1 Introduction

A Note on the Automorphism Group of the Bielawski–Pidstrygach Quiver

\ArticleName

A Note on the Automorphism Group
of the Bielawski–Pidstrygach Quiver

\Author

13566-590 São Carlos - SP, Brasil \Emailigorre@icmc.usp.br, tacchella@icmc.usp.br

\ArticleDates

Received August 29, 2012, in final form April 26, 2013; Published online April 30, 2013

\Abstract

We show that there exists a morphism between a group introduced by G. Wilson and a quotient of the group of tame symplectic automorphisms of the path algebra of a quiver introduced by Bielawski and Pidstrygach. The latter is known to act transitively on the phase space of the Gibbons–Hermsen integrable system of rank , and we prove that the subgroup generated by the image of together with a particular tame symplectic automorphism has the property that, for every pair of points of the regular and semisimple locus of , the subgroup contains an element sending the first point to the second.

\Keywords

Gibbons–Hermsen system; quiver varieties; noncommutative symplectic geometry; integrable systems

\Classification

37K10; 16G20; 14A22

## 1 Introduction

Let and be two positive natural numbers and denote by the complex vector space of matrices with entries in . The space

 Vn,r:=Matn,n(C)⊕Matn,n(C)⊕Matn,r(C)⊕Matr,n(C)

can be viewed (using the identifications provided by the trace form) as the cotangent bundle of the vector space , thus it comes equipped with the canonical holomorphic symplectic form

 ω(X,Y,v,w)=Tr(dX∧dY+dv∧dw). (1)

The group acts on by

 g.(X,Y,v,w)=(gXg−1,gYg−1,gv,wg−1). (2)

This action is Hamiltonian, and the corresponding moment map is

 μ(X,Y,v,w)=[X,Y]+vw. (3)

For every complex number the action of on is free, hence we can perform the symplectic quotient

 Cn,r:=μ−1(τI)/GLn(C). (4)

This family of smooth, irreducible affine algebraic varieties plays an important rôle in various fields. They are examples of Nakajima quiver varieties [12], and they also arise in the work of Nekrasov and Schwarz [13] on the moduli space of instantons on a non-commutative . Finally, and most importantly from the perspective of the present work, they can be seen as a completion of the phase space of a family of integrable systems that generalize the well-known rational Calogero–Moser model.

### 1.1 The Gibbons–Hermsen system

Let us briefly remind the reader of the definition of this integrable system, as it was introduced by Gibbons and Hermsen in the paper [6]. Just like the (complexified) Calogero–Moser model, the system describes the motion of point particles in the complex plane interacting pairwise according to a potential proportional to the second inverse power of their distance. In addition to the Calogero–Moser case, however, each particle is endowed with some additional “internal” degrees of freedom, parametrized by a vector in an auxiliary vector space and by its canonical conjugate in the dual space . The Hamiltonian of the system is given by

 H(x,p,v,ξ)=12n∑i=1p2i+12n∑i,j=1ξi(vj)ξj(vi)(xi−xj)2. (5)

For each particle there is the constraint (notice that these quantities are constants of the motion); moreover, two pairs and are considered equivalent if and for some . When these requirements completely fix the additional degrees of freedom and we recover the classic rational Calogero–Moser system.

As it was proved by Gibbons and Hermsen, the Hamiltonian system described above is completely integrable and its phase space can be identified with the manifold . Let us explain shortly how goes the proof of the complete integrability. Consider, for each and , the following function on the space :

 Jk,α=TrYkvαw. (6)

These functions are invariant with respect to the action (2), so that they descend to well defined functions on the quotient space ; the Hamiltonian (5) coincides, up to scalar multiples, with . The equations of motion determined by are

 ˙X=Yk−1vαw+Yk−2vαwY+⋯+vαwYk−1, ˙Y=0, ˙v=Ykvα, ˙w=−αwYk,

where . From this we can deduce that the Gibbons–Hermsen flows are complete. In fact, since is constant, the equations for and are linear with constant coefficients. This implies that the solutions of the last two equations are linear combinations of polynomials and exponentials, forcing the solution of the first equation to be of the same form.

The Poisson brackets defined by the symplectic form (1) are given by

 {Xij,Ykℓ}=δjkδiℓand{vij,wkℓ}=δjkδiℓ,

all the others being equal to zero. Then a short calculation shows that the Poisson bracket between two functions of the form (6) is

 {Jm,α,Jℓ,β}=Jm+ℓ,[α,β], (7)

where is the matrix commutator. Notice that these are the same relations holding in the Lie algebra of polynomial loops in : explicitly, the correspondence is given by

 Jk,α↔zkα. (8)

From (7) follows in particular that if and only if . It is then possible to find among the functions (6) a total of independent and mutually commuting first integrals (e.g. by taking and matrices spanning the space of diagonal matrices). These results imply the complete integrability of the Gibbons–Hermsen system.

### 1.2 Some reminder about non-commutative symplectic geometry

In [3], Bielawski and Pidstrygach study the varieties (4) in the case using the methods of non-commutative symplectic geometry [4, 7], starting from the quiver

 QBP=\xymatrix (9)

Recall that a quiver is simply a directed graph, possibly with loops and multiple edges. To every quiver one can associate its double obtained by keeping the same vertices and adding, for each arrow , a corresponding arrow going in the opposite direction. The path algebra (over ) of a quiver , denoted , is the complex associative algebra which is generated, as a linear space, by all the paths in and whose product is given by composition of paths (or zero when two paths do not compose).

Now let denote , the path algebra of the double of . Denote also by the group of (tame) non-commutative symplectomorphisms of this algebra (see Definition 2 in the next section). One of the main results of [3] is that this group acts transitively on . This is to be compared with the well-known result for the case , first obtained by Berest and Wilson in [2], according to which the group of automorphisms of the first Weyl algebra

 A1=C⟨a,a∗⟩/(aa∗−a∗a−1)

acts transitively on the Calogero–Moser varieties . This group can also be interpreted from the perspective of non-commutative symplectic geometry in the following way. Let denote the quiver with one vertex and one loop on it. The path algebra of its double is just the free associative algebra on the two generators and . The group of non-commutative symplectomorphisms of this algebra is the group of automorphisms of preserving the commutator , and this group is isomorphic to by a result of Makar-Limanov [9, 10]. Hence the rank 1 case fits into the same picture, by replacing the quiver (9) with .

It turns out that in this case these non-commutative symplectomorphisms have a very natural interpretation in terms of flows of the Calogero–Moser system. Indeed, a classic result of Dixmier [5] implies that the group is generated by a family of automorphisms labeled by a polynomial in (say with zero constant term), defined by the following action on the generators of :

 Φp(a)=a−p′(a∗),Φp(a∗)=a∗, (10)

together with the single automorphism defined by

 F0(a,a∗)=(−a∗,a), (11)

that we will call the formal Fourier transform. The action of these generators of on is given by

 Φp.(X,Y,v,w)=(X−p′(Y),Y,v,w),F0.(X,Y,v,w)=(−Y,X,v,w).

In particular the action of for a given polynomial corresponds exactly to the action of a linear combination of the (mutually commuting) Calogero–Moser flows, i.e. the flows of the Hamiltonian functions (for ) on , with “times” .

### 1.3 The main results of this note

Given the above, it is natural to ask if a similar picture holds also in the rank 2 case; namely, if the action of the group considered by Bielawski and Pidstrygach on can be made more concrete by interpreting its elements as flows of the Gibbons–Hermsen Hamiltonians (6). One difficulty here is given by the fact that, while the Calogero–Moser Hamiltonians generate an abelian Lie algebra, the Hamiltonians (6) generate a non-abelian one that cannot be trivially exponentiated to get a Lie group. In other words, when the maps of the form , where is a polynomial map , do not form a group. One way to avoid this problem would be to simply take the group of all holomorphic maps , but this group contains elements giving non-polynomial flows on which cannot be realized by the action of an element in .

In the unpublished notes [16], G. Wilson suggests to consider instead the subgroup of defined by

 Γalg:=Γalgsc×PGL2(C[z]), (12)

where is the subgroup consisting of maps of the form for some polynomial with no constant term and is seen as a subgroup of in the obvious manner. (The choice of this particular subgroup can be motivated also on purely algebraic grounds, as we will explain in Section 4.)

Now denote by the quotient of by the subgroup of scalar affine symplectic automorphisms, whose action on is trivial (see Definition 2 below). The aim of this note is to prove the following: {theorem} There exists a morphism of groups

 i:Γalg→PTAut(A;c), (13)

such that, if denotes the subgroup of generated by the image of and the symplectomorphism defined by extending the automorphism (11) from to in the following way:

 F(a,a∗,x,x∗,y,y∗):=(−a∗,a,−y∗,y,−x∗,x) (14)

and is the Zariski open subset of consisting of quadruples such that either  or  is regular semisimple i.e., diagonalizable with distinct eigenvalues, then for every pair of points there exists an element of which maps to .

Here should be seen as the rank 2 version of the analogue subset of the Calogero–Moser space consisting of quadruples for which either or are diagonalizable.

{remark}

Theorem 1.3 is not a real transitivity result since the action of does not preserve the subset . Unfortunately, it is not easy to understand if this action is transitive on the whole of or not. The main difficulty comes from the fact that the proof of transitivity in [3] for points outside of is not constructive; for this reason studying the action of on such points is much more difficult.

## 2 Preliminaries

For the remainder of this paper, will be fixed and equal to . In this case, as noticed in [3], we can obtain the manifold defined by (4) starting from the space of representations of the double of the quiver (9), in the following manner.

Let us denote by the complex vector space of linear representations of with dimension vector . A point in this space is a -tuple consisting of two matrices, two matrices and two matrices that represent, respectively, the arrows , , , , and in . This space is in bijection with via the following map:

 A↦X,B↦Y,X1↦v∙1,X2↦−v∙2,Y1↦w2∙,Y2↦w1∙, (15)

where by we denote the -th column of the matrix , and similarly by we denote the -th row of the matrix .

On the space there is a natural action of the group

 G(n,1)=(GLn(C)×GL1(C))/C∗≃GLn(C)

(where is seen as the subgroup of pairs of the form for some ) by change of basis. This action is Hamiltonian, with moment map given by

 ν(A,B,X1,X2,Y1,Y2)=([A,B]+X1Y2−X2Y1,Y1X2−Y2X1)∈g(n,1). (16)

It is easy to verify that, under the bijection (15), this action of on precisely coincides with the action of on given by (2). Finally, by comparing the two moment maps (3) and (16), we conclude that is exactly the same as the symplectic quotient , where denotes the coadjoint orbit of the point .

As in the introduction, we let stand for the path algebra of ; it is a non-commutative algebra over the ring , where the idempotents and correspond to the trivial paths at vertices and , respectively. For every we denote by the subgroup of that fixes . In particular we will be interested in , where

 c:=[a,a∗]+[x,x∗]+[y,y∗].
{definition}

The group will be called the group of non-commutative symplectic automorphisms of  [3, 8].

In what follows we will be interested in the following types of elements of . {definition} An automorphism of will be called:

• strictly triangular if it fixes the arrows of (i.e. , and );

• strictly op-triangular if it fixes the arrows of (i.e. , , ).

An explicit description of strictly triangular symplectic automorphisms of is derived in [3]. Namely, let  be the free algebra on two generators over  and define

 L2:=F2C+[F2,F2] (17)

as a quotient of complex vector spaces. Call and (the image in of) the two generators of . Notice that is just the vector space of necklace words in and (modulo scalars). Then to every we can associate the automorphism defined on the generators of  by

 a↦a, a∗↦a∗+∂f∂a, x↦x, x∗↦x∗+y∂f∂b, y↦y, y∗↦y∗+∂f∂bx,

where the substitution is understood and the -linear maps

 ∂∂a,∂∂b:L2→F2

are the “necklace derivations” defined e.g. in [4, 7]. Explicitly, they act as usual derivations, except that the letters in a necklace word must be cyclically permuted in order to always bring the cancelled letter at the front.

{example}

Let and . Then

 ∂∂af1=ab+ba,and∂∂af2=aab+aba+baa.

More generally

 ∂∂a(anb)=an−1b+an−2ba+⋯+aban−2+ban−1.

Notice that the result lives in , not in ; in particular it is not a necklace word, but a genuine word in the generators. {theorem}[Proposition 7.2 in [3]] Every is symplectic, and every symplectic automorphism that fixes , and lies in the image of .

A completely analogous description holds for strictly op-triangular symplectic automorphisms. Indeed, let denote the same vector space (17), but call now and (the image of) the two generators of . For every , let be the strictly op-triangular automorphism of  defined by

 a↦a+∂f∂a∗, a∗↦a∗, x↦x+∂f∂b∗y∗, x∗↦x∗, y↦y+x∗∂f∂b∗, y∗↦y∗,

where . We claim that every is symplectic, and every symplectic automorphism of that fixes , and is of this form. This can easily be proved by recycling exactly the same arguments used in [3] to prove Theorem 2. Alternatively, it is easy to verify that an automorphism is strictly triangular if and only if the automorphism is strictly op-triangular, where is the symplectic automorphism defined by (14). Thus we could simply define

 Λ′(f(a∗,b∗))=F∘Λ(−f(a,b))∘F−1. (18)

Another subgroup of easy to deal with is provided by the affine automorphisms, i.e. affine trasformations of the linear subspace spanned by , , , , and in . An automorphism of this kind which moreover preserves is completely specified by a pair where is an element of (the group of unimodular affine transformation of ) acting on the subspace spanned by and , while is an element of acting as follows on the subspace spanned by the other arrows:

Following [3], we denote by the subgroup consisting of these affine symplectic automorphisms.

{definition}

The group of tame symplectic automorphisms of , denoted , is the subgroup of generated by strictly triangular and affine symplectic automorphisms.

Notice that the automorphism defined by (14) belongs to ; it corresponds to the pair determined by and . It then follows immediately from the relation (18) that can also be generated by the strictly op-triangular automorphisms and by the affine symplectic ones.

Let be the subgroup of consisting of symplectic affine automorphism of the form where is the identity of and belongs to the center of (i.e.  for some ). Then it is easy to see that the action of on is trivial; hence the action of on descends through the quotient .

{definition}

We denote the quotient by .

An essential rôle in the sequel will be played by the following result, first proved by Nagao in [11] and later rederived in a more general context using the Bass–Serre theory of groups acting on graphs [14]. Let be a field, and denote by the subgroup of lower triangular matrices in and by the subgroup of lower triangular matrices in . {theorem}[Nagao] The group coincides with the free product with amalgamation . Suppose now that . Then, as is well known, we have that , where is the (normal) subgroup of lower unitriangular matrices (= unipotent elements in ) and is the subgroup of diagonal matrices. Exactly the same result holds also for : namely, the latter group is isomorphic to the semidirect product of its normal subgroup consisting of matrices of the form for some (which is in fact isomorphic to the abelian group ) and its subgroup of diagonal matrices, which is again . It follows that every element of can be uniquely written as a product of the form with and . Since

 (α00β)(10p1)(α−100β−1)=(10βαp1),

we see that, abstractly, the action of on defining the above semidirect product structure is given by

 (α,β).p=βαp. (19)

## 3 Proof of the results

Our strategy to define the morphism (13) is the following. First, we identify the action on of some strictly op-triangular automorphism in with the action of a unipotent matrix of the form via the flow induced by some particular Hamiltonians of the Gibbons–Hermsen system (using Theorem 3 below). In this way we obtain an embedding of the group in which is easily extended to the whole subgroup . The subgroup consisting of invertible scalar matrices can also be embedded in using affine automorphisms acting only on the subspace spanned by , , and . By Theorem 2 these embeddings extend to a unique morphism of groups . Finally we use to induce the desired morphism .

An automorphism acts on , and hence on , in the following way. For every arrow in , is a non-commutative polynomial in the arrows of ; in particular we can evaluate it on a point (by mapping each arrow to its matrix representation), and this gives a matrix . Then sends to the point

 (φ(a)(p),φ(a∗)(p),φ(x)(p),φ(y∗)(p),φ(y)(p),φ(x∗)(p)). (20)
{example}

If is the strictly triangular automorphism with then

 ∂f∂a=ab+baand∂f∂b=aa,

so that acts by the formula

 φ.(A,B,X1,X2,Y1,Y2)=(A,B+AX1Y1+X1Y1A,X1,X2+A2X1,Y1,Y2+Y1A2).

We are now going to prove a result that enables us to identify the action of some Hamiltonians functions on with the action of some triangular (or op-triangular) automorphisms in . This correspondence will be established in much more generality than what is needed in the sequel, since it may be of independent interest.

Let us define a linear map from the complex vector space defined in (17) to the ring of regular functions on as follows. Any element of can be written as a linear combination of necklace words with and not all zero. We set

 H(f):=TrXk1(ve12w)ℓ1⋯Xkn(ve12w)ℓn (21)

(where ) and extend this by linearity to the whole of . Similarly, we can define a map from to the ring of regular functions on by sending the generic necklace word in and to

 H′(f):=TrYk1(ve21w)ℓ1⋯Ykn(ve21w)ℓn, (22)

where .

{theorem}

The action determined by the flow at unit time of the Hamiltonian function resp.  on coincides with the action (20) of the automorphism resp. .

###### Proof.

By a straightforward, if tedious, calculation one can verify that the flow of is given by solving the following system of differential equations:

 ˙X=0, (23a) ˙Y=−n∑j=1kj∑ij=1Xkj−ij(ve12w)ℓjXkj+1(ve12w)ℓj+1⋯Xkj−1(ve12w)ℓj−1Xij−1, (23b) ˙v=n∑j=1ℓj∑ij=1(ve12w)ℓj−ijXkj+1(ve12w)ℓj+1⋯Xkj(ve12w)ij−1ve12, (23c) ˙w=−e12wn∑j=1ℓj∑ij=1(ve12w)ℓj−ijXkj+1(ve12w)ℓj+1⋯Xkj(ve12w)ij−1, (23d)

where is understood as a cyclic index modulo , i.e.  and . These equations can be easily integrated. Indeed, equation (23a) and “half” of equations (23c) and (23d) tell us that , and are constants; then the time derivatives of , and involve only these constants, so that the flows are linear in time. Thus the non-trivial part of the flow is given by

 Y(t)=Y−tn∑j=1kj∑ij=1Xkj−ij(v∙1w2∙)ℓj⋯(v∙1w2∙)ℓj−1Xij−1, v∙2(t)=v∙2+tn∑j=1ℓj∑ij=1(v∙1w2∙)ℓj−ijXkj+1⋯Xkj(v∙1w2∙)ij−1v∙1, w1∙(t)=w1∙−tw2∙n∑j=1ℓj∑ij=1(v∙1w2∙)ℓj−ijXkj+1⋯Xkj(v∙1w2∙)ij−1.

Using the map (15) we can see the above as the following flow on :

 A(t)=A,X1(t)=X1,Y1(t)=Y1, B(t)=B−tn∑j=1kj∑ij=1Akj−ij(X1Y1)ℓj⋯(X1Y1)ℓj−1Aij−1, X2(t)=X2−tn∑j=1ℓj∑ij=1(X1Y1)ℓj−ijAkj+1⋯Akj(X1Y1)ij−1X1, Y2(t)=Y2−tY1n∑j=1ℓj∑ij=1(X1Y1)ℓj−ijAkj+1⋯Akj(X1Y1)ij−1.

Evaluating at we recover exactly the action of the automorphism , as can be easily verified. A completely analogous calculation shows that the same relationship holds in the “opposite” case between and . ∎

{remark}

The Poisson brackets between the Hamiltonians (21), (22) are easily calculated in the following manner. The vector spaces and can be seen as subspaces of the path algebra , where is the quiver with a single vertex and two loops and on it. Then the four matrices define a point in the representation space and the maps and are just the restrictions to and , respectively, of the map

 ψ:C¯¯¯¯Q8[C¯¯¯¯Q8,C¯¯¯¯Q8]→C[Rep(¯¯¯¯Q8,(n))]GLn(C)

defined by Ginzburg in [7]. There it is proved that is in fact a Lie algebra morphism, so that the Poisson bracket between and (or ) is simply the image of the necklace Lie bracket under . It follows in particular that all the Hamiltonians in the image of Poisson-commute (and similarly for ); however in general.

Notice that the usual Hamiltonians (6) of the Gibbons–Hermsen system can only give a polynomial flow on when is either the identity (in which case , as a consequence of the moment map equation ) or a nilpotent matrix. In what follows we will consider in particular the Hamiltonians (but see Remark 3 below). Under the correspondence (8), such Hamiltonians correspond to matrices of the form . The exponential of a linear combination of matrices of this kind,

 ∑kpkzke21

is the lower unitriangular matrix , where is the polynomial with coefficients . Theorem 3 then suggests that these elements of should correspond to the op-triangular automorphisms in . We are now going to prove Theorem 1.3 by building the morphism along those lines.

###### Proof of Theorem 1.3.

In view of Theorem 2, the first goal is to define two morphisms of groups

 j1:GL2(C)→TAut(A;c)andj2:B2(C[z])→TAut(A;c),

that agree on . We define by sending to the affine symplectic automorphism determined by the pair , where is the identity in . To define , notice first that the subgroup of consisting of strictly op-triangular automorphisms of the form for some polynomial is isomorphic to . Moreover, let be any diagonal matrix in ; then a simple calculation shows that, for every ,

 (I,d)∘Λ′(−p(a∗)b∗)∘(I,d−1)=Λ′(−βαp(a∗)b∗).

This is exactly the action (19) defining the semidirect product structure of , hence we can define as the unique morphism of groups sending a lower unitriangular matrix to the automorphism and a diagonal matrix to the affine automorphism .

With these definitions it is immediate to verify that agrees with on ; then by the universal property of amalgamated free products there exists a unique morphism of groups whose restriction to , resp. , coincides with , resp. . It is clear that  descends to a well-defined morphism of groups . We extend to as follows. Let us define a morphism of groups by sending the generic scalar matrix (where ) to the automorphism , whose only nontrivial action on the generators is

 a↦a−∂∂a∗p(a∗). (24)

It is easy to verify that such an automorphism commutes with every element in the image of , since it commutes with both elements in the image of and affine automorphisms of the form . Thus we can define  by mapping the generic element to the product in .

Now consider the subgroup of generated by the image of and the affine symplectic automorphism defined by (14). Clearly, acts on by restriction of the action of . Recall from [3] that the strategy to prove the transitivity of the latter action is first to move every point of into the submanifold

 Mn:=\set(X,Y,v,w)∈Cn,2v∙2=0,w2∙=0

(isomorphic to the Calogero–Moser space), and then use the fact that