A generalization of Lazard's elimination theorem
aa r X i v : . [ m a t h . R T ] D ec A GENERALIZATION OF LAZARD’S ELIMINATION THEOREM
ELIZABETH JURISICH AND ROBERT WILSON
Abstract.
Using the classical Lazard’s elimination theorem, we obtain a decompositiontheorem for Lie algebras defined by generators and relations of a certain type. This is apreprint version of the paper appearing in Communications in Algebra Volume 32, Issue10, 2004. Introduction
This paper grew out of, and has its main application in, the theory of generalized Kac-Moody Lie algebras over a field Φ. Borcherds initiated the study of these algebras in [2],and applied the theory in his proof of the Conway-Norton conjectures [3]. Generalized Kac-Moody Lie algebras may be defined by generators and relations (see [7]). While studyingthese algebras it is natural to consider an algebra of the form L ( V ⊕ W ) /I where L ( X ) denotes the free Lie algebra on the vector space X , V and W are vector spaceswith V ∩ W = (0), L ( V ⊕ W ) is graded by giving nonzero elements of V degree zero andnonzero elements of W degree one, and where I is an ideal generated by homogeneouselements of degree zero or one. Theorem 1 gives the structure of such an algebra.We write U ( r ) for the universal enveloping algebra of the Lie algebra r . The algebra U ( r ) acts on r via the adjoint action; a · b denotes the image of b in r under the action of a ∈ U ( r ).Our main result, Theorem 1, generalizes and is proved using the following theorem ofLazard [10], [4, Proposition 10]. Let M = U ( L ( V )) · W ⊂ L ( V ⊕ W ). Note that M is a U ( L ( V ))-module and so is an L ( V )-module. Theorem 1 (Lazard’s Elimination Theorem) . The ideal of L ( V ⊕ W ) generated by W isisomorphic to L ( M ) and therefore L ( V ⊕ W ) ∼ = L ( V ) ⋉ L ( M ) . R. Block has pointed out to us that this theorem is proven in [4] only for finite dimen-sional V and W . Work of Block and Leroux [1] shows that the theorem holds in general.Several generalizations of Lazard’s elimination theorem are known: [1] (giving a generalcategory theoretic result which in the special case of a free Lie algebra gives Lazard’s theo-rem), [6] (treating certain generalized Kac-Moody Lie algebras), [7] (treating all generalizedKac-Moody Lie algebras), [11] (treating the case in which I is generated by elements of de-gree zero). All of these results on Lie algebras are contained in Theorem 1. In the specialcase pertinent to the Conway-Norton conjectures this theorem yields the decomposition appearing in [6]. This decomposition simplifies part of the proof of the Conway-Nortonconjectures [6, 9]. [5], which treats the case in which I is generated by a collection, θ , ofelements of the form [ r, s ] where r, s ∈ a basis for V ⊕ W , follows from Theorem 1 only forcertain θ (those in which, for every pair r, s , we have r, s ∈ V S W and at least one of r, s is in V ).Section 2 contains the statement and proof of Theorem 1. Section 3 discusses applicationsto generalized Kac-Moody Lie algebras.2. Main Result If r is a Lie algebra and S ⊂ r we let h S i r denote the ideal of r generated by S . Of course h S i r = U ( r ) · S . Lemma 1. If U ⊇ W then L ( U ) / h W i L ( U ) ∼ = L ( U/W ) .Proof. Write U = V ⊕ W , so V ∼ = U/W . Then L ( U ) / h W i L ( U ) = L ( V ⊕ W ) / h W i L ( U ) = ( L ( V ) ⋉ h W i L ( U ) ) / h W i L ( U ) ∼ = L ( V ) ∼ = L ( U/W ) . (cid:3) Now, as in Lazard’s Elimination Theorem, let M = U ( L ( V )) · W ⊆ L ( V ⊕ W ) . Then,as a corollary of that theorem we obtain:
Lemma 2. (1) U ( L ( V ⊕ W )) = U ( L ( M )) U ( L ( V ))(2) U ( L ( V ⊕ W )) = U ( L ( V )) + U ( L ( M )) M U ( L ( V )) .Proof. By the elimination theorem L ( V ⊕ W ) = L ( V ) ⋉ L ( M ) (where we identify h W i L ( V ⊕ W ) with L ( M )), so (1) follows by the Poincar´e-Birkhoff-Witt Theorem. Also, as Φ is the basefield U ( L ( M )) = Φ + U ( L ( M )) L ( M )= Φ + U ( L ( M )) M so (2) follows from (1). (cid:3) Now let A ⊂ L ( V ) ⊂ L ( V ⊕ W ), B ⊂ M ⊂ L ( M ) ⊂ L ( V ⊕ W ). Thus, if L ( V ⊕ W ) isgraded by giving nonzero elements of V degree zero and nonzero elements of W degree one,then A is an arbitrary subspace of elements of degree zero and B is an arbitrary subspaceof elements of degree one. Write g = L ( V ) / h A i L ( V ) , M = { [ M, h A i L ( V ) ] + U ( L ( V )) · B } and N = M/M .The following theorem, our main result, gives the structure of the quotient algebra L ( V ⊕ W ) / h A, B i L ( V ⊕ W ) . Theorem 2. (1)
The space N is a g -module. GENERALIZATION OF LAZARD’S ELIMINATION THEOREM 3 (2) L ( N ) is isomorphic as a g -module to the ideal of L ( V ⊕ W ) / h A, B i L ( V ⊕ W ) generatedby the image of W . (3) L ( V ⊕ W ) / h A, B i L ( V ⊕ W ) ∼ = g ⋉ L ( N ) .Proof. To prove (1), note M = U ( L ( V )) · W is an L ( V )-module as is U ( L ( V )) · B . Since M and h A i L ( V ) are L ( V )-modules, so is [ M, h A i L ( V ) ]. Thus N is an L ( V )-module. Since[ h A i L ( V ) , M ] ⊂ M , N is a g -module.We now prove (2): h A, B i L ( V ⊕ W ) = h A i L ( V ⊕ W ) + h B i L ( V ⊕ W ) = U ( L ( V ⊕ W )) · A + U ( L ( V ⊕ W )) · B. By Lemma 2 this is equal to U ( L ( V )) · A + U ( L ( M )) M U ( L ( V )) · A + U ( L ( M )) U ( L ( V )) · B = h A i L ( V ) + h [ M, h A i L ( V ) ] i L ( M ) + h U ( L ( V )) · B i L ( M ) = h A i L ( V ) + h M i L ( M ) . (*)Now the ideal of L ( V ⊕ W ) / h A, B i L ( V ⊕ W ) generated by W is( L ( M ) + h A, B i L ( V ⊕ W ) ) / h A, B i L ( V ⊕ W ) ∼ = L ( M ) / ( L ( M ) ∩ h A, B i L ( V ⊕ W ) ) . By equation (*) this is equal to L ( M ) / h M i L ( M ) . By Lemma 1 this is L ( M/M ) = L ( N ).Furthermore, L ( V ⊕ W ) / h A, B i L ( V ⊕ W ) ∼ = ( L ( V ) ⋉ L ( M )) / h A, B i L ( V ⊕ W ) . By equation(*) this is isomorphic to L ( V ) / h A i L ( V ) ⋉ L ( M ) / h M i L ( M ) = g ⋉ L ( N ) , so we have proven (3). (cid:3) Applications
Let I be an index set which is finite or countably infinite and let R ⊂ I × I . Let n bethe Lie algebra with generators X = { x i | i ∈ I } and relations (ad x i ) n ij x j for ( i, j ) ∈ R .We may assume that R does not contain diagonal elements ( i, i ) ∈ I because [ x i , x i ] = 0in L ( X ).If J ⊂ I , let n J denote the subalgebra of n generated by the x i for i ∈ J . Theorem 1gives: Theorem 3.
Let n , I, R be as above. Suppose that for some choice of S, T ⊂ I , I = S ∪ T (disjoint union), and i, j ∈ T with i = j implies ( i, j ) / ∈ R . Then n ∼ = n S ⋉ L ( U ( n S ) · W ) where W denotes the vector space spanned by the x i for i ∈ T .Proof. Let V be the vector space with basis x i , i ∈ S . Take A = { (ad x i ) n ij x j | i, j ∈ S } and B = { (ad x i ) n ij x j | i ∈ S, j ∈ T } . Then Theorem 1 gives the above decomposition, wherewe write N as the n S -module in L ( V ⊕ W ) / h A i L ( V ⊕ W ) generated by W . (cid:3) ELIZABETH JURISICH AND ROBERT WILSON
Let l be a generalized Kac-Moody algebra associated to a symmetrizable matrix ( a ij ) i,j ∈ I .By Proposition 1.5 [6] one has l = n + ⊕ h ⊕ n − . Because the radical (the maximal gradedideal not intersecting h ) is zero (see [6],[7]), the subalgebras n ± can be written as n above,choosing X = { x i = e i | i ∈ I } for n + and X = { x i = f i | i ∈ I } for n − where e i and f i , i ∈ I are the Chevalley generators of l . The Serre relations (ad x i ) n ij x j = 0 occur whenever a ii >
0, or when a ii ≤ a ij = 0. If we take R to be the set corresponding to theoccurrence of Serre relations and S, T as in Theorem 2, then applying Theorem 2 to both n + and n − gives Theorem 3.19 of [7]: Corollary 4.
Corollary 3 Let l be a generalized Kac-Moody algebra associated to a sym-metrizable matrix ( a ij ) i,j ∈ I . Let R denote the set { ( i, j ) | a ii > or a ii ≤ and a ij = 0 } ⊂ I × I . Choose S, T so that I = S ∪ T (disjoint union), and i, j ∈ T implies ( i, j ) / ∈ R . Let l be the subalgebra of l generated by the e i and f i with i ∈ S . Then l = u + ⊕ ( l + h ) ⊕ u − ,where u − is the free Lie algebra on the direct sum of the standard highest weight l -modules U ( n − S ) · f j for j ∈ T and u + is the free Lie algebra on the direct sum of the standard lowestweight l -modules U ( n + S ) · e j for j ∈ T . Theorem 5.1 of [6] is a special case of this Corollary. As noted in [7] one can iteratethis decomposition until l is a semi-simple or Kac-Moody subalgebra. The results of [5]on free partially commutative Lie algebras may be obtained as the case where all a ii < l from the identity for the subalgebra l . Conversely, one can prove Corollary 3 using thedenominator and character formulas for generalized Kac-Moody algebras. (This is theproof in [7].) Other applications include computing the homology of the Lie algebra overa standard module, and determining a class of completely reducible modules [8]. References [1] R. Block and P. Leroux,
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