -modules. Then correspondingly L ′ = ( X + I ) ⊗ H L = X ⊗ H L ⊕ I ⊗ H L ≃ L + I , where we have set I := I ⊗ H L . Under the identification L ′ ≃ O b ⊗ L , one has I = m b ⊗ L so that this reads as L ′ ≃ k b ⊗ L + m b ⊗ L = L + m b ⊗ L . RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 19
In other words, L ′ = L ⋉ I is the semidirect sum of the subalgebra L withthe ideal I , yielding the isomorphism L ′ / I ≃ L of topological Lie algebras. ByProposition 2.2 (iii) , we argue that the above isomorphism generalizes to extendedannihilation algebras e L ′ / I ≃ e L .6.2. Current representations.
Let L be a Lie pseudoalgebra over H and ∗ : L ⊗ V → ( H ⊗ H ) ⊗ H V be a pseudoaction of L on the left H -module V . If H ⊂ H ′ , then we may construct both the current Lie pseudoalgebra L ′ = Cur H ′ H L and the corresponding current H ′ -module V ′ = Cur H ′ H V := H ′ ⊗ H V . Proposition 6.2.
There exists a unique pseudoaction of L ′ on V ′ extending H ′ -bilinearly (1 ⊗ H a ) ∗ (1 ⊗ v ) = X i ( f i ⊗ g i ) ⊗ H ′ (1 ⊗ H v i ) , (6.1) where a ∈ L, v ∈ V, f i , g i ∈ H and a ∗ v = X i ( f i ⊗ g i ) ⊗ H v i . (6.2) Then (6.1) defines a Lie pseudoalgebra representation of L ′ on V ′ if and only if (6.2) gives a Lie pseudoalgebra representation of L on V .Proof. The fact that if V is a representation of L then V ′ is a representation of L ′ is a trivial check. The converse depends on the fact that( H ′ ⊗ H ′ ⊗ H ′ ) ⊗ H ′ V ′ ≃ ( H ′ ⊗ H ′ ⊗ H ′ ) ⊗ H ′ ( H ′ ⊗ H V ) ≃ ( H ′ ⊗ H ′ ⊗ H ′ ) ⊗ H V, and the linear isomorphism( H ′ ⊗ H ′ ⊗ H ′ ) ⊗ H ′ V ′ H ′ ⊗ H ′ ⊗ V ′ described in Corollary 2.1, under such identification, restricts on ( H ⊗ H ⊗ H ) ⊗ H ′ V to the analogous isomorphism( H ⊗ H ⊗ H ) ⊗ H V → H ⊗ H ⊗ V for V , so that checking that (6.1) satisfies the axioms of a Lie pseudoalgebra rep-resentation — on the set of H ′ -linear generators of the form 1 ⊗ H a, a ∈ L, and1 ⊗ H v, v ∈ V — is the same as checking that (6.2) does. However the axioms foran H ′ -Lie pseudoalgebra representations are invariant under H ′ -linear combination,and we are done. (cid:3) Remark . In simpler words, if V is a Lie pseudoalgebra representation of L , theCur H ′ H L -module Cur H ′ H V is obtained via extending by H ′ -bilinearity the pseudoac-tion (6.1) obtained by replacing ⊗ H with ⊗ H ′ in each a ∗ v , where a ∈ L, v ∈ V .If we choose, as usual, a basis { ∂ , . . . , ∂ r , ∂ r +1 , . . . , ∂ N + r } of d ′ in such a waythat { ∂ r +1 , . . . , ∂ N + r } is a basis of d ⊂ d ′ , then we may use the PBW basis of H ′ to uniquely express each element v ∈ Cur H ′ H V as v = X K ∈ Z r ×{ }⊂ Z N + r ∂ ( K ) ⊗ H v K , as the d -components in elements of the PBW basis may be moved on the right of ⊗ H and H ′ is a free right H -module generated by elements ∂ ( K ) , K ∈ Z r × { } ⊂ Z N + r .Notice that coefficients v K ∈ V depend on the choice of the basis, but their H -linearspan only depends on v . If M ⊂ V ′ is an L ′ -submodule, denote now by M the H -submodule of V generated by all coefficients m K of elements m ∈ M . Lemma 6.2. M is an L -submodule of V satisfying H ′ ⊗ H ( L · M ) ⊂ M ⊂ H ′ ⊗ H M .Proof. Every m ∈ M is an H ′ -linear combination of its coefficients, thus M ⊂ H ′ ⊗ H M . As for the other inclusion, express m ∈ M in the form m = X L ∈ Z r × ⊂ Z N + r ∂ ( L ) ⊗ H m L , where L = ( l , . . . , l r , , . . . , ∈ Z N + r and m L ∈ V . If a ∈ L satisfies a ∗ m K = X K ∈ × Z N ⊂ Z N + r ( ∂ ( K ) ⊗ ⊗ H u K + L , where, as before, K = (0 , . . . , , k r +1 , . . . , k N + r ), we want to show that all elementsof the form 1 ⊗ H u K + L lie in M . This follows immediately by computing(1 ⊗ a ) ∗ m = X K,L ( ∂ ( K ) ⊗ ∂ ( L ) ) ⊗ H ′ (1 ⊗ H u K + L ) , and using Lemma 2.1, along with Remark 2.3.This shows that 1 ⊗ H ( L · M ) ⊂ M , so that also H ′ ⊗ H ( L · M ) ⊂ M . However1 ⊗ H ( L · M ) ⊂ M implies L · M ⊂ M , so that M ⊂ V is an L -submodule. (cid:3) Corollary 6.1.
Let V be an irreducible L -module with a non-trivial L -action. Then V ′ = Cur H ′ H V is an irreducible L ′ = Cur H ′ H L -module.Proof. Let (0) = M ⊂ V ′ be a submodule, M ⊂ V its coefficient submodule.Then M is a nonzero L -submodule of V , so that M = V . Thus, by Lemma 6.2, H ′ ⊗ H ( L · V ) ⊂ M ⊂ H ′ ⊗ H V . However, L · V ⊂ V is an L -submodule of V ,which cannot equal (0) as the L -action of V is non-trivial. Then L · V = V and M = H ′ ⊗ H V = V ′ . (cid:3) Remark . Notice that if M ⊂ V is a maximal L -submodule such that the actionof L on V /M is non-trivial, then H ′ ⊗ H M is maximal in Cur H ′ H V = H ′ ⊗ H V .Indeed, Cur H ′ H ( V /M ) is irreducible by the above corollary; as the functor Cur H ′ H isexact, it commutes with taking quotient. Then Cur H ′ H ( V /M ) ≃ V ′ / ( H ′ ⊗ H M ),whence maximality of H ′ ⊗ H M in V ′ . Corollary 6.2.
Let M ( Cur H ′ H V = V ′ be a maximal L ′ -submodule. Then — either M ⊂ V is a maximal L -submodule and M = H ′ ⊗ H M ; — or M = V and the action of L ′ on V ′ /M is trivial.Proof. If M ⊂ V is a proper non-maximal L − submodule, then we may locate M ( M ( V , and M ⊂ H ′ ⊗ H M ( H ′ ⊗ H M ( V ′ , thus showing M is notmaximal.Viceversa, assume that M ( V is a maximal L − submodule. Then M ⊂ H ′ ⊗ H M ( V . By maximality of M , we get M = H ′ ⊗ H M . If instead M = V , then H ′ ⊗ H ( L · V ) ⊂ M ( V ′ . Then L · V ( V , so that the action of L on V /L · V istrivial. However, V ′ /M is a quotient of H ′ ⊗ H V /L · V = Cur H ′ H V /L · V showingthe action of L ′ on V ′ /M is trivial. (cid:3) RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 21
Our goal in this paper is figuring out to what extent the only irreducible repre-sentations of Cur H ′ H L are obtained by taking currents of irreducible representationsof L , when L is a simple primitive Lie pseudoalgebra.7. An exceptional representation of
Cur H ′ H H ( d , , ω )Let L ′ = Cur H ′ H L where L = H ( d , χ, ω ) = He satisfies[ e ∗ e ] = X k ( ∂ k ⊗ ∂ k ) ⊗ H e. (7.1)Here ∂ = ∂ − χ ( ∂ ), and we will be using the notation introduced in Sections 4.3and 5.4. Notice that L ′ = H ′ ⊗ H He may (and will) be identified with H ′ e .Now assume that R = (Π + ⊠ U, ρ ) is a finite-dimensional irreducible d + ⊕ sp ( d , ω )-module. We aim to find conditions characterizing the values of t ∈ d ′ \ d makingthe following modification of the tensor module pseudoaction given in (5.3): e ∗ t (1 ⊗ v ) = X ij ( ∂ i ∂ j ⊗ ⊗ H ′ ((1 ⊗ ρ ( f ij ) v ) (7.2) − X k ( ∂ k ⊗ ⊗ H ′ (1 ⊗ ρ ( ∂ k + ad sp ∂ k ) v − ∂ k ⊗ v )+ (1 ⊗ ⊗ H ′ (1 ⊗ ρ ( c ) v ) , + ( t ⊗ ⊗ H ′ (1 ⊗ v ) , into a Lie pseudoalgebra representation of L ′ = Cur H ′ H L . Clearly, removing thelast summand in the right-hand side yields the current L ′ -module Cur H ′ H V H χ,ω ( R ).In analogy with Section 5.4, use the definition (ad χ x )( ∂ ) := [ x, ∂ ] + χ ( ∂ ) x toextend ad χ to a map d ′ → Hom( d , d ′ ). Recall that s = P k χ ( ∂ k ) ∂ k . Proposition 7.1.
Let t be an element of d ′ \ d . Expression (7.2) defines a Liepseudoalgebra representation of L ′ precisely when [ t, s ] = 0 and ad χ t preserves d and lies in sp ( d , ω ) .Proof. Explicitly compute e ∗ t ( e ∗ t (1 ⊗ v )) − ( σ ⊗ id H ) ⊗ H e ∗ t ( e ∗ t (1 ⊗ v )) − [ e ∗ e ] ∗ t (1 ⊗ v ) . In order for ∗ t to be a Lie pseudoalgebra action of H ( d , χ, ω ), this must vanish. Allterms indeed cancel out, with the exception of X k (cid:16) ∂ k ⊗ ([ t, ∂ k ] + χ ( ∂ k ) t ) + ([ t, ∂ k ] + χ ( ∂ k ) t ) ⊗ ∂ k (cid:17) = 0 . (7.3)We may rewrite (7.3) as0 = X k (cid:0) ∂ k ⊗ ([ t, ∂ k ] + χ ( ∂ k ) t ) + ([ t, ∂ k ] + χ ( ∂ k ) t ) ⊗ ∂ k (cid:1) (7.4) − ⊗ X k χ ( ∂ k ) · ([ t, ∂ k ] + χ ( ∂ k ) t ) (7.5) − X k χ ( ∂ k ) · ([ t, ∂ k ] + χ ( ∂ k ) t ) ⊗ . The three summands lie in d ′ ⊗ d ′ , k ⊗ d ′ , d ′ ⊗ k respectively, so they need to vanishseparately. The summand in (7.4) may be rewritten as0 = X k (cid:0) ∂ k ⊗ (ad χ t )( ∂ k ) + (ad χ t )( ∂ k ) ⊗ ∂ k (cid:1) , (7.6)showing ad χ t preserves d and lies in sp ( d , ω ). The summand in (7.5) only vanishesif [ t, X k χ ( ∂ k ) ∂ k ] + X k χ ( ∂ k ) χ ( ∂ k ) t = 0 . (7.7)However the second term must vanish as P k ∂ k ⊗ ∂ k = − P k ∂ k ⊗ ∂ k so that(7.5) translates into [ t, s ] = 0. The third summand is now dealt with similarly andcancels out. (cid:3) Remark . When χ = 0, the above conditions translate into the fact that s normalizes d and ad s ∈ sp ( d , ω ). Example 7.1.
Let d ⊂ d ′ be abelian Lie algebra, χ = 0. Then [ t, s ] = 0 and ad χ t =0 ∈ sp ( d , ω ) hold for every t ∈ d ′ . In particular (7.2) defines a Lie pseudoalgebrarepresentation for every t ∈ d ′ \ d . Example 7.2.
Let h, e, f be the standard generators of d ′ = sl , and choose d = h h, e i . Then [BDK, Example 8.14] shows that the H -linear span of(2 − h ) ⊗ e + e ⊗ h ∈ W ( d )is a subalgebra isomorphic to H ( d , χ, ω ), where ω ( e ∧ h ) = 1 and χ = ι e ω .The only elements in d ′ commuting with s = 2 e are multiples of e , so thereis no t ∈ d ′ \ d satisfying [ t, s ] = 0. Consequently, (7.2) does not define a Liepseudoalgebra representation for any choice of t ∈ d ′ \ d , and all finite irreducibleCur H ′ H H ( d , χ, ω )-modules are, in this case, current representations.Whenever t ∈ d ′ \ d satisfies the conditions of Proposition 7.1, we shall denote by V H χ,ω,t, d ′ ( R ) = V H χ,ω,t, d ′ (Π + , U ) the corresponding Lie pseudoalgebra representationof Cur H ′ H H ( d , χ, ω ). Theorem 7.1.
Let t, t ′ ∈ d ′ be elements satisfying the conditions of Proposi-tion 7.1 and Π + , U be representations of the Lie algebras d + , sp ( d , ω ) respectively.Also denote by π : d + → d the canonical projection. Then V H χ,ω,t ′ , d ′ (Π + , U ) and V H χ,ω,t, d ′ ((Π + ) π ∗ ( ι t ′− t ω ) , U ) are isomorphic Cur H ′ H H ( d , χ, ω ) -modules.Proof. Set t ′ = t + δ . As [ s, t ] = [ s, t ′ ] = 0, then [ s, δ ] = 0. Also, ad χ t ′ and ad χ t both lie in sp ( d , ω ), hence ad χ δ ∈ sp ( d , ω ) by linearity of ad χ . Thus s satisfiesconditions (1) of Lemma 4.1, which forces ι δ ω to be a traceform of d and χ ( δ ) = 0.As δ = X k ω ( δ ∧ ∂ k ) ∂ k = − X k ω ( δ ∧ ∂ k ) ∂ k , RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 23 then( δ ⊗ ⊗ H ′ (1 ⊗ v ) = − X k ( ∂ k ⊗ ⊗ H ′ (( ι δ ω )( ∂ k ) ⊗ v ) (7.8)= − X k (( ∂ k + χ ( ∂ k )) ⊗ ⊗ H ′ (( ι δ ω )( ∂ k ) ⊗ v )= − X k ( ∂ k ⊗ ⊗ H ′ (( ι δ ω )( ∂ k ) ⊗ v ) − (1 ⊗ ⊗ H ′ ((( ι δ ω )( X k χ ( ∂ k ) ∂ k )) ⊗ v )= − X k ( ∂ k ⊗ ⊗ H ′ (( ι δ ω )( ∂ k ) ⊗ v ) , as ( ι δ ω )( X k χ ( ∂ k ) ∂ k ) = ω ( δ ∧ s ) = − χ ( δ ) = 0 . We may now rewrite the action of Cur H ′ H H ( d , χ, ω ) on V H χ,ω,t ′ , d ′ (Π + , U ) as follows: e ∗ t ′ (1 ⊗ v ) = e ∗ t (1 ⊗ v ) + ( δ ⊗ ⊗ H ′ (1 ⊗ v ) (7.9)= e ∗ t − X k ( ∂ k ⊗ ⊗ H ′ (( ι δ ω )( ∂ k ) ⊗ v )= X ij ( ∂ i ∂ j ⊗ ⊗ H ′ ((1 ⊗ f ij .v ) − X k ( ∂ k ⊗ ⊗ H ′ (1 ⊗ (( ∂ k .v + ( ι δ ω )( ∂ k ) v ) + (ad sp ∂ k ) .v )) − ∂ k ⊗ v )+ (1 ⊗ ⊗ H ′ (1 ⊗ c.v ) , + ( t ⊗ ⊗ H ′ (1 ⊗ v ) . As ι δ ω is a traceform on d , and the action of c ∈ d + is left unchanged, the con-tribution ( ι δ ω )( ∂ k ) may be absorbed in the d + representation, thus yielding thepseudoaction of Cur H ′ H H ( d , χ, ω ) on V H χ,ω,t, d ′ ((Π + ) π ∗ ( ι δ ω ) , U ). (cid:3) Irreducible representations of current Lie pseudoalgebras not oftype H
The usual strategy [BDK1, BDK2, BDK3] towards describing irreducible repre-sentations of a simple Lie pseudoalgebra L is by locating singular vectors for theaction of the corresponding annihilation algebra, i.e., vectors that are killed by theaction of L > . In this paper, our point of view is that, as far as reasonable, the ac-tion of L ′ is recovered from that of L ⊂ L ′ , and singular vectors for the two actionscoincide: in order for this philosophy to hold, one must introduce a few tweaksfor current Lie pseudoalgebras of type H, so that we will treat other cases first.Throughout this section, L will be a primitive Lie pseudoalgebra from Examples4.1-4.4, and L ′ = Cur H ′ H its current pseudoalgebra.We choose S ⊂ L , ℓ ∈ N , as in Section 4 and endow the annihilation algebra L with the filtration L n = F n + ℓ − X ⊗ H S . As S ⊂ L ≃ ⊗ H L ⊂ H ′ ⊗ H L =Cur H ′ H L = L ′ still generates L ′ as an H ′ -module, we may choose it in order buildup a similar filtration for its annihilation algebra L ′ = X ′ ⊗ H ′ L ′ , that we will freelyidentify with X ′ ⊗ H L .Notice that the value of ℓ ≤ L and L ′ and that I = I ⊗ H L = I ⊗ H S ⊂ L ′ − ℓ ⊂ L ′− , as I ⊂ F X ′ . We will employ the notation I i = I i ⊗ H L = I i ⊗ H S ⊂ L ′ , i ≥
0, where I i is the i -fold power of the ideal I ⊂ X ′ , so that I = L ′ , I = I .The computation of the normalizer of L n in e L is accomplished in [BDK1, BDK2,BDK3]. Our present goal is to find a large subalgebra N ′ ⊂ e L ′ normalizing L ′ n , n ≥
0. We already know that [ L ′ , L ′ n ] ⊂ L ′ n .8.1. I normalizes L ′ n .Proposition 8.1. I i is an ideal of L ′ . Also [ I i , I j ] ⊂ I i + j and I i ⊂ L ′ i − s .Proof. Structure constants in Lie pseudobracket of L only involve elements in H .As ( I ) . d ⊂ I , then also ( I i ) .H ⊂ I i , thus proving the first two claim. The lastclaim is clear, as I i ⊂ F i − X ′ . (cid:3) Proposition 8.2. I = I normalizes L ′ n for each n .Proof. By Proposition 4.1, S has been chosen in such a way that structure constantsare contained in ( d + k ) ⊗ ( d + k ). Notice that ( I ) . d ⊂ I , and (F k X ′ ) . d ⊂ (F k X ′ ) . d ′ ⊂ F k − X . Thus[ I , L ′ n ] = [ I ⊗ H S, F n + ℓ − X ′ ⊗ H S ] ⊂ ( I · F n + ℓ − X ′ ) ⊗ H S. However, this lies in F n + ℓ − X ′ ⊗ H S = L ′ n as I ⊂ F X ′ , see Remark 2.4. (cid:3) Normalizing elements not contained in L ′ when L = H ( d , χ, ω ) . Wewill henceforth assume that L is not isomorphic to H ( d , χ, ω ).We have seen above that we have a surjective homomorphisms algebras ι ∗ ⊗ H id L : L ′ → L along with a splitting ς ⊗ H id L : L → L ′ . Moreover, L ′ may be identifiedwith O ⊗ L , where O = k [[ x , . . . , x r ]], and the Lie bracket of L ′ , under theseidentification, extends O -bilinearly that of L .As L is a primitive Lie H -pseudoalgebra, which is not isomorphic to H ( d , χ, ω ),then L is simple, so thatDer( O b ⊗L ) = (Der O ⊗ id L ) ⋉ ( O b ⊗ Der L ) , see for instance [BDK, Proposition 6.12(ii)]. Here L is the annihilation algebra ofone of the primitive Lie pseudoalgebras, so Der L equals— k ad E + ad L when L = S ( d , χ ).— ad L ≃ L in all other cases.Notice that L is isomorphic to g , when L is a simple finite-dimensional Lie algebraover g , whereas L ≃ W N , (resp. K N ) when L = W ( d ) (resp. K ( d , θ )). Further-more, the non-inner derivation E stabilizes ad L and provides an explicit splittingof Der L / ad L ≃ k E . Set D to equal E when L = S ( d , χ ) and 0 when L = W ( d ) or K ( d , θ ). Then we may summarize the above information in the following Proposition 8.3.
Let L = H ( d , χ, ω ) be a primitive simple Lie pseudoalgebra, L ′ = Cur H ′ H L , and denote by L , L ′ the corresponding annihilation Lie algebras.Then Der L ′ = ((Der O ⊗ id L ) + ( O⊗ D )) ⋉ ( O b ⊗ ad L ) . (8.1)Let ∂ ∈ d ⊂ d ′ ⊂ d ⋉ L ′ = e L ′ . Then the adjoint action of ∂ on L ′ is a derivationof L ′ , and we may consider its projection π (ad ∂ ) ∈ (Der O ⊗ id L ) + O ⊗ D on thefirst summand of the above decomposition. As (8.1) is a semidirect sum, then π is a Lie algebra homomorphism, whence d ∋ ∂ π (ad ∂ ) is as well. When ∂ ∈ d ,denote now by b ∂ the element ∂ − [ π (ad ∂ )] ∈ e L ′ , where we denote by [ π (ad ∂ )] theunique element in L ′ inducing the inner derivation π (ad ∂ ) ∈ O b ⊗ ad L = ad L ′ .The action of b ∂ on L ′ coincides with π (ad ∂ ), so that η : d ∋ ∂ b ∂ ∈ (Der O ⊗ id L ) + O ⊗ D is a Lie algebra homomorphism. We denote by b d the Lie subalgebra consisting ofall b ∂, ∂ ∈ d . Remark . Of course, when L ≃ g is a simple Lie algebra over k , then d = (0),and η is the trivial map. Remark . The special case H = H ′ , L = L ′ has already been treated in [BDK1,BDK2, BDK3]. Here O = k , so that Der O = 0 and elements from b d act on L asmultiples of D .— b d coincides with e d ⊂ e L from [BDK1, BDK2] when L equals either W ( d ) or K ( d , θ ). As D = 0, the subalgebra b d ⊂ e L centralizes L .— b d is the same as b d ⊂ e L in [BDK1] when L = S ( d , χ ).As I ⊂ L ′ is an ideal, and the left action of d on L ′ stabilizes I , we concludethat all elements b ∂, ∂ ∈ d , stabilize I . Then Proposition 8.4.
Denote by (Der O ) the family of all derivations of O mapping O to its maximal ideal m . Then the adjoint action of b ∂ ∈ e L on L lies in (Der O ) ⊗ id L + O ⊗ D for all ∂ ∈ d .Proof. The derivation D stabilizes I = m b ⊗ L . Furthermore, elements in Der O⊗ id L stabilize I iff the corresponding derivation stabilizes I . (cid:3) Corollary 8.1.
Elements in b d normalize L ′ n and centralize L .Proof. Identify L ′ with X ′ ⊗ H S ≃ ( O b ⊗ X ) ⊗ H S ≃ O b ⊗ L . Under this identification L ′ n = F n + ℓ − X ′ ⊗ H S corresponds to X i + j = n m i b ⊗ L j . Now, D stabilizes each L j , so that O ⊗ D stabilizes all of the above summands.Elements from (Der O ) stabilize m i and do nothing on the tensor factor lying in L , so that they also stabilize all of the above summands. As the adjoint action of b d on L ′ lies in (Der O ) ⊗ id L + O ⊗ D , and D centralizes L , the claim is proved. (cid:3) Theorem 8.1.
For each n ≥ , the normalizer of L ′ n in e L ′ contains L ≥ + I + b d .Proof. We have proved above that the normalizer contains L ′ + I + b d . When L = W ( d ) , S ( d , χ ), then L ′ = P j m j b ⊗ L − j so that this equals L + I + b d , and L = L ≥ .When L = K ( d , θ ), recall that the filtrations {L n } and {L ≥ n } do not coincide.However, the normalizer of L ′ n also contains 1 ⊗ E ′ . Now use L ≥ = L + E ′ toobtain 1 ⊗ E ′ + L ′ + I + b d = L ≥ + I + b d . (cid:3) Finite-dimensional irreducible representations of ( L ≥ + I + b d ) / L ′ n . As N ′ = L ≥ + I + b d normalizes L ′ n in e L ′ , if V ′ is an e L ′ -module and R ⊂ V ′ is afinite-dimensional subspace killed by L ′ n , then the action of N ′ must stabilize R .However, if the action of N ′ on R is irreducible, a large part of N ′ will have to acttrivially on R . Proposition 8.5.
When n ≥ , the subalgebra ( L > + I + L ′ n ) / L ′ n is a solvableideal of N ′ / L ′ n .Proof. First of all, L ′ n ⊂ L ′ ⊂ L + I ⊂ L ≥ + I ⊂ N ′ , so that L ′ n is an ideal of N ′ . We already know that I is an ideal of the whole e L ′ , so I + L n is an ideal of N ′ as well. Furthermore, [ I j , I k ] ⊂ I j + k forces thelower central sequence of I to lie in I n +1 ⊂ L ′ n +2 − ℓ ⊂ L ′ n after n + 1 steps, so that( I + L ′ n ) / L ′ n is a nilpotent, hence solvable, ideal of N ′ / L ′ n . We are thus left withshowing that L > projects to a solvable ideal of N ′ / ( I + L ′ n ).Now, L > is a subalgebra of L ⊂ L ′ and the adjoint action of L ≥ + I + L ′ n maps it to L > + I + L ′ n , whence L > projects to an ideal of N ′ / ( I + L ′ n ). Again,[ L ≥ j , L ≥ k ] ⊂ L ≥ j + k and L ≥ i ⊂ L ⌊ ( i − / ⌋ , forces the lower central sequence of L > to lie inside L n ⊂ L ′ n ⊂ I + L ′ n within 2 n + 1 steps. Thus L ′ n projects to a solvableideal of N ′ / ( I + L ′ n ), proving the claim. (cid:3) Proposition 8.6.
Let R be a finite-dimensional irreducible N ′ -module, with atrivial action of L ′ n , where n ≥ . Then L > + I acts trivially on R .Proof. We already know that L > + I projects to a solvable ideal of the finite-dimensional Lie algebra N ′ / L n . Then we may use Proposition 6.1 and the argumentin [BDK1, Lemma 3.4] applied to both the semisimple part and the center of thereductive Lie algebra L ⊂ N in order to show that ( L > + I ) ∩O b ⊗ L ⊥ acts triviallyon R .However, both L > and m b ⊗ L ⊥ lie inside O b ⊗ L ⊥ , so they need to act triviallyon R . Recall now that I = m b ⊗ L ⊥ + m ⊗ D . We may then use Proposition 6.1to argue that elements from m ⊗ D ⊂ [ O b ⊗ L > , m b ⊗ L ⊥ ] arise as Lie brackets ofelements acting trivially on R . We thus conclude that both L > and I act triviallyon R . (cid:3) Irreducible representations and tensor modules.
As usual, L is a prim-itive simple Lie H -pseudoalgebra not isomorphic to H ( d , χ, ω ) and L ′ = Cur H ′ H L isthe corresponding current simple Lie pseudoalgebra.Let V ′ be a finite irreducible representation of L , which has a nontrivial pseu-doaction by definition. Then ker n V ′ := { v ∈ V ′ | L ′ n v = 0 } is a finite dimensionalvector space for all choices of n , and is nonzero for sufficiently large values of n , as V ′ is a discrete continuous representation of L ′ . As N ′ normalizes L ′ n , its actionon V ′ preserves ker n V ′ . Pick a minimal nonzero, hence irreducibile, N ′ -submodule R ⊂ V ′ . Then the action of N ′ = L ≥ + I + b d factors through the quotient N ′ / ( L > + I ) ≃ ( b d + L ≥ ) / L > , which is isomorphic to the direct sum Lie algebra d ⊕ L by Corollary 8.1. In particular, L ′ ⊂ L > + I acts trivially on R , showing R ⊂ ker V ′ . Theorem 8.2.
Let V ′ be a finite irreducible representation of L ′ = Cur H ′ H L , where L = H ( d , χ, ω ) is a primitive finite simple Lie pseudoalgebra. Then there exists RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 27 an irreducible finite-dimensional b d ⊕ L -module R such that V ′ is a quotient of thecurrent tensor module Cur H ′ H V ( R ) by a maximal submodule, where V ( R ) is a tensormodule for L as in Sections 5.1-5.3 or a finite-dimensional representation of theLie algebra L when d = (0) .Proof. We only deal with the d = (0) case, as the Lie algebra case is completelyanalogous. Recall that ker V ′ is finite-dimensional. If 0 = R ⊂ ker V ′ is aminimal, hence irreducible, N ′ -submodule, then the induced module Ind e LN ′ R mapssurjectively on V ′ by irreducibility. As e L ′ = d + N ′ , then Ind e LN ′ R ≃ U ( ∂ ′ ) ⊗ R ,and it is thus a finite left H ′ -module by multiplication on the first tensor factor.We know by Proposition 8.6 that L > + I acts trivially on R , so we may useTheorem 3.1 to write down the Lie pseudoalgebra action of L ′ on H ′ ⊗ R . Thisyields(1 ⊗ H s ) ∗ (1 ⊗ u ) = X K ∈ N N + r ( S ( ∂ ( K ) ) ⊗ ⊗ H ′ (1 ⊗ ( x K ⊗ H ′ s ) · u ) (8.2)= X K ∈ × N N ( S ( ∂ ( K ) ) ⊗ ⊗ H ′ (1 ⊗ ( x K ⊗ H ′ s ) · u ) , (8.3)where s ∈ S , u ∈ R and S ⊂ L is as in Section 4. As the only S ( ∂ ( K ) ) involved in(8.3) lie in d , and the corresponding Fourier coefficients x K ⊗ H ′ s all lie in L ⊂ L ′ ,we may use Proposition 6.2 to conclude that the pseudoaction s ∗ (1 ⊗ u ) = X K ∈ N N ( S ( ∂ ( K ) ) ⊗ ⊗ H (1 ⊗ ( x K ⊗ H ′ s ) · u )defines a Lie pseudoalgebra representation of L on H ⊗ R , which coincides with thecorresponding L -tensor module V ( R ). Then the L ′ -representation defined by (8.2)coincides with Cur H ′ H V ( R ). (cid:3) Corollary 8.2.
Every finite irreducible representation of L ′ = Cur H ′ H L , where L = H ( d , χ, ω ) is a primitive Lie pseudoalgebra, is of the form Cur H ′ H V , for somefinite irreducible L -module V .Proof. By Theorem 8.2, every finite irreducible L ′ -module V ′ arises as a quotientof Cur H ′ H V ( R ) for an opportune choice of a tensor module V ( R ) for the primitiveLie pseudoalgebra L . If V ( R ) is irreducible, then Cur H ′ H V ( R ) is irreducible as wellby Corollary 6.1. Thus V ( R ) ≃ Cur H ′ H V ( R ) and we are done.If V ( R ) is not irreducible, then we may find a maximal L ′ -submodule M ⊂ Cur H ′ H V ( R ) such that V ′ ≃ V ( R ) /M . As the L ′ -action on V ′ is nontrivial, we mayapply Corollary 6.2 and conclude that M = Cur H ′ H M , where M ⊂ V ( R ) is amaximal L -submodule. Then V ′ = (Cur H ′ H V ( R )) / (Cur H ′ H M ) ≃ Cur H ′ H ( V ( R ) /M )and V := V ( R ) /M is an irreducible L -module. Finiteness, i.e, Noetherianity, of V follows easily from finiteness of V ′ and exactness of the current functor. (cid:3) Irreducible representations of current Lie pseudoalgebras oftype H
In this section, we review the strategy of Section 8 in the case of the simple Liepseudoalgebra Cur H ′ H H ( d , χ, ω ), highlighting the relevant differences, which dependon the presence of nontrivial central elements in the corresponding annihilation algebras. Once again, we employ the notation introduced in Sections 4.3 and 5.4,so that L = H ( d , χ, ω ) = He satisfies[ e ∗ e ] = X k ( ∂ k ⊗ ∂ k ) ⊗ H e. Its annihilation algebra
P ≃ P N is an irreducible central extension of the simplelinearly compact Lie algebra H N from the Cartan list. More precisely, Z ( P ) isone-dimensional, spanned by e − χ ⊗ H e , and H := P /Z ( P ).9.1. Derivations of O b ⊗ P N . It is well known that Der H = k ad E + ad H , wheread E is the semisimple derivation of H N inducing the standard grading. As P ≃ P N is itself a graded Lie algebra, and its grading is compatible with that of its quotient H N , we will also denote by ad E the corresponding derivation of P N . Then Proposition 9.1.
One has
Der P = k ad E ⋉ ad P . Furthermore, Der( O b ⊗P ) = (Der O ⊗ id P ) ⋉ ( O b ⊗ Der P ) = (Der O ⊗ id P ⋉ O ⊗ ad E ) ⋉ O b ⊗ ad P . Lemma 9.1.
Let g be a Lie algebra, d ∈ Der g a derivation whose image is con-tained in the center Z ( g ) . If g = [ g , g ] , then d = 0 .Proof. One has d [ x, y ] = [ d ( x ) , y ] + [ x, d ( y )]. As Im d ⊂ Z ( g ), then d [ x, y ] = 0 forevery x, y ∈ g . (cid:3) Proof of Proposition 9.1. As H ≃ H N is a linearly compact simple Lie algebra,we already know from [BDK, Proposition 6.12 (ii)] that Der( O b ⊗H ) = Der O ⊗ id H + O b ⊗ Der H .Let δ ∈ Der( O b ⊗P ). Every derivation of a Lie algebra stabilizes its center,so δ induces a derivation e δ of ( O b ⊗P ) /Z ( O b ⊗P ) ≃ O b ⊗H . Then there exist d ∈ Der O , w ∈ H , c ∈ k , such that e δ ( φ ⊗ x ) = d ( φ ) ⊗ x + φ ⊗ ([ w, x ] + c ad E ( x )) , Choose a lifting w ∈ P of w ∈ H . Then δ ( φ ⊗ x ) ≡ d ( φ ) ⊗ x + φ ⊗ ([ w, x ] + c ad E ( x )) mod Z ( O b ⊗P ) , so that δ − ( d ⊗ φ ⊗ (ad w + c ad E )) is a derivation of O b ⊗P mapping everythingto the center. However, P equals its derived Lie algebra, so the same holds of O b ⊗P and we may use Lemma 9.1 to conclude. The case r = 0 , O = k takes care of thefirst claim. (cid:3) The normalizer of P ′ n , n > , in f P ′ . Let L = H ( d , χ, ω ), L ′ = Cur H ′ H L and denote by P , P ′ the corresponding annihilation algebras. We have seen that ι ∗ : X ′ → X induces a projection P ′ → P which admits a splitting; we thus obtain asubalgebra P ⊂ P ′ and the Lie bracket on P ′ ≃ O b ⊗ P , where O = ( H ′ H + ) ⊥ ⊂ X ′ ,extends O -bilinearly that of P . If I = ( H ′ H ) ⊥ , then X ′ = X + I and P ′ = P ⋉ I ,where I = I ⊗ H L . Notice that I corresponds to m b ⊗P under the identification P ′ ≃ O b ⊗P . As usual, we denote by {P j } , {P ′ j } the filtrations of P , P ′ defined asin Section 3.2, see also the beginning of Section 8. We already know that I and P ′ both normalize each P ′ n . We also denote by P k the graded component of P ofdegree k ≥ RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 29
Choose ∂ ∈ d ⊂ d ′ ⊂ d ⋉ P ′ =: f P ′ . Then the adjoint action of ∂ on P ′ definesa derivation of P ′ ≃ O b ⊗P . Using Proposition 9.1 we understand that projectingDer O b ⊗P on the first summand of the semidirect decompositionDer( O b ⊗P ) = (Der O ⊗ id P + O ⊗ ad E ) ⋉ O b ⊗ ad P defines a Lie algebra homomorphism, so that we may find γ ( ∂ ) ∈ P ′ so that theadjoint action of the difference b ∂ = ∂ − γ ( ∂ ) ∈ e P ′ on P ′ = O b ⊗P is induced byan element of Der O acting on the first tensor factor plus a suitable O -multiple ofad E . Notice that γ ( ∂ ) is only determined modulo Z ( P ′ ) = O ⊗ Z ( P ), but we maychoose it in a unique way if we demand that γ ( ∂ ) ∈ O b ⊗L − . Denote by b d the Liesubalgebra of e L ′ generated by elements b ∂, ∂ ∈ d . Notice that this is not k -linearlygenerated by the above elements. Proposition 9.2.
The adjoint action of b d on P ′ normalizes P and centralizes P .The map d ∋ ∂ [ b ∂ ] ∈ b d / ( b d ∩ Z ( P ′ )) is a Lie algebra isomorphism. Proposition 9.3.
The Lie algebra N ′ = b d + P ≥ + I + Z ( P ′ ) normalizes P ′ n forevery n > .Proof. We know that P ′ , I , b d and Z ( P ′ ) all normalize P ′ n , so that their sum does sotoo. However, from P ′ = P + I follows P ′ ⊂ P + I . As the filtration on P inducedby the grading coincide with our standard filtration, see Section 4.2, we also obtain P = P ≥ . We conclude that N ′ = b d + P ′ + I + Z ( P ′ ) = b d + P ≥ + I + O b ⊗ Z ( P )normalizes P ′ n . Notice that P = P ≥ ⊂ P ′ is a subalgebra and that I , Z ( P ′ ) areideals, so the sum P ≥ + I + Z ( P ′ ) is a subalgebra of P ′ . Also, b d is a subalgebraof b P ′ and normalizes P ≥ + I + Z ( P ′ ), so that N ′ is a subalgebra of f P ′ . (cid:3) Lemma 9.2.
The sum N ′ = ( b d + P + (span k h , x , . . . , x r i ⊗ Z ( P )) ⊕ ( P > + m b ⊗ P ⊥ + m b ⊗ Z ( P )) is a direct sum decomposition as vector subspaces. Moreover, the second sum-mand on the right-hand side is an ideal of N ′ . The quotient N ′ / ( P > + m b ⊗ P ⊥ + m b ⊗ Z ( P )) is thus isomorphic to ( b d + P + O b ⊗ Z ( P )) / ( m b ⊗ Z ( P )) . Proof.
Equality follows from P ≥ = P + P > and I = m b ⊗ P = m b ⊗ ( P ⊥ + Z ( P ))by noticing that Z ( P ′ ) = O b ⊗ Z ( P ) and O = span k h , x , . . . , x r i + m .The second summand on the right-hand side is a subalgebra of P ′ as P ⊥ = P ≥− .It is an ideal of N ′ since P > , P ⊥ , Z ( P ) are all homogeneous for the grading, andare thus b d + P -stable. (cid:3) Remark . Recall that b d = D + [ D , D ], where D = span k h ∂, ∂ ∈ d i . The Liealgebra b d is thus a finite-dimensional subalgebra of d + O b ⊗ Z ( P ) ⊂ d ⋉ P ′ ⊂ d ′ ⋉ P ′ =: f P ′ . We know that b d centralizes P , so that b d + P = b d ⊕ P is adirect sum of Lie algebras. However, b d does not necessarily centralize P ′ = O b ⊗ P ,though it normalizes each m j b ⊗ P k . Lemma 9.3.
Let L = H ( d , χ, ω ) , L ′ = Cur H ′ H L , and e P = d ⋉ P , f P ′ = d ′ ⋉ P ′ bethe corresponding extended annihilation algebras. Then the quotient Lie algebra N ′ k := ( b d + P + O b ⊗ Z ( P )) / ( m k b ⊗ Z ( P )) is isomorphic to — d ⊕ sp ( d , ω ) when k = 0 ; — d + ⊕ sp ( d , ω ) when k = 1 ,Thus, N ′ is an abelian extension of both N ′ and N ′ , and the intersection [ N ′ , N ′ ] ∩ ( O / m ) b ⊗ Z ( P ) is only contained in ( m / m ) b ⊗ Z ( P ) when χ = 0 and ω is exact.Proof. Case k = 0: First of all, the sum b d + P is a direct sum of Lie algebras, as b d centralizes P . Its intersection with O b ⊗ Z ( P ) may only lie in b d , as Z ( P ) lies indegree −
2, and is therefore trivial. The statement then follows from Remark 9.1.Case k = 1 follows from Proposition 2.2 (iii) . The last claim is then immediate. (cid:3) The following consequence of the Cartan-Jacobson theorem is well known.
Lemma 9.4.
Let → a i → e π → g → be an extension of the Lie algebra g by the abelian ideal a . If ( V, ρ ) is a finite-dimensional irreducible representation of e , then there exists a splitting (as vectorspaces) s : g → e such that ρ ◦ s : g → gl ( V ) is a Lie algebra homomorphism.Proof. By the Cartan-Jacobson theorem, each element from a ∈ i ( a ) act on V viamultiplication by a scalar ξ ( a ), thus yielding a linear form ξ : a → k whose kernelker ξ = a ∩ ker ρ has codimension at most 1 in a . If we set a = a / ker ξ, e = e / ker ξ, then e is an extension of g by the abelian ideal a , and ρ factors through e with anontrivial action of a on V .If a = 0, or equivalently ξ = 0, then choose any section s : g → a as vectorspaces. Then [ s ( g ) , s ( h )] = s ([ g, h ]) mod a for every g, h ∈ g , and as a ⊂ ker ρ weobtain that ρ ◦ s : g → gl ( V ) is a Lie algebra homomorphism.We may thus assume without loss of generality that ξ = 0 and dim a = 1.Nonzero elements in a act via multiplication by nontrivial scalars on V , so theycannot be commutators in e ; this forces the adjoint action of e on a to be trivial, sothat e is a central extension of g , and [ e , e ] to intersect a trivially, so that the centralextension is not irreducible, and must admit a Lie algebra splitting s : g → e . Anylifting s : g → e of s will then satisfy [ s ( g ) , s ( h )] = s ([ g, h ]) mod ker ξ , whence φ = ρ ◦ s : g → gl ( V ) is a Lie algebra homomorphism. (cid:3) Remark . If s : g → e is a splitting of (9.1), then every g ∈ e may be decomposedas g = s ( π ( g )) + ( g − s ( π ( g ))) , where g − s ( π ( g )) ∈ i ( a ). When φ = ρ ◦ s : g → gl ( V ) is a Lie algebra homomor-phism, then ρ ( g ) = φ ( π ( g )) + ξ ( i − ( g − s ( π ( g )))) id V . In other words, each finite-dimensional irreducible representation V of an abelianextension e of g may only differ from an opportune irreducible action of g on V bymultiples of id V .9.3. Finite dimensional irreducible representations of N ′ / P n , n ≥ . Oursetting is as in Section 7. L = H ( d , χ, ω ), L ′ = Cur H ′ H and e P , e P ′ are the cor-responding extended annihilation algebras. V ′ is an irreducible representation of L ′ , so that the L ′ -action on V ′ is non-trivial and ker n V ′ = { v ∈ V ′ | P ′ n v = 0 } is a nonzero finite-dimensional vector subspace of V ′ when for sufficiently largevalues of n . Then N ′ stabilizes ker n V ′ , as N ′ normalizes P ′ n . If R is a nonzero RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 31 N ′ -submodule of ker n V ′ , then Ind e P ′ N ′ R projects to V ′ . Without loss of generality,we may assume R to be N ′ -irreducible. The following fact is then going to proveuseful. Lemma 9.5.
Let R be an irreducible finite-dimensional N ′ -module with a trivialaction of P ′ n . Then P > + m b ⊗ P ⊥ + m b ⊗ Z ( P ) acts trivially on R .Proof. The descending central series of all three summands eventually sits inside P ′ n , so that their sum is an ideal, by Lemma 9.2, projecting to the radical of N ′ / P ′ n . However, the adjoint action of P on the first two summands decomposesas a sum of nontrivial irreducible representations, whereas the third summand liesin [ m b ⊗ P ⊥ , m b ⊗ P ⊥ ]. We may then use [BDK1, Lemma 3.4] to conclude. (cid:3) Proposition 9.4.
Let R be an irreducible finite-dimensional N ′ -module with atrivial action of P ′ n . Then the action of N ′ on R factors through N ′ / ( P > + m b ⊗ P ⊥ + m b ⊗ Z ( P )) ≃ ( b d ⊕ P ) + ( O / m ) b ⊗ Z ( P ) = N ′ . If ρ : N ′ → gl ( R ) denotes the above action, one may find an irreducible action φ : N ′ → gl ( R ) such that ρ ( x ) − φ ( x + m b ⊗ Z ( P ′ )) is a scalar multiple of id R forevery x ∈ N ′ . We may summarize the last proposition as follows: if V ′ is a finite irreduciblerepresentation of the Lie pseudoalgebra L ′ , then one may find a finite dimensionalirreducible N ′ -module R such that V ′ is a quotient of Ind e P ′ N ′ R , and the action of N ′ on R is uniquely described by that of d ⊕ P , along with a suitable scalar actionof the abelian ideal ( m / m ) b ⊗ Z ( P ), which however does not affect irreducibility of R . As e P ′ = d ′ + N ′ , then Ind e P ′ N ′ R is isomorphic to U ( d ′ ) ⊗ R , where the d ′ -actionis obtained by left multiplication on the first tensor factor. Theorem 9.1.
Let V ′ be a finite irreducible representation of the current Lie pseu-doalgebra L ′ = Cur H ′ H H ( d , χ, ω ) . Then there exists a finite-dimensional irreducible d + ⊕ sp ( d , ω ) -module R = Π ⊠ U such that V ′ is a quotient of the left H ′ -module H ′ ⊗ R , endowed with the pseudoaction e ∗ (1 ⊗ v ) = X ij ( ∂ i ∂ j ⊗ ⊗ H ′ (1 ⊗ f ij .v ) − X k ( ∂ k ⊗ ⊗ H ′ (1 ⊗ ( ∂ k + ad sp ( ∂ k )) .v − ∂ k ⊗ v )+ (1 ⊗ ⊗ H ′ (1 ⊗ c.v )+ ( t ⊗ ⊗ H ′ (1 ⊗ v ) . where v ∈ R and either t = 0 or t ∈ d ′ \ d .Proof. Let R ⊂ ker V ′ be a (finite-dimensional) irreducible N ′ -summand. Thenthe N ′ -action on R factors through the quotient N ′ , and one may use Lemma 9.4 tofind a section of N ′ in N ′ in order to express the above N ′ -action on R by means ofan irreducible action of N ′ ≃ d + ⊕ sp ( d , ω ), at least up to adding multiples of id R .One may then proceed similarly to [BDK3, Proposition 6.3], while expressing theaction of N ′ on v ∈ R as the sum of the N ′ -action and scalar multiples of v . The extra terms that arise, with respect to the ordinary tensor module pseudoaction,are of two kinds: first, there are terms of the form r X i =1 ( S ( ∂ i ) ⊗ ⊗ H ′ (1 ⊗ ( x i ⊗ c ) .v ) , (9.2)where c = e − χ ⊗ H e ∈ P ∈ Z ( P ) is the linear generator of the center of P , that occurbecause of the possibly nontrivial (scalar) action of ( m / m ) ⊗ Z ( P ) on R . Secondly,there are extra scalar terms which originate from expressing the N ′ -action via anirreducible N ′ -action. Adding things up yields a total contribution of the form( t ⊗ ⊗ H ′ (1 ⊗ v ) , where t is an element of d ′ whose projection to the vector space quotient d ′ / d provides a complete description of the central action of ( m / m ) ⊗ Z ( P ) by meansof a linear functional on m / m ≃ ( d ′ / d ) ∗ .Notice that when m b ⊗ Z ( P ) acts trivially, then both extra terms vanish and t = 0. If instead the m b ⊗ Z ( P ) is nontrivial, then the second summation lies in( d ⊗ ⊗ H ′ (1 ⊗ v ), and cannot cancel with the first contribution. (cid:3) We denote the above L ′ -module by V H χ,ω,t, d ′ ( R ). Remark . If t = 0, then V H χ,ω, , d ′ ( R ) ≃ Cur H ′ H V H χ,ω ( R ), where V H χ,ω ( R ) is as inSection 5.4. When t = 0, then ad χ preserves d and lies in sp ( d , ω ). In particular t := k t + d is a Lie subalgebra of d ′ strictly containing d , and V H χ,ω,t, d ′ ( R ) =Cur H ′ U ( t ) V H χ,ω,t, t ( R ).10. Singular vectors and irreducibility
Let L be one of the finite primitive simple Lie pseudoalgebras listed in Section4, and consider a finite representation V ′ of the current Lie pseudoalgebra L ′ =Cur H ′ H L . We know that ker n V ′ ⊂ V ′ is nonzero for sufficiently large values of n and is stabilized by the action of N ′ = b d + L ≥ + I ⊂ e L ′ . Moreover, every finite-dimensional irreducible N ′ -submodule of ker n V ′ lies in ker V ′ and has a trivialaction of the ideal Z = L > + m ⊗ L ⊥ + m k ⊗ L , where k = 2 when L = H ( d , χ, ω )and k = 1 otherwise. If L = H ( d , χ, ω ), then N ′ / Z ≃ b d ⊕ L is isomorphic to— d ⊕ gl d when L = W ( d );— d ⊕ sl d when L = S ( d , χ );— d ⊕ sp (ker θ, d θ ) when L = K ( d , θ ).When L = H ( d , χ, ω ), instead, N ′ / Z is an abelian extension of d + ⊕ sp ( d , ω ). Definition 10.1.
Let V ′ be a representation of a simple current Lie pseudoalgebra L ′ . An element v ∈ V ′ is a singular vector if Z .v = 0. The set of all singular vectorssing V ′ := { v ∈ V ′ | Z .v = 0 } ⊂ ker V ′ is a subspace of V ′ .We have already showed that sing V ′ is a finite-dimensional N ′ -submodule of V ′ as soon as V ′ is a finite representation of L with a nontrivial action, e.g., when V ′ is a finite irreducible L ′ -module. Proposition 10.1.
Let L be a primitive Lie pseudoalgebra, and S ⊂ L, ℓ ∈ N , asin Section 4. If V ′ is a finite Lie pseudoalgebra representation of L ′ = Cur H ′ H L , RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 33 then (1 ⊗ H S ) ∗ (sing V ′ ) ⊂ (F ℓ H ⊗ k ) ⊗ H ′ (F H ) · (sing V ′ ) + r X k =1 ( ∂ i ⊗ ⊗ H ′ (sing V ′ ) , and the second summand is possibly nonzero only when L = H ( d , χ, ω ) .Proof. Follows by using Theorem 3.1 with a PBW basis of H ′ corresponding to abasis { ∂ , . . . , ∂ N + r } of d ′ such that ∂ r +1 , . . . , ∂ N + r is a basis of d . (cid:3) We have already seen that Ind e L ′ N ′ R ≃ H ′ ⊗ R is a finite L ′ -module projectingto V ′ . It is (isomorphic to) a current representation Cur H ′ H V ( R ) when the secondsummand vanishes, and to V t, d ′ ( R ) , t ∈ d ′ \ d , otherwise.10.1. Singular vectors in current modules.
Whenever V is an L -module, weset ker V := { v ∈ V | L ∗ v = 0 } . It is clearly an H -submodule of V . Here wecompute singular vectors in the current representation Cur H ′ H V of Cur H ′ H L . It isconvenient to treat the case L = H ( d , χ, ω ) separately. Proposition 10.2.
Let L = H ( d , χ, ω ) be a primitive Lie pseudoalgebra as inSection 4. If V is an L -module, then sing Cur H ′ H V = 1 ⊗ H sing V + H ′ ⊗ H ker V .Proof. Let u = P K ∈ N r × ∂ ( K ) ⊗ H u K ∈ H ′ ⊗ H V . If a ∗ u K = P L ∈ × N N ( ∂ ( L ) ⊗ ⊗ H u K + L , then (1 ⊗ H a ) ∗ u = X K,L ( ∂ ( L ) ⊗ ∂ ( K ) ) ⊗ H ′ u K + L . Using Proposition 10.1 and Corollary 2.3, we argue that if u is singular, then u K + L = 0 whenever K = 0 or, equivalently, u K ∈ ker V ; similarly, u ∈ sing V . (cid:3) We may now step on to the case L = H ( d , χ, ω ). Lemma 10.1.
Let V be a Lie pseudoalgebra representation of L = H ( d , χ, ω ) = He .Then C ( V ) := { v ∈ V | e ∗ v ∈ (1 ⊗ ⊗ H V } equals ker V .Proof. Clearly, C ( V ) ⊂ sing V , as u ∈ V is singular precisely when e ∗ u ∈ (F H ⊗ k ) ⊗ H V . Then u ∈ C ( V ) implies sp ( d , ω ) .u = 0 and ∂u − ∂.u = 0 for all ∂ ∈ d ,where ∂.u denotes the action of ∂ ∈ d + on u ∈ sing V . However ∂.u = ∂u for every ∂ ∈ d implies([ ∂, ∂ ′ ]+ ω ( ∂ ∧ ∂ ′ ) c ) .u = [ ∂, ∂ ′ ] + .u = ∂. ( ∂ ′ .u ) − ∂ ′ . ( ∂.u ) = ∂ ( ∂ ′ u ) − ∂ ′ ( ∂u ) = [ ∂, ∂ ′ ] u, that is ω ( ∂ ∧ ∂ ′ ) c.u = 0 for all ∂, ∂ ′ ∈ d . As ω is nondegenerate, this forces c.u = 0,whence e ∗ u = 0. (cid:3) Proposition 10.3.
Let V be a Lie pseudoalgebra representation of H ( d , χ, ω ) .Then sing Cur H ′ H V = 1 ⊗ H sing V + H ′ ⊗ H ker V .Proof. We shall prove sing Cur H ′ H V = 1 ⊗ H sing V + d ′ ⊗ H C ( V ) + H ′ ⊗ H ker V .Then the claim follows using the previous lemma. First of all, Proposition 10.1implies (1 ⊗ H e ) ∗ (sing V ′ ) ⊂ ((F H ⊗ k ) + ( k ⊗ d ′ )) ⊗ H ′ V ′ . Now proceed as in Proposition 10.2 and compute (1 ⊗ H e ) ∗ u . If { ǫ i , ≤ i ≤ r } denotes the standard canonical basis of N r , then Corollary 2.3 yields u ǫ i ∈ C ( V ),and a ∗ u K = 0 if K = 0 , ǫ i , whence u K ∈ ker V . Similarly, u ∈ sing V . (cid:3) Corollary 10.1.
Let V be a finite irreducible module over the primitive Lie pseu-doalgebra L . Then sing Cur H ′ H V = 1 ⊗ H sing V .Proof. Follows from ker V = 0. (cid:3) Remark . Recall that the embedding ι : H → H ′ is a pure ring homomorphism,so that m ⊗ H m provides an injective H -linear homomorphism M → Cur H ′ H M for every left H -module M . Thus each Lie H -pseudoalgebra embeds H -linearly inthe corresponding current Lie pseudoalgebras, and similarly for representations.In particular, the subspace 1 ⊗ H sing V in the previous corollary is isomorphicto sing V , both as a vector space and as a d ⊕ L -module (resp. d + ⊕ L -modulewhen L is of type H).10.2. Irreducibility of V H χ,ω,t, d ′ ( R ) . Let V H χ,ω,t, d ′ ( R ) , t ∈ d ′ \ d , be a Lie pseudoal-gebra representation of L ′ = Cur H ′ H H ( d , χ, ω ) as introduced in Section 7. We willnow investigate irreducibility of V H χ,ω,t, d ′ ( R ) by explicitly computing its singularvectors. Theorem 10.1.
Let d ⊂ d ′ be Lie algebras, and t ∈ d ′ \ d an element satisfying theconditions of Proposition 7.1. Then sing V H χ,ω,t, d ′ ( R ) = k ⊗ R . As a consequence,the L ′ -module V H χ,ω,t, d ′ ( R ) , t ∈ d ′ \ d , is irreducible as soon as R is an irreducible d + ⊕ sp ( d , ω ) -module.Proof. The representation V H χ,ω,t, d ′ ( R ) = H ′ ⊗ R of L ′ = Cur H ′ H H ( d , χ, ω ) is definedby the Lie pseudoalgebra action e ∗ t (1 ⊗ v ) = X ij ( ∂ i ∂ j ⊗ ⊗ H ′ (1 ⊗ f ij .v ) (10.1) − X k ( ∂ k ⊗ ⊗ H ′ (1 ⊗ ( ∂ k + ad sp ∂ k ) .v − ∂ k ⊗ v )+ (1 ⊗ ⊗ H (1 ⊗ c.v ) + ( t ⊗ ⊗ H ′ (1 ⊗ v ) . Expression (10.1) may be right-straightened to e ∗ t (1 ⊗ v ) = X i,j (1 ⊗ ∂ i ∂ j ) ⊗ H ′ (1 ⊗ f ij .v ) (10.2)+ X k (1 ⊗ ∂ k ) ⊗ H ′ (1 ⊗ ( ∂ k + ad sp ( ∂ k )) .v − ∂ k ⊗ v ) − (1 ⊗ t ) ⊗ H ′ (1 ⊗ v )+ terms in ( k ⊗ F H ) ⊗ H ′ (F H ′ ⊗ V ) . Now pick a basis t = ∂ , . . . , ∂ N + r of d ′ so that ∂ r +1 , . . . , ∂ N + r is a basis of d . Usethe corresponding PBW basis to express0 = u = X L ∈ Z N + r ∂ ( L ) ⊗ u L . (10.3)Plugging (10.3) into e ∗ t u yields a right-straightened expression. If n is the maximalvalue of | L | such that u L = 0, choose among all such L = ( l , l , . . . , l N + r ) with | L | = n one with the highest value of l . Then the term multiplying 1 ⊗ t∂ ( L ) in e ∗ t u equals 1 ⊗ u L . If u is singular and L = 0, this must vanish by Proposition 10.1,leading to a contradiction. We conclude that sing V t, d ′ = k ⊗ R . The remainingclaim follows easily. (cid:3) RREDUCIBLE MODULES OVER FINITE SIMPLE LIE PSEUDOALGEBRAS IV 35
Classification of irreducible modules over non primitive simpleLie pseudoalgebras
We summarize all previous results in the following theorem
Theorem 11.1.
Let d ⊂ d ′ be finite-dimensional Lie algebras, H ⊂ H ′ their uni-versal enveloping algebras endowed with the canonical cocommutative Hopf algebrastructure. The following is a complete list of finite irreducible representations of thecurrent Lie pseudoalgebra L ′ = Cur H ′ H L , where L is a primitive Lie pseudoalgebra: — Cur H ′ H V , where V is a finite irreducible L -module; — V H χ,ω,t, d ′ ( R ) , where L = H ( d , χ, ω ) , R is a finite-dimensional irreduciblerepresentation of d + ⊕ sp ( d , ω ) , and t ∈ d ′ \ d satisfies(i) ad χ t preserves d and lies in sp ( d , ω ) ;(ii) [ s, t ] = 0 , where s satisfies χ = ι s ω .The only nontrivial isomorphisms between the above irreducible modules are thosedescribed in Theorem 7.1.Proof. We are only left with proving that representations in the list are pairwisenon-isomorphic. This is done by comparing the N ′ -actions on the space of singularvectors contained in each representation.First of all, sing Cur H ′ H V = sing V , and irreducible representations of a simpleprimitive Lie pseudoalgebra are all told apart by their singular vectors, viewed asa d ⊕ L -module. This takes care of the cases L = W ( d ) , S ( d , χ ) , K ( d , θ ).When L = H ( d , χ, ω ), no current irreducible representation is isomorphic toa non-current one as the action of central elements lying in m b ⊗ Z ( L ) is trivialin the former case and nontrivial in the latter. Furthermore, any isomorphism V H χ,ω,t, d ′ ( R ) ≃ V H χ,ω,t, d ′ ( R ) induces an isomorphism of the corresponding sin-gular vectors, viewed as a d + ⊕ sp ( d , ω )-module. However, singular vectors areall constant and lie in a single d + ⊕ sp ( d , ω )-component, which is isomorphic to R , R , respectively. Thus, for equal values of t , the Cur H ′ H H ( d , χ, ω )-modules V H χ,ω,t, d ′ ( R ) , V H χ,ω,t, d ′ ( R ) are isomorphic precisely when R , R are isomorphic rep-resentations of d + ⊕ sp ( d , ω ).Finally recall that the action on V H χ,ω,t, d ′ ( R ) of central elements lying in m b ⊗ Z ( L )is via multiplication by scalars, and defines a linear functional m / m → k . However, m / m ≃ ( d ′ / d ) ∗ and this linear functional corresponds to a unique class in the vectorspace quotient d ′ / d , which coincides with [ t ]. This shows that if V H χ,ω,t, d ′ ( R ) and V H χ,ω,t ′ , d ′ ( R ′ ) are isomorphic, then elements t, t ′ project to the same class of thequotient d ′ / d . Then their difference δ = t ′ − t ∈ d satisfies the conditions in Lemma4.1, so that Theorem 7.1 yields V H χ,ω,t ′ , d ′ ( R ′ ) = V H χ,ω,t, d ′ ( R ′ ⊗ ( k π ∗ ( ι δ ω ) ⊠ k )), andone falls back to the previous case. (cid:3) References []BDK B. Bakalov, A. D’Andrea, and V. G. Kac,
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