A question of Joseph Ritt from the point of view of vertex algebras
aa r X i v : . [ m a t h . QA ] S e p A QUESTION OF JOSEPH RITT FROM THE POINT OF VIEWOF VERTEX ALGEBRAS
TOMOYUKI ARAKAWA, KAZUYA KAWASETSU, AND JULIEN SEBAG
Abstract.
Let k be a field of characteristic zero. This paper studies a problemproposed by Joseph F. Ritt in 1950. Precisely, we prove that(1) If p > i ∈ N , the nilpotency index of theimage of T i in the ring k { T } / [ T p ] equals ( i + 1) p − i .(2) For every pair of integers ( i, j ), the nilpotency index of the image of T i U j in the ring k { T } / [ T U ] equals i + j + 1. main result k be a field of characteristic zero. Let us denote by k { T } (resp. k { T, U } )the differential k -algebra obtained by endowing the k -algebra k [ T i ; i ∈ Z > ] (resp. k [ T i , U j ; i, j ∈ Z > ]) with the differential ∂ defined by ∂T i = T i +1 (resp. ∂T i = T i +1 and ∂U j = U j +1 ) for every integer i > i, j ) ∈ Z > ofintegers). In [Rit50, Appendix/5], J. F. Ritt asked the following question: Question. (1)
Let p > be an integer. Let [ T p ] be the differential ideal gen-erated by T p . For p > , i > what is the least q ( i ) such that T q ( i ) i ≡ T p ] ? (2) In k { T, U } , what is the least power q ( i, j ) of T i U j such that ( T i U j ) q ( i,j ) ≡ T U ] ?. For i = 1, Ritt states in loc. cit. , with no proof, that q ( i ) = 2 p −
1. In [O’K60],K. B. O’Keefe gave a proof of this formula and has shown that, for i = 2 and p >
2, one has q ( i ) = 3 p −
2. See also [Mea55, Mea73, O’K66] for complements orconnected problems. All these works use differential algebra, and are broadly basedon the reduction process due to H. Levi (see [Lev42]). Despite all these results, tothe best of our knowledge, Ritt’s question is remained open, in general, up to 2014.In [Pog14], G. A. Pogudin indeed provides an answer by showing that(1.1) q ( i ) = ( i + 1) p − i for every integer i ∈ Z > . The guideline of his proof consists in injecting thedifferential algebra k { T } / [ T p ] into a Grassmann algebra, endowed with a structureof differential algebra, and testing the vanishing of the image of T qi in that algebra(see [Pog14, Lemma 2,Theorem 3]). In the direction of the second question, onealso deduce from the Levi reduction process the following formula (see [Lev42, III],or, e.g., see [BS])(1.2) q ( i, j ) = i + j + 1 . C , to identify C { T } / [ T p ] with the Feigin-Stoyanovsky principal subspace [SF94] of the level ( p −
1) vacuum representationof the affine Kac-Moody algebra b sl , which is a commutative vertex algebra. Thisoperation allows us to reduce formula (1.1) to a simple fact from representationtheory. In [MP12], the Feigin-Stoyanovsky principal subspace is generalized to thenotion of principal subalgebras of lattice vertex algebras. They are isomorphic tothe free vertex algebras in the sense of [B86, R02] (see [Kaw15]). The answer to thesecond question can be similarly given using the theory of free vertex algebras andlattice vertex operators. 2. Vertex algebras V be a vector space over k . A field on V is a formal power series f ( z ) ∈ End( V )[[ z, z − ]] such that f ( z ) v ∈ V (( z )) for any v ∈ V . Here, V (( z )) is the spaceof formal Laurent series whose coefficients are elements of V .2.2. A vertex algebra is a vector space V equipped with(1) (Vacuum vector) | i ∈ V ,(2) (State-field correspondence) Y ( · , z ) : V → End( V )[[ z, z − ]],(3) (Translation operator) T ∈ End( V ),such that(1) T | i = 0,(2) Y ( | i , z ) = id V ,(3) Y ( v, z ) | i = v + O ( z ) ( v ∈ V ),(4) [ T, Y ( v, z )] = ∂ z Y ( v, z ) ( v ∈ V ),(5) (locality) for any u, v ∈ V , there exists N ∈ Z such that(2.1) ( z − w ) − N ( Y ( u, z ) Y ( v, w ) − Y ( v, w ) Y ( u, z )) = 0 . The biggest number N ∈ Z which satisfies (2.1) is called the locality bound for thepair u, v . The locality bound N for u, v is the same as the number satisfying u ( N − v = 0 , u ( n ) v = 0 n > N. Here we have employed the notation Y ( u, z ) = P n ∈ Z u ( n ) z − n − .We note that the multiplication v u ( n ) v is not associative in general. Themonomial u ( n )( u ( n )( · · · ( u m ( n m ) v ) · · · )) ∈ V with u , . . . , u m , v ∈ V is simplywritten as u ( n ) u ( n ) · · · u m ( n m ) v .3. Proof of Formula (1.1)3.1. Let k ′ be a field extension of k . Since, for every i >
0, the polynomials ∂ i ( T p )belong to k { T } , we observe that, for every integer q ∈ Z > , the relation T qi ∈ [ T p ]holds in k ′ { T } if and only if it holds in k { T } . Thus, we may replace k with analgebraic closure of k , and, then, by the Lefschetz principle, assume that k = C . QUESTION OF JOSEPH RITT FROM THE POINT OF VIEW OF VERTEX ALGEBRAS 3 g = sl with the standard basis { e, h, f } , and let b g = g [ t, t − ] ⊕ C K bethe affine Kac-Moody algebra associated with g = sl . The commutation relationsof b g are given by [ x m , y n ] = [ x, y ] m + n + nδ m + n, ( x | y ) K , [ K, b g ] = 0, where x m = xt m and ( x | y ) = tr( xy ) for x, y ∈ g , m ∈ Z . Let L p − ( g ) be the irreducible vacuumrepresentation of the affine Kac-Moody algebra of level p −
1, which is the uniquesimple quotient of the induced module U ( b g ) ⊗ U ( g [ t ] ⊕ C K ) C p − , where C p − is theone-dimensional representation of g [ t ] ⊕ C K on which g [ t ] acts trivially and K actsas multiplication by p −
1. As is well-known ([FZ92]), there is a unique vertexalgebra structure on L p − ( g ) such that the highest weight vector | i is the vacuumvector and Y ( x − | i , z ) = x ( z ) := X n ∈ Z x n z − n − , x ∈ g . Feigin-Stoyanovsky principal subspace W of L p − ( g ) is by definition thecommutative vertex subalgebra of L p − ( g ) generated by e ( z ). Let ∂ be the differ-ential of C [ e − , e − , e − , . . . , ] defined by ∂e − i = ie − i − for every i >
1. We have asurjective morphism C [ e − , e − , e − , . . . , ] ։ W, f f | i (3.1)of differential algebras. According to [SF94] (see also [CLM08a, CLM08b, Fei11,LiH]), the kernel J of the above map is the ideal generated by the ∂ n ( e p − ) for all n ∈ Z > . Therefore, we have an isomorphism of differential algebras given by: W = C [ e − , e − , e − , . . . , ] /J ∼ = C { T } / [ T p ] , e − i T i − / ( i − . (3.2) Remark . Let us stress that the character of W coincides with that of the Vi-rasoro (2 , p + 1)-minimal model vertex algebra Vir , p +1 ([SF94, Fei11]). Let usstress that, if X = Spec( C [ T ] / h T p i ), the k -algebra C { T } / [ T p ] is isomorphic to thealgebra O ( J ∞ X ) of the arc scheme J ∞ X associated with X . Let us then mentionthat the identification of O ( J ∞ X ) with gr Vir , p +1 has been previously establishedin [vEH].3.5. Formula (1.1) immediately follows from (3.2) and Proposition 3.6. Proposition 3.6.
We have e i ( p − − i | i 6 = 0 and e i ( p − − i = 0 on W for all i > .Proof. We have [ f i , e − i ] = − h + iK , and so, { f i , − h + iK, e − i } forms an sl -tripleinside b g . Since L ( g ) is integrable and ( − h + iK ) | i = i ( p − | i , | i generates an( i ( p − sl -triple. Therefore, e i ( p − − i | i 6 = 0and e i ( p − − i | i = 0. (cid:3) Proof of Formula (1.2)In order to give a proof of second question, we need to introduce lattice vertexalgebras and free vertex algebras.4.1. As in subsection 3.1, we may assume that k = C . TOMOYUKI ARAKAWA, KAZUYA KAWASETSU, AND JULIEN SEBAG L be an integral lattice with the Z -bilinear form ( · , · ) : L × L → Z on L . We set h = k ⊗ Z L with the extended k -bilinear form ( · , · ) : h × h → k and let k [ L ] = L α ∈ L ke α be the group algebra of L . Then the vector space V L = M (1) ⊗ k [ L ]admits a natural vertex superalgebra structure, called the lattice vertex superalge-bra associated with L . Here, M (1) is the Heisenberg vertex algebra (Fock space)attached to h . We have | i = 1 ⊗
1, where we write 1 for the vacuum vector of M (1).The state-field correspondence for 1 ⊗ e α is Y (1 ⊗ e α , z ) = E − ( − α, z ) E + ( − α, z ) ⊗ e α z α , where E ± ( − α, z ) = exp (cid:16) X n ∈± Z − α ( n ) n z − n (cid:17) ,z α is defined to be z ( α,β ) on M (1) ⊗ e β , and e α : k [ L ] → k [ L ] is defined by e α ( e β ) = η ( α, β ) e α + β with a certain cocycle η on L . The superalgebra V L is a vertex algebraif and only if L is an even lattice. Here L is called even if ( α, α ) is even for any α ∈ L .Let g be a finite-dimensional simple Lie algebra of type ADE with the root lattice Q . It is known that the affine vertex algebra L ( g ) of level 1 is isomorphic to thelattice vertex algebra V Q .4.3. Let C be a Z -basis of L . The vertex subsuperalgebra W = W ( C, L ) of V L generated by { e α ; α ∈ C } is called a principal subalgebra of V L [MP12].For instance, the Feigin-Stoyanovsky principal subspaces of L ( g ) are isomorphicto the principal subalgebras W (Φ , Q ) of lattice vertex algebras V Q , where Φ is abase of the root system of g .4.4. Let B be a set and N : B × B → Z a symmetric function. The free vertexsuperalgebra F = F ( B, N ) is freely generated by B such that for any a, b ∈ B , thenumber N ( a, b ) is the locality bound for the pair a, b . It has the following universalproperty: any vertex superalgebra generated by B satisfying(4.1) ( z − w ) − N ( a,b ) ( Y ( a, z ) Y ( b, w ) − Y ( b, w ) Y ( a, z )) = 0 with a, b ∈ B, is a sujective image of F (see [R02, Kaw15]). The free vertex algebras were firstmentioned in [B86] and constructed in [R02]. The construction in [R02] basicallyproceeds as follows: • Consider the free associative algebra A generated by the symbols a ( n ) with a ∈ B and n ∈ Z . • Take an appropriate completion b A of A so that we can take the quotientof b A by the two-sided ideal generated by the left-hand sides of (4.1), whichare infinite sums in general. • Again quotient it by the left ideal generated by a ( n ) for a ∈ B and n > F = F ( B, N ). • The set B is embedded in F by a a ( − QUESTION OF JOSEPH RITT FROM THE POINT OF VIEW OF VERTEX ALGEBRAS 5
See also [Kaw15] for an account of Roitman’s construction. There, the completionis taken as a generalization of the degreewise completion in the sense of [MNT10],which is used to define the universal enveloping algebras of vertex algebras.Let L = L ( B, N ) be the free abelian group generated by B with the Z -bilinearform ( · , · ) : L × L → Z defined by bilinearly extending the assignment ( a, b ) = − N ( a, b ) for a, b ∈ B . Then the free vertex (super)algebra F ( B, N ) is isomorphicto the principal subalgebra W ( B, L ) (see [R02, Kaw15]).4.5. We recall combinatorial bases of free vertex (super)algebras from [R02, MP12,Kaw15]. Let B be a set and N : B × B → Z a symmetric function. Suppose that B is totally ordered with the order < . The free vertex superalgebra F = F ( B, N )has the C -basis which consists of the monomials of the form a m ( n m + m − X i =1 N ( a m , a i )) · · · a ( n + N ( a , a )) a ( n ) | i with m > a a . . . a m ∈ B and n , n , . . . , n m ∈ Z < such that n i n i − if a i = a i − for 1 < i m .4.6. Let us take over the notation of the previous subsection and suppose from nowon that N ( a, a ) ∈ Z for every a ∈ B . In this case, the superalgebra F = F ( B, N )is a vertex algebra.Recall that a vertex algebra V is called commutative if a ( n ) b = 0 for any a, b ∈ V and n >
0. In this case, V is a differential algebra with the multiplication a · b = a ( − b and differential ∂ = T , the translation operator of V .By the construction of free vertex algebras, we see that F is commutative if andonly if N ( a, b ) F is commutative.Let R F be Zhu’s Poisson algebra associated with F : R F := F/C ( F ) , C ( F ) := span C { u ( − v ; u, v ∈ F } . Let u and v be elements of F . Note that u ( n ) v ∈ C ( F ) for any n −
2. We writeby ¯ u ∈ R F the image of u under the canonical surjection F → R F . The producton R F is defined by ¯ u ¯ v = u ( − v and the Poisson bracket is { ¯ u, ¯ v } = u (0) v .As we assume that F is commutative, the Poisson bracket is trivial and R F is acommutative associative algebra.By using the combinatorial basis given in Subsection 4.5, we observe that(4.2) R F ∼ = C [ B ] / ( ab ; a, b ∈ B, N ( a, b ) − C [ B ] is the polynomial algebra with the set B of independentvariables.4.7. In this section, we describe free vertex algebras as lifts of quotients of differen-tial algebras. After finishing this work, we found that recent preprint [LiH] provesTheorem 4.8 and Corollary 4.10 including super-cases using theory of principal sub-spaces of lattice vertex superalgebras. We however would like to keep the presentproofs which use the theory of free vertex algebras as it seems to be remarkablyshort. TOMOYUKI ARAKAWA, KAZUYA KAWASETSU, AND JULIEN SEBAG
Let B be a set and N : B × B → Z a symmetric function. Suppose that( a, a ) ∈ Z for every a ∈ B . We assume that N ( a, b ) a, b ∈ B so that F = F ( B, N ) is commutative. Recall that k { B } = k [ a i ; a ∈ B, i ∈ Z > ] denotesthe differential k -algebra with the differential ∂ . Theorem 4.8.
Let I be the differential ideal generated by { a − m − b ; a, b ∈ B, N ( a, b ) m − } . Then, we have (4.3) F ( B, N ) ∼ = k { B } /I. Proof.
By the universality of F , we have the surjection F → k { B } /I since k { B } /I is commutative. On the other hand, we have the surjection π : k { B } /I → W ( B, L )from k { B } /I to the principal subalgebra of the lattice vertex algebra V L with thelattice L = L ( B, N ). Since F ∼ = W ( B, L ), we have the assertions. (cid:3)
Remark . When the lattice L ( B, N ) is positive definite, Theorem 4.8 is provedin [P14, Theorem 2].
Corollary 4.10.
The free vertex algebra F = F ( B, N ) is isomorphic to the jet liftof Zhu’s Poisson algebra R F (4.4) F ( B, N ) ∼ = k { B } / ( a b ; a, b ∈ B, N ( a, b ) − if and only if N ( a, a ) ∈ { , − } for every a ∈ B and N ( a, b ) ∈ { , − } for everypair ( a, b ) with a = b ∈ B .Proof. Write J the divisor of the right-hand side of (4.4). As ∂ ( a a ) = 2 a a , wesee that I = J if N ( a, a ) ∈ { , − } for any a ∈ B and N ( a, b ) ∈ { , − } for all a = b ∈ B . Since ∂ ( a a ) = 2 a a + a a and ∂ ( a b ) = a b + a b , we have thesecond assertion. (cid:3) Example . Let B = { a, b } with N ( a, a ) = N ( b, b ) = 0 and N ( a, b ) = N ( b, a ) = −
1. Then R F ∼ = k [ a, b ] / ( ab ) as algebras and F ∼ = k { a, b } / ( a b ) as differentialalgebras, where F = F ( B, N ).4.12. We now apply Corollary 4.10 to give an answer to the second problem ofRitt.Let L be a lattice with the Z -basis C = { α, β } and the symmetric bilinear formdefined by ( α, α ) = 0 , ( β, β ) = 0 , ( α, β ) = 1 . Let us consider the lattice vertex algebra V L with h = C ⊗ Z L . Note that we have e γ .e δ ( z ) = z − ( γ,δ ) e δ ( z ) .e γ γ, δ ∈ L. Note also that(4.5) E + ( h, z )1 ⊗ e γ = 1 ⊗ e γ for any h ∈ h and γ ∈ L . Lemma 4.13. (cf. [LL12, Proposition 6.3.14])
For any γ, δ ∈ L , E + ( γ, z ) E − ( δ, w ) = (1 − w/z ) ( γ,δ ) E − ( δ, w ) E + ( γ, z ) . QUESTION OF JOSEPH RITT FROM THE POINT OF VIEW OF VERTEX ALGEBRAS 7
The vertex algebra V L is graded by conformal weights: V L = L ∆ ∈ Z ( V L ) ∆ ,where ( V L ) ∆ is the subspace spanned by the vectors of conformal weight ∆ and theconformal weight of h ( − n ) · · · h m ( − n m ) ⊗ e γ is given by n + · · · n m + ( γ, γ )2 . Note that e α ( n ) V ∆ ⊂ V ∆ − n − . (4.6)Now, the key point is that the free vertex algebra F ( B, N ) defined in Example1 is isomorphic to the following principal subspace W of V L : W = W L ( C ) = h ⊗ e α , ⊗ e β i ⊂ V L . It is defined by the assignment a ⊗ e α and b ⊗ e β . Theorem 4.14.
Let i, j be non-negative integers. Then in W we have ( e α ( − i − e β ( − j − n | i 6 = 0 if and only if n i + j. Proof.
The element ( e α ( − i − e β ( − j − n | i belongs to M (1) ⊗ C e n ( α + β ) ⊂ V L .Since the conformal weight of e n ( α + β ) is n , the conformal weight of any homoge-nous vector v of M (1) ⊗ C e n ( α + β ) is equal to or greater than n , and it equals to n if and only if v coincides with e n ( α + β ) up to constant multiplication. On the otherhand, ( e α ( − i − e β ( − j − n | i is homogenous of conformal weight ( i + j ) n , see(4.6). Hence ( e α ( − i − e β ( − j − n | i = 0 for n > i + j , and(4.7) ( e α ( − i − e β ( − j − i + j | i = c ⊗ e ( i + j )( α + β ) , c ∈ C . for some c ∈ C . It remains to show that c = 0. Since W is commutative, the vector( e α ( − i − e β ( − j − n | i coincides withRes (cid:16) z − i − · · · z − i − i + j w − j − · · · w − j − i + j e α ( z ) · · · e α ( z i + j ) e β ( w ) · · · e β ( w i + j ) | i (cid:17) . (4.8)Here Res f ( z , . . . , z i + j , w , . . . , w i + j ) denotes the coefficient of z − . . . z − i +1 w − . . . w − i + j in f . Using Lemma 4.13 repeatedly, we have up to some non-zero multiple from2-cocycle ε that e α ( z ) · · · e α ( z i + j ) e β ( w ) · · · e β ( w i + j )= z i + j · · · z i + ji + j i + j Y k,ℓ =1 (1 − w k /z ℓ ) E − ( − α, z ) · · · E − ( − α, z i + j ) E − ( − β, w ) · · · E − ( − β, w i + j ) · E + ( − α, z ) · · · E + ( − α, z i + j ) E + ( − β, w ) · · · E + ( − β, w i + j ) ⊗ e ( i + j ) α ( z . . . z i + r ) α e ( i + j ) β ( w . . . w i + j ) β . TOMOYUKI ARAKAWA, KAZUYA KAWASETSU, AND JULIEN SEBAG
Hence by (4.5), we find that up to some nonzero multiple from 2-cocycle (4.8) isequal toRes z j − · · · z j − i + j w − j − · · · w − j − i + j i + j Y k,ℓ =1 (1 − w k /z ℓ )(4.9) E − ( − α, z ) · · · E − ( − α, z i + j ) E − ( − β, w ) · · · E − ( − β, w i + j ) ⊗ e ( i + j )( α + β ) (cid:17) . Since it must be equal to 1 ⊗ e ( i + j )( α + β ) up to constant multiplication, from weightconsideration we conclude that (4.9) coincides withRes z j − · · · z j − i + j w − j − · · · w − j − i + j i + j Y k,ℓ =1 (1 − w k /z ℓ )1 ⊗ e ( i + j )( α + β ) . If j = 1, then this equals − ( i + j )! ⊗ e ( i + j )( α + β ) . In a similar way, we see that it isequal to ( − j P i + j,j ⊗ e ( i + j )( α + β ) , where P i + j,j is the number of arrangements of 0and 1 on the ( i + j ) × ( i + j )-square such that any row and any column have exactly j tuples of 1. Since P i + j,j = 0, we have Theorem 4.14. (cid:3) It follows immediately from Theorem 4.14 that the answer to the second questionof Ritt is given by i + j + 1. Acknowledgements.
This work was started when T. A. was visiting Universityof Lille from May, 2019 to July, 2019. He thanks the institute for its hospitality.T. A. is partially supported by by the Labex CEMPI (ANR-11-LABX-0007-01)and by JSPS KAKENHI Grant Numbers 17H01086, 17K18724. K. K. is partiallysupported by MEXT Japan “Leading Initiative for Excellent Young Researchers(LEADER)”, JSPS Kakenhi Grant numbers 19KK0065 and 19J01093.
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Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502JAPAN
E-mail address : [email protected] Priority Organization for Innovation and Excellence, Kumamoto University, Ku-mamoto 860-8555 JAPAN.
E-mail address : [email protected] Institut de recherche math´ematique de Rennes, UMR 6625 du CNRS, Universit´e deRennes 1, Campus de Beaulieu, 35042 Rennes cedex (France)
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