Bracket width of simple Lie algebras
aa r X i v : . [ m a t h . AG ] F e b BRACKET WIDTH OF SIMPLE LIE ALGEBRAS
BORIS KUNYAVSKI˘I AND ANDRIY REGETA
Abstract.
The notion of commutator width of a group, defined as the smallest numberof commutators needed to represent each element of the derived group as their product,has been extensively studied over the past decades. In particular, in 1992 Barge and Ghysdiscovered the first example of a simple group of commutator width greater than one amonggroups of diffeomorphisms of smooth manifolds.We consider a parallel notion of bracket width of a Lie algebra and present the firstexamples of simple Lie algebras of bracket width greater than one. They are found amongthe algebras of polynomial vector fields on smooth affine varieties. Introduction
The notion of width in various contexts appears in many algebraic structures. It is partic-ularly well studied in group theory where the commutator width got special attention, beingrelated to many important properties of the class of groups under investigation. Recall thatgiven a group G , the commutator width is defined as supremum of the lengths ℓ ( g ), g run-ning over the derived subgroup [ G, G ], where ℓ ( g ) is the smallest number of commutatorsneeded to represent g as their product. Examples of groups of commutator width greaterthan one were known to Miller more than 100 years ago, and such examples can easily bediscovered even among perfect groups (i.e., groups G coinciding with [ G, G ]). However, ittook quite a time to discover simple groups of commutator width greater than one. As oftenhappens, the insight came from outside: Barge and Ghys [BG92] have found their examplesamong groups of differential-geometric nature, applying deep results from that area. Thegroups they considered were infinite (in any finite simple group every element is a singlecommutator, as predicted by Ore’s conjecture settled in 2010 [LOST10], after more thanhalf-a-century efforts). The interested reader can find more details in the survey [GKP18]and the references therein.The group case served for us as a prototypical example for investigating the similar notionof bracket width in the parallel universe of Lie algebras. It is defined in full analogy with thecommutator width of a group. Given a Lie algebra L over a field k , we define its bracketwidth as supremum of the lengths ℓ ( a ), a running over the derived algebra [ L, L ], where ℓ ( a )is defined as the smallest number m of Lie brackets [ x i , y i ] needed to represent a in the form a = m X i =1 [ x i , y i ] . The bracket width applies in studying different aspects of Lie algebras, ranging from problemsmotivated by logic (elementary equivalence), see [Rom16], to those coming from differential
Research of the first author was supported by the ISF grants 1623/16 and 1994/20. It started during thevisit of the first author to the MPIM (Bonn) and continued during the visit of the second author to Bar-IlanUniversity. The support of these institutions is gratefully acknowledged. eometry, see [LT13]. Our focus is on simple Lie algebras. Here one can observe certainparallelism with the commutator width of groups: there is little hope to find an example ofa finite-dimensional simple Lie algebra of bracket width greater than one, though in generalthe problem is open (there are many cases where the width is known to be equal to one, andit is known that it cannot exceed two [BN11]); see [GKP18, Section 6] for more details.Examples of simple Lie algebras of width greater than one should thus be sought among infinite-dimensional algebras. There are several natural families of such to be checked first.Four families of Lie algebras of Cartan type all consist of the algebras of width one, in light ofthe results of Rudakov [Rud69]. The case of (subquotients of) Kac–Moody algebras is open,to the best of our knowledge. In the present paper, we start the study of another classicallyknown family, algebras Vec( X ) of polynomial vector fields on affine irreducible varieties X .It is well-known that the Lie algebra Vec( X ) is simple if and only if X is smooth (see [Jor86],[Sie96, Proposition 1]). Thus our primary objects of interest are smooth affine curves andsurfaces. It turns out that already among the Lie algebras of vector fields on smooth affinecurves there are algebras of width greater than one. The simplest example appears in thecase where X is an affine hyperelliptic curve for which the algebra Vec( X ) was studied indetail by Billig and Futorny [BF18]. Here is our first main result. Theorem A. (Theorem 1) Let h ( x ) be a separable monic polynomial of odd degree greaterthan one. Let C h ⊂ A be given by the equation y = 2 h ( x ) . Then the bracket width of theLie algebra
Vec( C h ) is greater than one. When considering surfaces, we focus on the so-called Danielewski surface D p ⊂ A givenby the equation xy = p ( z ) where p is a polynomial without multiple roots.These surfaces attracted a lot of attention over the last few decades. They were ini-tially used in [Dan89] to present a counterexample to a generalized version of the ZariskiCancellation Problem.In this paper we study the width of the Lie subalgebra h LNV(D p ) i of Vec(D p ) generatedby all locally nilpotent vector fields. As shown in [LR21, Theorem 1], h LNV(D p ) i is a simpleLie algebra. If deg p = 1, then D p is isomorphic to the affine plane A . It is well-known thatin this case h LNV(D p ) i is isomorphic to the Lie algebra of vector fields with zero divergence,and its bracket width is equal to one (see Proposition 1 below for a more general statement).Our aim is to study the bracket width of h LNV(D p ) i in the case deg p ≥
2. Note that theLie algebras h LNV(D p ) i and h LNV(D q ) i are isomorphic if and only if the varieties D p andD q are isomorphic (see [LR21, Theorem 2]). Remark 1.
By [Dai04, Lemma 2.10] two Danielewski surfaces D p and D q are isomorphic ifand only if there exists an automorphism of algebras F : C [ z ] ∼ −→ C [ z ] such that F ( p ) = cq for some c ∈ C ∗ . In particular, this implies that deg p = deg q . If deg p = deg q = 2, then D p and D q are isomorphic.We consider the cases deg p = 2 and deg p ≥ p holds (this isHypothesis (J) in Section 4 below whose flavour is somewhat reminiscent of the JacobianConjecture). Here is our next main result. Theorem B. (Proposition 9 and Theorem 2) i) Suppose that deg p = 2 . Then the bracket width h LNV(D p ) i is at most two. (ii) Suppose that deg p ≥ . Assume in addition that Hypothesis (J) holds. Then thebracket width of h LNV(D p ) i is greater than one.Notation and conventions. Our base field is k = C , the field of complex numbers. Allvarieties are assumed affine and irreducible. Accordingly, all algebraic groups are assumedlinear and connected. All vector fields are assumed polynomial.By Vec( X ) we denote the Lie algebra of vector fields on the variety X . If needed, weidentify vector fields with derivations of the algebra O ( X ) of regular functions on X . Avector field ν ∈ Vec( X ) is called locally nilpotent if for any f ∈ O ( X ) there exists s ∈ N such that ν s ( f ) = 0. By h LNV( X ) i we denote the Lie subalgebra of Vec( X ) generated byall locally nilpotent vector fields.Throughout below by ‘width’ we mean ‘bracket width’, unless specified otherwise.2. Some simple Lie algebras of width 1
We start with the following computation which is probably well known (see, e.g., the proofof Proposition 1 in [Rud69, Section 2]): in the simple algebra Vec( A n ) each element can berepresented as a single bracket. Indeed, for any µ = f ∂ x i + · · · + f k ∂ x ik ∈ Vec( A n )we can write f ∂ x i + · · · + f k ∂ x ik = [ ∂ x , g ∂ x i + · · · + g k ∂ x ik ] , where g i is a polynomial such that ∂g i ∂x = f i for every i = 1 , . . . , k .The same is true for the Witt algebra W n = Vec( T n ) = Der k [ x , x − , . . . , x n , x − n ]of vector fields on the n -dimensional torus: each element of W n can be represented as a singlebracket, see Proposition 3 below.Let us now consider the Lie algebraVec ◦ ( A n ) = { µ = f ∂ x + · · · + f n ∂ x n ∈ Vec( A n ) | Div µ = ∂f ∂x + · · · + ∂f n ∂x n = 0 } . According to [Sha81, Lemma 3], this algebra is simple.Note that Vec ◦ ( A n ) is of particular interest as the Lie algebra of the so-called ind-groupSAut( A n ) = { f = ( f , . . . , f n ) ∈ Aut( A n ) | det Jac( f ) = det (cid:20) ∂f i ∂x j (cid:21) ij = 1 } , see [Sha81] for details. Proposition 1.
The width of the Lie algebra
Vec ◦ ( A n ) is equal to 1.Proof. For µ, ν ∈ Vec( A n ) we have(1) Div[ µ, ν ] = µ (Div ν ) − ν (Div µ ) , see, e.g., [Sha81, Lemma 1]. Similarly, for each µ = f ∂ x i + · · · + f k ∂ x ik ∈ Vec ◦ ( A n ) we have f ∂ x i + · · · + f n ∂ x n = [ ∂ x , g ∂ x i + · · · + g n ∂ x n ] , here g i is a polynomial such that ∂g i ∂x = f i for every i = 1 , . . . , k . By (1), we have that0 = ∂ x (Div( g ∂ x + · · · + g n ∂ x n )) which implies that Div( g ∂ x + · · · + g n ∂ x n ) ∈ C [ x , . . . , x n ].Consider now σ = g∂ x ∈ Vec( A n ), where g ∈ C [ x , . . . , x n ] is such that Div( σ ) = ∂g∂x = − Div( g ∂ x + · · · + g n ∂ x n ). We have Div( g ∂ x i + · · · + g n ∂ x n + σ ) = 0 and f ∂ x i + · · · + f n ∂ x n = [ ∂ x , g ∂ x i + · · · + g n ∂ x n + σ ] . This shows that any element in Vec ◦ ( A n ) can be represented as a single bracket. (cid:3) Our next step consists in extending this sort of arguments to a wider class of smooth affinevarieties. This is summarized in the next two propositions.Denote A = Spec k [ x ]. Proposition 2.
Let X be a smooth affine variety. Then the width of Vec( A × X ) is equalto one.Proof. Let ν ∈ Vec( X ), and let k ≥
0. Then x k ν ∈ Vec( A × X ) can be represented as asingle bracket:(2) [ ∂ x , k + 1 x k +1 ν ] = 1 k + 1 ( ∂ x ( x k +1 ) ν + x k +1 [ ∂ x , ν ]) = x k ν. Further, the same is true for any x k f ∂ x , where f ∈ O ( X ) and k ≥ ∂ x , k + 1 x k +1 f ∂ x ] = 1 k + 1 ( ∂ x ( x k +1 ) f ∂ x + x k +1 [ ∂ x , f ∂ x ]) = x k f ∂ x . Since any µ ∈ Vec( A × X ) can be represented as a sum of elements of the form x k ν , ν ∈ Vec( X ), k ≥
0, and elements of the form x k f ∂ x , f ∈ O ( X ), k ≥
0, we conclude that µ = [ ∂ x , δ ] for a suitable δ ∈ Vec( A × X ). (cid:3) Denote T = A \ { } = Spec k [ x, x − ]. Proposition 3.
Let X be a smooth affine variety. Then the width of Vec( T × X ) is equalto one.Proof. We have(4) x k f ∂ x = [ x l ∂ x , k − l + 1 x k − l +1 f ∂ x ] , where k − l + 1 = 0, l ∈ N , k ∈ Z and f ∈ O ( X ). Furthermore, observe that any x k ν ∈ Vec( T × X ), where ν ∈ Vec( X ) and k ∈ Z , can also be represented as a singlebracket:(5) [ x l ∂ x , k − l + 1 x k − l +1 ν ] = 1 k − l + 1 ( x l ∂ x ( x k − l +1 ) ν + x k − l +1 [ x l ∂ x , ν ]) = x k ν, where l ∈ N , k ∈ Z , and k − l + 1 = 0.Let ∂ ∈ Vec( T × X ). It can be represented as a finite sum of elements of the form x k ν ,where ν ∈ Vec( X ), k ∈ Z , and elements of the form x k f ∂ x , where f ∈ O ( X ) and k ∈ Z .Define S ⊂ N as the set of all exponents k contained in all summands of ∂ . Now, if l − / ∈ S and 2 l − / ∈ S , then by (4) and (5) we can represent ∂ as a bracket [ x l ∂ x , δ ] for a suitable δ ∈ Vec( T × X ). Since we have infinitely many choices for l ∈ N , we conclude that any ∂ ∈ Vec( T × X ) can be represented as a single bracket. (cid:3) orollary 1. Let G be a connected linear algebraic k -group. Suppose that G is not semisim-ple. Then the width of Vec( G ) is equal to 1.Proof. The fact that the underlying variety of such a G can be written as a product of either A or T and some smooth affine variety is perhaps known to experts. We are not aware ofany precise reference and therefore present an argument.Its first part is borrowed from [Pop15] where it was used in a slightly different context.Let U denote the unipotent radical of G . Then by a theorem of Rosenlicht (see [Ros56,Theorem 10], [Gro58, Propositions 1,2]), the underlying variety of G is isomorphic to U × G/U where L = G/U is (the underlying variety of) a reductive group.If U = { } , then by the same theorem we have U ∼ = G a × U/ G a where the underlyingvariety of the additive group G a is the affine line A , and we fall under the assumptions ofProposition 2.If U = { } , the group G = L is reductive. Then the underlying variety of G is a directproduct of the underlying variety of a torus T and the underlying variety of a semisimplegroup H , see [Pop21, Theorem 1]. As G is not semisimple by the hypothesis of the theorem,we have dim T >
0, so that Proposition 3 is applicable.In both cases, we conclude that the width of Vec( G ) is equal to one. (cid:3) Vector fields on smooth affine curves
We start with the case of rational curves.
Proposition 4.
Let C be a smooth affine rational curve. Then the width of the Lie algebra Vec( C ) is at most two.Proof. It is well known that C is isomorphic to A \ { p , . . . , p k } for some k ≥ p = 0. Therefore, we can assume that O ( C ), viewed as a vector space, has a basis { x i , x j , x − p ) j , . . . x − p k ) jk } . Note that [ ∂ x , x i ∂ x ] = ix i − ∂ x and[ ∂ x , x − p r ) j r ∂ x ] = − j r ( x − p r ) j r +1 ∂ x . Therefore, any element of the form
P ∂ x , where P does not contain as a summand monomialsproportional to one of the elements x , x − p ) , . . . , x − p k ) , can be represented as a bracket ofthe element ∂ x and some suitable element from Vec( C ).Now, [ x∂ x , x − p i ∂ x ] = − x − p i ∂ x − p i ( x − p i ) ∂ x . It follows that for any µ ∈ Vec( C ) there exist ν, δ ∈ Vec( C ) such that µ = [ ∂ x , ν ] + [ x∂ x , δ ].The claim follows. (cid:3) Conjecture 1.
Let C = A \ { p , . . . , p k } , where k ≥ . Then the width of Vec( C ) equalstwo. Remark 2.
We expect that for C ≃ A \ { , } the element (cid:0) x + x − (cid:1) ∂ x ∈ Vec( C ) is notrepresentable as a single bracket. Here is a calculation bringing some evidence in support ofthis hypothesis. ssume that(6) (cid:18) x + 1 x − (cid:19) ∂ x = [ µ, ν ] , where µ, ν ∈ Vec( C ). Without loss of generality we can assume that the monomial summandsof µ and ν of highest degree in x are not proportional. Otherwise, we can replace ν by ν − cµ for a suitable constant c ∈ C ∗ . Further, since[ x m ∂ x , x n ∂ x ] = ( n − m ) x m + n − ∂ x , we have µ = ax∂ x + b∂ x + P s> c s x s ∂ x + P l> c l ( x − l ∂ x and ν = P m> c m x m ∂ x + P n> c n ( x − n ∂ x .Now, O ( C ) viewed as a vector space has a basis { x i , x − j | i ∈ Z , j ∈ N } and we have[ 1( x − a ) n ∂ x , x − a ) m ∂ x ] = ( n − m ) 1( x − a ) n + m +1 ∂ x for any a ∈ C . Then it is not difficult to see that µ = ax∂ x + b∂ x + X s> c s x s ∂ x and ν = X n> ˜ c n ( x − n ∂ x . The existence of the coefficients a , b , c s , ˜ c n in this formula seems unlikely though we did notsucceed in proving this.For non-rational affine curves C we are able to exhibit examples where the width of theLie algebra Vec( C ) is greater than one. These examples, vector fields on affine hyperellipticcurves, are borrowed from the paper of Billig and Futorny [BF18] where many properties ofthese algebras had been established. Below we closely follow [BF18, Section 5], includingnotation.Let H = { y = 2 h ( x ) } where h ( x ) is a separable monic polynomial of odd degree 2 m + 1 ≥ A = O ( H ) = k [ x, y ] / h y − h ( x ) i . As a vector space, A ∼ = k [ x ] ⊕ yk [ x ], and we have anisomorphism of Lie algebras Vec( H ) ∼ = Der k ( A ). Lemma 1. [BF18, Proposition 5.1] Vec( H ) is a free A -module of rank 1 generated by τ = y∂ x + h ′ ( x ) ∂ y . (cid:3) An important role is played by the following filtration on A . Define the degree of amonomial in A = k [ x ] ⊕ yk [ x ] bydeg( x k ) := 2 k, deg( x k y ) := 2 k + 2 m + 1 . Let A s be the space spanned by the monomials of degree ≤ s . Then we have a filtration A ⊂ A ⊂ A ⊂ . . . We have A s A k ⊂ A s + k .For f ∈ A define its degree as the degree of its (unique) leading monomial LT( f ) (whichis the term of highest degree in the expansion of f in the basis { x k , x k y } ).We have deg( f g ) = deg( f ) + deg( g ).Further, consider the grading on A induced by this filtration. Let gr A = A ⊕ A /A ⊕ A /A ⊕ . . . be the associated graded algebra. Lemma 2. [BF18, page 3425] i) Each graded component is of dimension at most 1 and gr A ∼ = k [ x, y ] / (cid:10) y − x m +1 (cid:11) . (ii) There is an embedding ψ : k [ x, y ] / (cid:10) y − x m +1 (cid:11) ֒ → k [ t ] ,x t , y m +1 t m +1 , allowing one to identify gr A with a subalgebra of k [ t ] generated by t and t m +1 . (iii) We have a multiplicative map LT : A → gr A . (cid:3) Further, the filtration and grading on A introduced above give rise to a filtration and agrading on the algebra D = Vec( H ). Recall that by Lemma 1, we have D = Aτ . For anymonomial u ∈ A , u = 1, we have τ ( A ) = 0 and deg τ ( u ) = deg( u ) + 2 m −
1. Hence for anynonzero gτ ∈ D and any non-constant f ∈ A we havedeg( gτ ( f )) = deg( f ) + deg( g ) + 2 m − . Define deg( gτ ) := deg( g ) + 2 m − . This allows one to define a filtration(0) ⊂ D m − ⊂ D m ⊂ D m +1 ⊂ . . . where D s is the subspace of elements of degree ≤ s , and the associated graded algebra gr D . Lemma 3. [BF18, Lemma 5.2] gr D acts on gr A by derivations. One can identify gr D witha gr A -submodule of Der k [ t ] generated by t m ∂ t . (cid:3) Here are some more properties of D . Lemma 4. [BF18, Theorem 5.3]
Let = η ∈ D . Then (1) Ker ad( η ) = kη. (2) η / ∈ Im ad( η ) . (3) D has no semisimple elements. (4) D has no nilpotent elements. (cid:3) (We say that η is semisimple if ad( η ) has an eigenvector.) Theorem 1.
The width of D is greater than one.Proof. Let η ∈ D be such that deg η = 2 m −
1. Suppose that there exist ν, ξ ∈ D such that[ ν, ξ ] = η . Then ν and ξ are not proportional. Then we have the inequality(7) deg[ ν, ξ ] ≥ deg ν + 2 m − . This key observation was made on page 3426 of [BF18] without proof. For the reader’sconvenience, we present a proof here following the advice of
Yuly Billig .Let ν = f τ , ξ = gτ , where f, g ∈ A . Without loss of generality we can assume that theleading monomials LT( f ) and LT( g ) are different (if they are not, we can subtract a scalarmultiple of one of f, g from the other). By Lemma 3, we can embed gr D into Der k [ t ]. Thenthe images of ν and ξ under this map are of the form t k d/dt and t l d/dt with different k and l . Moreover, by the same Lemma 3, the image of the bracket [ ν, ξ ] is [ t k d/dt, t l d/dt ], which s equal to ct k + l − d/dt , where c = 0 because k = l . Since by Lemma 3 the image of gr D inDer k [ t ] is generated by t m d/dt , we have l ≥ m , so that k + l − t k d/dt, t l d/dt ] ≥ deg t k d/dt + 2 m − . Applying Lemma 3 once again, we get the same inequality for the degrees of leading mono-mials: deg[LT( f ) , LT( g )] ≥ deg LT( f ) + 2 m − , which proves inequality (7).It remains to note that k = deg ν > d/dt is not contained in the image of gr D in Der k [ t ]. Together with inequality (7), this givesdeg[ ν, ξ ] > m − η, contradiction. (cid:3) Vector fields on Danielewski surfaces
This section is devoted to the proof of Theorem B. Its more precise statement will be givenbelow in Section 4.4, see Proposition 9 and Theorem 2.4.1.
Vector fields.
Recall that we consider the surface D p ⊂ A given the equation xy = p ( z ) where p is a polynomial without multiple roots. Our main object is the simple Liealgebra h LNV(D p ) i generated by all locally nilpotent vector fields. We assume deg p ≥ p = 1, then D p ≃ A , and in this case the width of h LNV(D p ) i is equal toone by Proposition 1 as h LNV( A ) i = Vec ◦ ( A ) (see [Sha81, Lemma 2]).We start with some generalities.Recall that Vec(D p ) can be viewed as a rank 2 projective module over the ring of regularfunctions O (D p ) which is generated by(8) ν x := p ′ ( z ) ∂∂y + x ∂∂z , ν y := p ′ ( z ) ∂∂x + y ∂∂z , ν z := x ∂∂x − y ∂∂y with the unique relation xν y − yν x = p ′ ( z ) ν z .We have a linear map (see [LR21, page 9])(9) Θ : O (D p ) → Vec(D p )with kernel k . For f ∈ O (D p ) we have(10) θ f = ( p ′ ( z ) f y + xf z ) ∂∂x − ( p ′ ( z ) f x + yf z ) ∂∂y + ( yf y − xf x ) ∂∂z , see [KL13, page 9]. In particular, we get(11) θ f ( x ) = − f ′ ( x ) ν x , θ f ( y ) = f ′ ( y ) ν y , θ f ( z ) = f ′ ( z ) ν z , where ν x , ν y , ν z have been defined in (8).Note that ν x , ν y are locally nilpotent, and so are θ f ( x ) = − f ′ ( x ) ν x and θ f ( y ) = f ′ ( y ) ν y .Moreover, [ ν x , ν y ] = p ′′ ( z ) ν z , and so p ′′ ( z ) ν z ∈ h LNV(D p ) i . A precise description of thefunctions f ∈ O (D p ) such that θ f ∈ h LNV(D p ) i is given by the following result due to utzschebauch and Leuenberger , see [KL13, Theorem 3.26]. Note that we have acanonical decomposition of O (D p ) into a direct sum of vector spaces(12) O (D p ) = xk [ x, z ] ⊕ yk [ y, z ] ⊕ k [ z ] . Proposition 5.
For f ∈ O (D p ) we have θ f ∈ h LNV(D p ) i if and only if f is of the form f = X i> a i ( z ) x i + X j> b j ( z ) y j + ( r ( z ) p ( z )) ′ + γ where a i , b j , r ∈ k [ z ] and γ ∈ k . (cid:3) Denote by L p := Θ − ( h LNV(D p ) i ) ⊂ O (D p ) the subspace of the functions considered inProposition 5. This proposition shows that(13) L p = xk [ x, z ] ⊕ yk [ y, z ] ⊕ { ( rp ) ′ | r ∈ k [ z ] } ⊕ k, where the vector space { ( rp ) ′ | r ∈ k [ z ] } ⊕ k coincides with L p ∩ k [ z ].The next assertion, appearing as Proposition 4 in [LR21], will be used in the proof ofTheorem 2. Proposition 6.
Let δ ∈ h LNV(D p ) i and f ∈ O (D p ) . Then θ δf = [ δ, θ f ] . (cid:3) On the structure of
Aut(D p ) . For the proof of Theorem B we need some informationabout the automorphism group of D p . In this section we follow [LR21, Section 5]. For any g ∈ xk [ x ] and h ∈ yk [ y ] we define the following k + -actions on D p : λ x,g ( s ) := ( x, x − p ( z + sg ( x )) , z + sg ( x )) ,λ y,h ( s ) := ( y − p ( z + sh ( y )) , y, z + sh ( y )) . This gives rise injective homomorphisms of groups λ x : xk [ x ] + → Aut(D p ) , λ y : yk [ y ] + → Aut(D p )defined by g λ x,g := λ x,g (1) and h λ y,h := λ y,h (1). The images of these maps are denotedby U x and U y , respectively. In addition, there is a faithful action ρ of the multiplicative group k ∗ on D p given by ρ ( t )( x, y, z ) := ( tx, t − y, z ) . We denote by T ⊂ Aut(D p ) the image of ρ . Further, let I be the subgroup of Aut(D p )generated by the involution τ : ( x, y, z ) ( y, x, z ). Conjugation with I permutes U x andU y , and I normalizes T . Let Γ be the subgroup of Aut(D p ) generated by γ : ( x, y, z ) ( x, µy, az + b ), where µ, a, b ∈ k are such that µp ( z ) = p ( az + b ). It is easy to check thatΓ normalizes U x and U y and commutes with T . If p is generic in the sense that no affineautomorphism of k permutes the roots of p , then Γ is trivial. The next proposition followsfrom [LR21, Proposition 5]. Proposition 7.
Assume deg p ≥ . Then Aut(D p ) = (U x ∗ U y ) ⋊ (( T × Γ) ⋊ I ) , where U x ∗ U y is the free product of U x and U y . .3. Action of
Aut(D p ) on vector fields. For any isomorphism ϕ : D p → D p we defineAd( ϕ ) : Vec(D p ) ∼ −→ Vec(D p ) by Ad( ϕ )( ν ) := ϕ ∗− ◦ ν ◦ ϕ ∗ , where we view ν as a derivation of O (D p ) and ϕ ∗ : O (D p ) → O (D p ) denotes the pull-back of ϕ . The next lemma follows from [LR21, Lemma 2] and [KL13, Remark 10]. Lemma 5.
Let ϕ ∈ h U x , U y , T, I i ⊆ Aut(D p ) . Then the diagram O (D p ) Vec(D p ) O (D p ) Vec(D p ) ΘΘ ϕ ∗ ≃ Ad( ϕ ) − ≃ is commutative, i.e., Ad( ϕ ) − θ f = θ ϕ ∗ ( f ) for any f ∈ O (D p ) . Remark 3.
We claim that any ϕ ∈ Aut(D p ) sends a locally nilpotent vector field ν on D p to a locally nilpotent vector field. Indeed, let f ∈ O (D p ) be a regular function. There exists k ∈ N such that ν k ( ϕ ∗ ( f )) = 0 and hence,(Ad( ϕ ) ν ) k ( f ) = ( ϕ ∗− ◦ ν ◦ ϕ ∗ ) k ( f ) = ϕ ∗− ( ν k ( ϕ ∗ ( f ))) = 0 . This proves the claim. Further, as Ad( ϕ ) sends any locally nilpotent vector field to a locallynilpotent vector field, it follows that Ad( ϕ ) preserves the Lie subalgebra h LNV(D p ) i ⊂ Vec(D p ).Consider the space L p ⊂ O (D p ) defined in the previous section (see formula (13)). Proposition 8.
For any ϕ ∈ Aut(D p ) , the function z does not belong to ϕ ∗ ( L p ) . In partic-ular, if ϕ ∈ h U x , U y , T, I i ⊆ Aut(D p ) , then ϕ ∗ ( L p ) = L p .Proof. Let f ∈ L p . Assume first ϕ ∈ h U x , U y , T, I i ⊂ Aut(D p ). Then by Lemma 5, we haveAd( ϕ ) θ f = θ ( ϕ ∗ ) − ( f ) . Since Remark 3 implies that Ad( ϕ ) θ f belongs to h LNV(D p ) i , we have( ϕ ∗ ) − ( f ) ∈ L p .Let now γ ∈ Γ. Then γ sends ( x, y, z ) to ( x, µy, az + b ) (see Section 4.2 for details). Hence, γ ∗ ( z ) = az + b , which does not belong to L p by (13). Now, because any automorphism ofAut(D p ) can be written as a product of γ ∈ Γ and ϕ ∈ h U , U y , T, I i (see Proposition 7), weconclude that ( γ ◦ ϕ ) ∗ ( z ) = ϕ ∗ ( γ ∗ ( z )) = ϕ ∗ ( az + b ) L p as ( ϕ ∗ ) − ( L p ) = L p . The prooffollows. (cid:3) Proof of Theorem B.
We now consider two separate cases: deg p = 2 and deg p ≥ Proposition 9.
Suppose that deg p = 2 . Then the width of the simple Lie algebra h LNV(D p ) i is at most two.Proof. First note that x∂ x − y∂ y = [ ν x , ν y ] ∈ h LNV(D p ) i . Hence, for any f of the form X i> a i ( z ) x i + X j> b j ( z ) y j ∈ L p we can find g = X i> i a i ( z ) x i − X j> j b j ( z ) y j ∈ L p uch that ( x∂ x − y∂ y )( g ) = f . Therefore, by Proposition 6 we have(14) θ f = θ ( x∂ x − y∂ y )( g ) = [ x∂ x − y∂ y , θ g ] . Further, ( r ( z ) p ( z )) ′ + γ can be represented as ν x ( r ( z ) x ) = ( p ′ ( z ) ∂∂y + x ∂∂z )( r ( z ) x ) = ( r ( z ) p ( z )) ′ and hence(15) θ ( r ( z ) p ( z )) ′ = [ ν x , θ r ( z ) x ] . By Proposition 5, any µ ∈ h LNV(D p ) i can be represented as a sum of θ f from (14) and θ ( r ( z ) p ( z )) ′ from (15), and the assertion of the proposition follows. (cid:3) Conjecture 2.
Suppose that deg p = 2 . Then the width of h LNV(D p ) i is equal to two. Remark 4.
Our working hypothesis is that θ x + z cannot be represented as a single bracket.Before going over to the case deg p ≥
3, we need to formulate an important conditionunder which we shall be able to prove our main result.We use the embedding O (D p ) ֒ → O (D p ) x = k [ x, x − , z ] which allows us to write any f ∈ O (D p ) in the form(16) f = X i a i ( z ) x i , a i ( z ) ∈ k [ z ] . For any f, g ∈ O (D p ) we then defineJac( f, g ) = f z g x − f x g z . The following condition will play a special role in the sequel.
Hypothesis (J). If Jac( f, g ) = 1 , then f and g are variables, i.e., there exists an automor-phism ϕ of D p such that ϕ ∗ ( f ) = x and ϕ ∗ ( g ) = z . Theorem 2.
Suppose that deg p ≥ and D p satisfies Hypothesis (J) . Then the width of theLie algebra h LNV(D p ) i is greater than one.Proof. We will show that θ x cannot be represented as a single bracket. Assume to thecontrary that there exist ν, µ ∈ h LNV(D p ) i such that [ ν, µ ] = θ x . Then by the isomorphism(9) and Proposition 5, there exist f, g ∈ L p such that [ θ f , θ g ] = θ x . Further, by Proposition6, we have [ θ f , θ g ] = θ θ f ( g ) .By formula (10), taking into account Proposition 5 and formula (16), we can write f inthe form f = X i a i ( z ) x i , where a ( z ) = ( r ( z ) p ( z )) ′ + γ and a i ( z ) , r ( z ) ∈ k [ z ] , γ ∈ k. Similarly, g can be written in the form g = X j b j ( z ) x j , where b ( z ) = ( s ( z ) p ( z )) ′ + ˜ γ and b j ( z ) , s ( z ) ∈ k [ z ] , ˜ γ ∈ k. Then θ f = xf z ∂∂x − ( p ′ ( z ) f x + yf z ) ∂∂y − xf x ∂∂z . ence, θ f ( g ) = x Jac( f, g ) , where Jac( f, g ) = f z g x − f x g z . Since θ f ( g ) should be of degree 1 in x , it follows that θ f ( g ) = cx for some c ∈ k \ { } .Therefore, Jac( f, g ) ∈ k ∗ .Hypothesis (J) now implies that f and g are variables, so that after applying an automor-phism of D p we have f = x , g = z . However, since by Proposition 8 the function z does notbelong to ϕ ∗ ( L p ) for any ϕ ∈ Aut(D p ), we get a contradiction. (cid:3) Concluding remarks
As the reader may have noticed, the results presented above reflect only first steps inunderstanding the width of simple Lie algebras. In this short section, we summarize ourvision of eventual forthcoming steps to be considered. For brevity, below we shorten ‘algebrasof width greater than one’ to ‘wide algebras’.The first immediate question to ask is
Question 1.
What is the actual width of the algebras appearing in Theorems A and B? Canit be made as large as possible?
For better understanding of the width behaviour, it is highly desirable to enlarge the bankof examples of wide simple Lie algebras. A natural way (suggested by
Yuly Billig ) forgeneralizing the examples of Theorem A is towards the Krichever–Novikov algebras relatedto vector fields on punctured curves [Sch14]. This class of infinite dimensional algebras isextremely rich. These algebras arise from meromorphic objects (functions, vector fields,forms of certain weights, matrix valued functions, etc.) which are holomorphic outside afixed set of points. This construction includes Lie algebras and associative algebras (andalso superalgebras, Clifford algebras, etc.). In some natural situations algebras belonging tothis family are known to be simple (see [Sch14, Proposition 6.99]) and look as a source foreventual wide algebras.Note that the Lie algebra L = Vec( X ) captures the geometry of X in the sense that theisomorphism of Lie algebras Vec( X ) ∼ = Vec( Y ) implies that X and Y are isomorphic (here X and Y are arbitrary normal varieties), see [Gra78], [Sie96]. This gives rise to the followingvague question. Question 2.
What geometric properties of X imply that L = Vec( X ) is wide? Pursuing the geometric flavour of the notion of width, one can ask
Question 3.
Does there exist a Lie-algebraic counterpart of the Barge–Ghys example from [BG92] mentioned in the introduction?
This requires going over to the category of smooth vector fields on smooth manifolds. Notethat even simpler looking problems discussed in [LT13] are not yet settled being related tosubtle differential-geometric considerations.One can also look for additional sources of wide simple Lie algebras. A possible candidate(also suggested by
Yuly Billig ) is the following one: uestion 4. Let K denote the Lie algebra obtained from the matrix (cid:18) (cid:19) in the sameway as Kac–Moody Lie algebras are obtained from generalized Cartan matrices, see [Kac68, § . Is K wide? Here is a challenging general question.
Question 5.
Do there exist simple Lie algebras of infinite width?
Note that in sharp contrast with the group case, where there are examples of finitelygenerated simple groups of infinite width, the width of any finitely generated Lie algebra isfinite, see [Rom16].Finally, one can ask a ‘metamathematical’ question.
Question 6.
Let L be a ‘generic’ (‘random’, ‘typical’) simple Lie algebra. Is L wide? Of course, any eventual answer will heavily depend on what is meant by the euphemismsused in the statement. However, the absence of semisimple and nilpotent elements in theLie algebra Vec( C h ) appearing in Theorem A (see Lemma 4), is a witness of the absence ofany analogue of the triangular decomposition. This gives some evidence for the following(‘metamathematical’) working hypothesis: ‘amorphous’ (less structured) simple Lie algebrastend to be wide. In a sense, this is supported by the cases of the algebras of Cartan type andVec ◦ ( A n ) whose automorphisms groups were computed in [Rud69] and [KR17], respectively. Acknowledgements . We thank Joseph Bernstein, Yuly Billig, Zhihua Chang, Adrien Dubouloz,Be’eri Greenfeld, Hanspeter Kraft, Leonid Makar-Limanov, Anatoliy Petravchuk, VladimirPopov, Oksana Yakimova, and Efim Zelmanov for useful discussions regarding various as-pects of this work.
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Department of Mathematics, Bar-Ilan University,Ramat Gan 5290002, Israel
Email address : [email protected] Institut f¨ur Mathematik, Friedrich-Schiller-Universit¨at Jena,Jena 07737, Germany
Email address : [email protected]@gmail.com