aa r X i v : . [ m a t h . QA ] J un ACTA MATHEMATICA SPALATENSIA
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A note on symmetric orderings
Zoran ˇSkoda
Abstract
Let ˆ A n be the completion by the degree of a differential operator of the n -th Weyl algebra with generators x , . . . , x n , ∂ , . . . , ∂ n . Consider n elements X , . . . , X n in ˆ A n of the form X i = x i + ∞ X K =1 n X l =1 n X j =1 x l p K − ,lij ( ∂ ) ∂ j , where p K − ,lij ( ∂ ) is a degree ( K −
1) homogeneous polynomial in ∂ , . . . , ∂ n ,antisymmetric in subscripts i, j . Then for any natural k and any function i : { , . . . , k } → { , . . . , n } we prove X σ ∈ Σ( k ) X i σ (1) · · · X i σ ( k ) ⊲ k ! x i · · · x i k , where Σ( k ) is the symmetric group on k letters and ⊲ denotes the Fock actionof the ˆ A n on the space of (commutative) polynomials. Keywords: Weyl algebra, symmetric orderingMath. Subj. Class.: 16S30,16S32
In an earlier article [3], we derived a universal formula for an embedding ofthe universal enveloping algebra U ( g ) of any Lie algebra g with underlyingrank n free module over a commutative ring k containing the field Q ofrational numbers into a completion ˆ A n, k of the n -th Weyl algebra over k .1 efinition 1. The n -th Weyl algebra A n, k over a commutative ring k is theassociative k -algebra defined by generators and relations as follows: A n, k := k h x , . . . , x n , ∂ , . . . , ∂ n i / h [ x i , x j ] , [ ∂ i , ∂ j ] , [ x i , ∂ j ] − δ ji , i, j, = 1 , . . . , n i . We use the “contravariant” notation for the generators of A n, k ([7], 1.1)and δ ij is the Kronecker symbol. The reader should recall the usual interpre-tation of the Weyl algebra elements as regular differential operators [2, 3]. Inother words, the elements of A n, k act on the polynomial algebra k [ x , . . . , x n ],consisting of commutative polynomials via the physicists’ Fock action heredenoted by ⊲ : A n, k ⊗ k [ x , . . . , x n ] → k [ x , . . . , x n ]. By definition, generators x i act as the multiplication operators by x i and ∂ j act as partial derivatives.The unit polynomial 1 ∈ k [ x , . . . , x n ] is interpreted as the vacuum state.Complete A n, k along the filtration given by the degree of differential ope-rator ([3, 7, 8]); the completion will be denoted ˆ A n, k . Thus, the elements inˆ A n, k can be represented as arbitrary power series in ∂ , . . . , ∂ n with coeffici-ents (say on the left) in the polynomial ring k [ x , . . . , x n ].For a fixed basis X g , . . . , X g n of g , denote by C kij ∈ k for i, j, k ∈ { , . . . , n } the structure constants defined by[ X g i , X g j ] = n X k =1 C kij X g k . (1)Constants C kij are antisymmetric in lower indices and satisfy a quadraticrelation reflecting the Jacobi identity in g . According to [3], there is a uniquemonomorphism of k -algebras ι : U ( g ) → ˆ A n, k extending the formulas X g i ι ( X g i ) = n X l =1 x l ∞ X N =0 ( − N N ! B N ( C N ) li , (2)where B N is the n -th Bernoulli number and C is an n × n matrix with valuesin k , defined by C ij = n X k =1 C ijk ∂ k . The monomorphism ι does not depend on the choice of the basis; over R and C the formula (2) appeared to be known much before ([1, 5]) and, suitablyinterpreted, corresponds to the Gutt’s star product [4]. A simple differentialgeometric derivation of the formula (2) over R is explained in detail in [7],2ection 1.2. Similarly, Sections 7-9 of [3] provide a geometrical derivation informal geometry over any ring containing rationals. See also [6] for anotherpoint of view. Expression (2) is related to the part of Campbell-Baker-Hausdorff series linear in the first argument ([3], Sections 7-9). Denote by e g : k [ x , . . . , x n ] → U ( g ) , x α · · · x α k k ! X σ ∈ Σ( k ) X g α σ · · · X g α σk (3)the standard symmetrization (or coexponential) map (of vector spaces), wherethe symmetric group on k letters is denoted Σ( k ). Via monomorphism ι , theexpression on the right-hand side of (3) can be interpreted in ˆ A n, k . If weapply the resulting element of ˆ A n, k on 1 using (the formal completion of) theFock action, we recover back the left-hand side of (3). In other words ([3, 8]),(( ι ◦ e g )( q )) ⊲ q, q ∈ k [ x , . . . , x n ] , (4)where ⊲ denotes the Fock action by differential operators.In this paper, it is proven that already the tensorial form, X i ˜ X i := n X l =1 x l ∞ X N =0 A N ( C N ) li , (5)of the universal formula (2), with A = 1, guarantees in characteristic 0 thatprecisely the symmetrically ordered noncommutative expressions1 k ! X σ ∈ Σ( k ) X α σ (1) · · · X α σ ( k ) , interpreted via the embedding (5), and after acting upon the vacuum, recoverback the commutative product x α · · · x α k . The coefficients A N in (5) may bearbitrary for N > A = 1, instead of the choice A N = ( − N N ! B N for all N , and X i may be generators of an arbitrary finitely generated associative k -algebra U , instead of the motivating choice X i = X g i ∈ U ( g ).Even more generally, we may replace A N ( C N ) li in (5) by any expressionof the form p N − ,lij ( ∂ , . . . , ∂ n ) ∂ j provided that p N − ,lij = p N − ,lij ( ∂ , . . . , ∂ n ) isa homogeneous polynomial of degree ( N −
1) in ∂ , . . . , ∂ n , antisymmetricunder interchange of i and j . Note that the previous case involving U ( g )may be recovered by setting p N − ,lij = ( − N B N N ! n X s =1 ( C N − ) ls C sij .
3e do not discuss when the correspondence (5) (or its generalization invol-ving p N − ,lij ) extends to a homomorphism U → ˆ A n, k of algebras (in physicsliterature also called a realization of U ). If U is tautologically defined as thesubalgebra of ˆ A n, k generated by the expressions ˜ X i ∈ ˆ A n, k , we alert the re-ader that the corresponding PBW type theorem often fails and the dimensionof the space of degree k > X i genericallyexceeds the dimension of the space of symmetric polynomials of degree k .In the rest of the article below, X i -s are defined as elements in ˆ A n, k fromthe start, hence we proceed without a distinction between X i and ˜ X i . Theorem 2.
Assume k is a field of characteristic different from . Let X i = x i + n X l =1 x l ∞ X N =1 n X j =1 p N − ,lij ( ∂ , . . . , ∂ n ) ∂ j , i = 1 , . . . , n, (6) be n distinguished elements of ˆ A n, k , where p N − ,lij ( ∂ , . . . , ∂ n ) are arbitraryhomogeneous polynomials of degree ( N − in ∂ , . . . , ∂ n , antisymmetric inlower indices i, j . Let α : { , . . . , k } → { , . . . , n } be any function. Then, inthe index notation, α i = α ( i ) , X σ ∈ Σ( k ) X α σ (1) · · · X α σ ( k ) ⊲ k ! x α · · · x α k . (7) Proof.
We prove the theorem by induction on degree k . For k = 1 allterms with N ≥ k , we write the sum (7) over all permutations in Σ( k ) in adifferent way. We use the fact that the set of permutations of n elementsΣ( n ) is in the bijection with the set of pairs ( i, ρ ) where 0 ≤ i ≤ k and ρ ∈ Σ( k − i, ρ ) σ, σ ( k ) := i, k = 1 ,ρ ( k − , k > ρ ( k − < i,ρ ( k −
1) + 1 , k > ρ ( k − ≥ i. For example, (3 , (2 , , , , (3 , , , , , i : { , . . . , k − } → { , . . . , i − , i + 1 , . . . , k } byΘ i ( j ) := (cid:26) j, j < i,j + 1 , j ≥ i. Clearly now σ ( j + 1) = Θ i ( ρ ( j )) for 1 ≤ j < k .We may thus renumber the sum X σ ∈ Σ( k ) X α σ (2) · · · X α σ ( k ) as the double sum k X i =1 X α ( i ) · X ρ ∈ Σ( k − X ( α ◦ Θ i )( ρ (1)) · · · X ( α ◦ Θ i )( ρ ( k − By the assumption of induction, X ρ ∈ Σ( k − X ( α ◦ Θ i )( ρ (1)) · · · X ( α ◦ Θ i )( ρ ( k − ⊲ k − x ( α ◦ Θ i )(1) · · · x ( α ◦ Θ i )( k − The function Θ i takes all values between 1 and k except i exactly once.Therefore, the left-hand side of (7) may be rewritten as( k − k X i =1 X α ( i ) ⊲ ( x α (1) · · · x α ( i − x α ( i +1) · · · x α ( k ) ) . (8)Substituting the expression (6) for X α ( i ) in (8) we immeditaly observe twosummands. Let δ be the Kronecker symbol. Then the first summand is( k − k X i =1 n X r =1 x r δ rα ( i ) · ( x α (1) · · · x α ( i − x α ( i +1) · · · x α ( k ) ) = k ! x α (1) · · · x α ( k ) , yielding the desired right-hand side for (7). Hence for the step of inductionon k it is sufficient to show that the remaining summand( k − k X i =1 ∞ X N =1 n X l =1 x l n X s =1 p N − ,lα ( i ) s ∂ s ( x α (1) · · · x α ( i − x α ( i +1) · · · x α ( k ) )5anishes. This follows if for any N > k X i =1 n X s =1 p N − ,lα ( i ) s ∂ s ( x α (1) · · · x α ( i − x α ( i +1) · · · x α ( k ) ) = 0 . (9)Let s ∈ { , . . . , n } and M ( s ) = { j ∈ { , . . . , i − , i + 1 , . . . , k }| s = α ( j ) } . Byelementary application of partial derivatives, ∂ s ( x α (1) · · · x α ( i − x α ( i +1) · · · x α ( k ) ) = X j ∈ M ( s ) Y r ∈{ ,...,k }\{ i,j } x α ( r ) . (10)In particular, the contributions from s / ∈ { α (1) , . . . , α ( i − , α ( i +1) , . . . , α ( k ) } ,that is for M ( s ) = ∅ , vanish and ∂ s ( x α (1) · · · x α ( i − x α ( i +1) · · · x α ( k ) ) = 0.Thus, for fixed i , the overall sum over all s ∈ { , . . . , n } becomes a newsum over all j ∈ { , . . . , i − , i + 1 , . . . , k } and each j = i appears preciselyonce, namely for s = α ( j ). For fixed pair ( i, s ), notice that the summandsdo not depend on j ∈ M ( s ), but we do not use this fact. By antisymmetry, p N − ,lα ( i ) α ( i ) = 0 if char k = 2, hence we are free to add any terms multiplied by p N − ,lα ( i ) α ( i ) . For fixed i , we conclude n X s =1 p N − ,lα ( i ) s ∂ s ( x α (1) · · · x α ( i − x α ( i +1) · · · x α ( k ) ) = k X j =1 p N − ,lα ( i ) α ( j ) Y r ∈{ ,...,k }\{ i,j } x α ( r ) . Regarding that Q r ∈{ ,...,k }\{ i,j } x α ( r ) is a symmetric tensor in i, j , and p N − ,lα ( i ) α ( j ) is antisymmetric under exchange of i and j , their contraction must be zero, k X i =1 k X j =1 p N − ,lα ( i ) α ( j ) Y r ∈{ ,...,k }\{ i,j } x α ( r ) = 0 . Therefore, (9) follows, and consequently the step of induction on k . (cid:3) The reader may want to understand the reindexing and cancellation ar-guments following formula (10) on an example where α is not injective. Sup-pose n = 3, k = 4, and α sends 1 , , , , , , P i =1 P s =1 p N − ,lα ( i ) s ∂ s (cid:16)Q r = i x α ( r ) (cid:17) has contributions as follows: for i = 1one obtains P s p N − ,l s ∂ s ( x x x ) = p N − ,l x x + 2 p N − ,l x x , for i = 2 and i = 3 equal contributions P s p N − ,l s ∂ s ( x x x ) = p N − ,l x x + p N − ,l x x + p N − ,l x x , and for i = 4 one obtains P s p N − ,l s ∂ s ( x x x ) = p N − ,l x x +2 p N − ,l x x . By the antisymmetry of p N − ,l , the double sum is 0.6 orollary 3. Under the assumptions of Theorem 2, there is a well-defined k -linear map ˜ e : k [ x , . . . , x n ] → ˆ A n, k extending the formulas ˜ e : x α · · · x α k X σ ∈ Σ( k ) X α σ (1) · · · X α σ ( k ) , (11) for all k ≥ and for all (nonstrictly) monotone α : { , . . . , k } → { , . . . , n } .Map ˜ e satisfies ˜ e ( P k ) ⊲ k ! P k (12) for all (commutative) polynomials P k = P k ( x α , . . . , x α n ) homogeneous ofdegree k . In particular, ˜ e is injective iff char k = 0 . In that case, the elements e ( x α · · · x α n ) are linearly independent. If char k = 0 , a modified map e : k [ x , . . . , x n ] → ˆ A n, k with normalization on k -homogeneous elements givenby e : x α · · · x α k k ! X σ ∈ Σ( k ) X α σ (1) · · · X α σ ( k ) , (13) is an injection. The map ˜ e is well-defined because the right-hand side in (11) is symme-tric in α , . . . , α k . Formula (7) can be restated as ˜ e ( − ) ⊲ k ! id. Notethat the expressions (11) do not span an associative subalgebra, but only asubspace e ( k [ x , . . . , x n ]) of the subalgebra k h X , . . . , X n i of ˆ A n, k generatedby X , . . . , X n , in general. Denote by π : k h X , . . . , X n i → k [ x , . . . , x n ]the vector space projection given by the Fock action on the vacuum vector1 ∈ k [ x , . . . , x n ], that is π ( P ) = P ⊲ P ∈ k h X , . . . , X n i . If char k = 0,the map e can be viewed as a k -linear section of the projection map π . Inparticular, e is an isomorphism onto its own image and Ker π ⊕ Im e = k h X , . . . , X n i . Acknowledgements.
I have proved the result in 2006 at IRB; the finalwriteup has been finished and submitted in the first days of my Autumn2019 stay at IH ´ES. In the final stage, I have been partly supported by theCroatian Science Foundation under the Project “New Geometries for Gravityand Spacetime” (IP-2018-01-7615). 7 iteratura [1]
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