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Forum of Mathematics, Sigma (2022), Vol. 10:e88 1β54
doi:10.1017/fms.2022.76
RESEARC H A R T I C L E
Euler continuants in noncommutative quasi-Poisson
geometry
Maxime Fairon 1,2 and David FernΓ‘ndez 3,4
1Department of Mathematical Sciences, Loughborough University, Epinal Way, LE11 3TU Loughborough, United-Kingdom;
E-mail: M.Fairon@lboro.ac.uk.
2School of Mathematics and Statistics, University of Glasgow, University Place, G12 8QQ Glasgow, United-Kingdom;
E-mail: Maxime.Fairon@glasgow.ac.uk.
3DMATH., UniversitΓ© du Luxembourg, Maison du Nombre, 6 Avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg;
E-mail: david.fernandez@uni.lu.
4FakultΓ€t fΓΌr Mathematik, UniversitΓ€t Bielefeld, UniversitΓ€tsstr. 25 33615 Bielefeld, Germany;
E-mail: dfernand@math.uni-bielefeld.de.
Received: 28 May 2021; Revised: 6 August 2022; Accepted: 6 September 2022
2020 Mathematics Subject Classiο¬cation: Primary β 16G20, 17B63; Secondary β 14A22, 53D30, 53D20
Abstract
It was established by Boalch that Euler continuants arise as Lie group valued moment maps for a class of wild
character varieties described as moduli spaces of points on P1by Sibuya. Furthermore, Boalch noticed that these
varieties are multiplicative analogues of certain Nakajima quiver varieties originally introduced by Calabi, which
are attached to the quiver Ξπon two vertices and nequioriented arrows. In this article, we go a step further by
unveiling that the Sibuya varieties can be understood using noncommutative quasi-Poisson geometry modelled on
the quiver Ξπ. We prove that the Poissonstructure car ried by these varieties is induced, via the KontsevichβRosenberg
principle, by an explicit Hamiltonian double quasi-Poisson algebra deο¬ned at the level of the quiver Ξπsuch that
its noncommutative multiplicative moment map is given in terms of Euler continuants. This result generalises the
Hamiltonian double quasi-Poisson algebra associated with the quiver Ξ1by Van den Bergh. Moreover, using the
method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to Ξπadmits a factorisation
in terms of ncopies of the algebra attached to Ξ1.
Contents
1 Introduction 2
1.1 Euler continuants ..................................... 2
1.2 Euler continuants as moment maps ............................ 3
1.3 Main results ........................................ 6
2 Noncommutative quasi-Poisson geometry 8
2.1 Hamiltonian double quasi-Poisson algebras ....................... 8
2.1.1 Double derivations ................................ 8
2.1.2 Double derivations for quivers .......................... 8
2.1.3 Double quasi-Poisson brackets .......................... 9
2.1.4 Multiplicative moment maps ........................... 11
2.2 Fusion of Hamiltonian double quasi-Poisson algebras .................. 11
Β© The Author(s), 2022. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative
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https://doi.org/10.1017/fms.2022.76 Published online by Cambridge University Press
2Maxime Fairon and David FernΓ‘ndez
2.3 π»0-Poisson algebras .................................... 13
2.4 Example from a one-arrow quiver ............................ 13
3 Euler continuants and Hamiltonian double quasi-Poisson algebras 14
3.1 Euler continuants with idempotents ............................ 14
3.2 The result ......................................... 16
3.3 The noncommutative bivector .............................. 18
3.4 The FlaschkaβNewell Poisson bracket .......................... 19
4 Proof of Theorem 3.4 21
4.1 Preparation for the proof ................................. 21
4.2 The double bracket (3.10) is quasi-Poisson ........................ 24
4.3 The element (3.11) is a multiplicative moment map ................... 29
5 Factorisation after localisation at several Euler continuants 38
5.1 The result ......................................... 38
5.2 The algebra Afus
π..................................... 39
5.3 Alternative description of the algebra Bloc
π........................ 41
5.4 Proof of Theorem 5.1 ................................... 42
5.5 Proof of Proposition 5.7 .................................. 42
5.5.1 Preparation .................................... 42
5.5.2 Proof of Proposition 5.7 ............................. 47
5.6 Towards the quasi-bisymplectic form ........................... 51
1. Introduction
1.1. Euler continuants
Fix a string S=βπ₯1. . . π₯πβ of πβ₯1 indeterminates, which are not necessarily invertible or commuting.
We deο¬ne the k-th Euler continuant (polynomial), denoted (π₯1, . . . , π₯π), by starting with the product
π₯1Β· Β· Β· π₯πand then taking the sum of all the distinct substrings of Sobtained by removing adjacent pairs
π₯βπ₯β+1in all possible ways, with +1 assigned to the empty substring. The ο¬rst instances of this family
were already written in 1764 by Euler [28, p. 55] as
(π₯1)=π₯1,
(π₯1, π₯2)=π₯1π₯2+1,
(π₯1, π₯2, π₯3)=π₯1π₯2π₯3+π₯3+π₯1,
(π₯1, π₯2, π₯3, π₯4)=π₯1π₯2π₯3π₯4+π₯1π₯2+π₯1π₯4+π₯3π₯4+1,
(π₯1, π₯2, π₯3, π₯4, π₯5)=π₯1π₯2π₯3π₯4π₯5+π₯1π₯2π₯3+π₯1π₯2π₯5+π₯1π₯4π₯5+π₯3π₯4π₯5+π₯1+π₯3+π₯5.
More succinctly, Euler continuants can be deο¬ned by the following recurrence:
(β
) =1,(π₯1)=π₯1,(π₯1, . . . , π₯π)=(π₯1, . . . , π₯πβ1)π₯π+ (π₯1, . . . , π₯πβ2)if πβ₯2.(1.1)
Originally, Euler introduced this class of polynomials to grasp the numerators and denominators of
continued fractions (hence the name βcontinuantsβ). For example,
π₯1+1
π₯2
=(π₯1, π₯2)
(π₯2), π₯1+1
π₯2+1
π₯3
=(π₯1, π₯2, π₯3)
(π₯2, π₯3), π₯1+1
π₯2+1
π₯3+1
π₯4
=(π₯1, π₯2, π₯3, π₯4)
(π₯2, π₯3, π₯4).
Since then, Euler continuants have appeared in many unexpected areas of research, and they have
become essential in mathematics. In the context of number theory, Euler continuants are closely related
to Euclidβs algorithm; see, for example, [34] for a more detailed account. The number of terms of the k-th
Euler continuant is the (π+1)-th Fibonacci number, while the Catalan numbers count the factorisations
of the continuant via triangulations of polygonsβsee [38], where a connection to Lodayβs free duplicial
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Forum of Mathematics, Sigma 3
algebras is also stated. In knot theory, the Conway polynomial of an oriented two-bridge link is related
to Euler continuants [36]; in contact and symplectic geometry, the braid varieties associated with two-
stranded braids are smooth varieties whose deο¬ning equations are closely linked to Euler continuants
[21]. Another appearance of Euler continuants occurs when we consider a matrix version of the Sternβ
Brocot tree [34], which is a classical construction of all nonnegative fractions whose numerators and
denominators are coprime. If we let
πΏ=ξ1 1
0 1 ξ, π
=ξ1 0
1 1 ξ,
each node in the matrix SternβBrocot tree can be represented as a sequence
π
π0πΏπ1π
π2πΏπ3Β·Β·Β·π
ππβ2πΏππβ1,where π0, ππβ1β₯0, π2, . . . , ππβ2β₯1,
and nis even. Then it is not diο¬cult to prove that the four entries of the obtained matrix can be rewritten
in terms of Euler continuants. To show this, it suο¬ces to note that
ξ1πΌ
0 1 ξ=ξ0 1
1 0 ξξ0 1
1πΌξ,ξ1 0
πΌ1ξ=ξ0 1
1πΌξξ0 1
1 0 ξ,(1.2)
for any constant πΌand then use that, given πβ₯2 and constants π₯πfor 1 β€πβ€π, we have
ξ0 1
1π₯πξξ0 1
1π₯πβ1ξΒ· Β· Β· ξ0 1
1π₯1ξ=ξ(π₯πβ1, . . . , π₯2) (π₯πβ1, . . . , π₯1)
(π₯π, . . . , π₯2) (π₯π, . . . , π₯1)ξ.(1.3)
This remarkable matrix identity can be proven by induction and it will be crucial in Β§1.2.
Finally, beyond the recurrence (1.1), an eο¬ective way to deο¬ne Euler continuants uses the determinant
of a tridiagonal matrix. For such a matrix, the only nonzero entries are given by the indeterminates π₯π
along the main diagonal, β1 on the ο¬rst diagonal below it and +1 on the ο¬rst diagonal above it. This
approach suggests how to introduce new families of continuants; for instance, by taking determinants of
other tridiagonal matrices. They have appeared in interesting contexts such as Coxeterβs frieze patterns,
PtolemyβPlΓΌcker relations and cluster algebras, or the discrete SturmβLiouville, Hill or SchrΓΆdinger
equations (see [37] and references therein).
Quite strikingly, Boalch [15] realised that Euler continuants naturally appear in the setting of certain
wild character varieties linked to the work of Sibuya [40]. In that case, the continuants can be understood
as Lie group valued moment maps [1,2]; we will explain these links in Β§1.2. In this article, we are
able to further prove that Euler continuants are noncommutative moment maps in the sense of Van den
Berghβs noncommutative quasi-Poisson geometryβsee Β§1.3. This is another step towards the programme
that we have initiated in [31] which aims at understanding the Poisson geometry of wild character
varieties in terms of Hamiltonian double quasi-Poisson algebras attached to quivers. Our programme is
based on two main tools. First, the interpretation of such varieties using graphs/quivers undertaken by
Boalch [11]β[16]. Second, the application of the KontsevichβRosenberg principle [35], whereby one
studies a noncommutative structure on an associative algebra whenever it induces the corresponding
standard algebro-geometric structures on representation spaces. Whereas in [31] we pursued this idea
by varying the number of vertices of the quivers under consideration (typically, complete graphs with
some additional data), in this article we ο¬x the quivers to have two vertices and we vary the number of
arrows between them.
1.2. Euler continuants as moment maps
Character varieties of Riemann surfaces have become central objects in modern mathematics. This is
essentially due to their dual nature: They can be deο¬ned as moduli spaces of monodromy data of regular
singular connections or as spaces of representations of the fundamental group. Partially motivated by
works on two-dimensional gauge theory concerning the AtiyahβBott symplectic form, Boalch started
https://doi.org/10.1017/fms.2022.76 Published online by Cambridge University Press
4Maxime Fairon and David FernΓ‘ndez
in [7,8] a groundbreaking programme to investigate the geometry of wild character varieties (note this
terminology appeared in [13]). These spaces generalise character varieties by considering moduli spaces
of monodromy data classifying irregular meromorphic connections on bundles over Riemann surfaces;
alternatively, wild character varieties parametrise fundamental group representations enriched by adding
Stokes data at each singularity. This led to the construction of many new holomorphic symplectic
manifolds [6,7], which turn out to admit hyperkΓ€hler metrics, as shown in [5]. These developments
prompted Boalch in [9] to begin an extensive study of wild character varieties from an alternative
algebraic perspective. Indeed, he realised that such varieties can be constructed as ο¬nite-dimensional
multiplicative symplectic quotients of smooth aο¬ne varieties, involving a distinguished holomorphic 2-
form. This point of view required the introduction of a complex-analytic version of quasi-Hamiltonian
geometry [1], which builds on the key operations of ο¬ssion and fusion. Whereas the former consists in
breaking the structure group into some relevant subgroupsβsee [10,13]βthe latter provides a way to
glue pieces of surfaces together out of building blocks given by conjugacy classes, pairs of pants and the
important ο¬ssion spaces [13]. In the rest of this subsection, we will come across one of these building
blocks: The (reduced) ο¬ssion space Bπ+1; see equation (1.5).
In [15], Boalch examined an interesting class of wild character varieties arising as speciο¬c moduli
spaces of points on P1; they were originally studied by Sibuya [40], and for certain values they give rise
to the prominent gravitational instantons. These wild character varieties are multiplicative analogues of a
family of HyperkΓ€hler varieties introduced by Calabi [20] in 1979 as higher-dimensional examples of the
EguchiβHanson spaces. If we denote by Ξπthe quiver with two vertices {1,2}and narrows ππ: 1 β2
(whose double Ξπhas nextra arrows ππ: 2 β1; see Figure 1), Calabiβs varieties can be described as
(Nakajima/additive) quiver varieties attached to Ξπ. They are obtained by Hamiltonian reduction from
the moduli space of representations of Ξπwith dimension vector π=(1,1). This prompted Boalch [15]
to make the crucial observation that Sibuyaβs varieties, as multiplicative analogues of Calabiβs varieties,
could also be deο¬ned using the quiver Ξπ. This point of view naturally leads to Euler continuants as we
explain now.
If πΊ:=GL2(C), let π+(resp. πβ) be the subgroup of unipotent upper (resp. lower) triangular
matrices, Tbe the maximal diagonal torus of Gand we let πΊβ¦=π+ππβdenote the subspace of
matrices in Gadmitting a Gauss decomposition. Recall that, for a matrix π=(ππ π ) β πΊ, we have that
πβπΊβ¦is equivalent to the condition π22 β 0, in which case we can write that
π=ξ1π12 πβ1
22
0 1 ξξπ11 βπ12πβ1
22 π21 0
0π22 ξξ 1 0
πβ1
22 π21 1ξ.(1.4)
Following Boalch [15] from now on (see also Palubaβs thesis [38] and even more generally [9,13]), we
introduce for any πβ₯1 the reduced ο¬ssion space Bπ+1β (πβΓπ+)π, which is the smooth complex
variety deο¬ned by
Bπ+1:=ξππ,π βπβ, π π, π βπ+for 1 β€πβ€π|ππ, π ππ, π Β·Β·Β·ππ, 1ππ,1βπΊβ¦ξ.(1.5)
1β’ β’ 2
.
.
.
.
.
.
π1
ππ
π1
ππ
Figure 1. The quiver Ξπwith 2πarrows is the double of Ξπ, which is only formed of the n arrows
ππ: 1 β2.
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Forum of Mathematics, Sigma 5
The condition of admitting a Gauss decomposition has a deep geometric meaning. Indeed, there is a
natural action of Ton Bπ+1by simultaneous conjugation, which preserves a 2-form, and the inverse ππ
of the diagonal part of the Gauss decomposition can be interpreted as a Lie group valued moment map,
in the sense of [1]. Furthermore, since we consider the Gauss decomposition of a product of unipotent
matrices, ππtakes value in πβ©SL2(C). This implies that, if we ο¬x the T-valued moment map ππto
π‘πΎ:=diag(πΎ, πΎβ1) β πβ©SL2(C)for generic πΎβCΓ, the corresponding geometric invariant theory
(GIT) quotient Mπ(π‘πΎ)=πβ1
π(π‘πΎ)//πadmits a symplectic form. The space Mπ(π‘πΎ)hence obtained is
an example of the Sibuya spaces mentioned earlier.
To relate the construction of Mπ(π‘πΎ)to quivers and Euler continuants, we let
ππ,π =ξ1 0
π΅π1ξ, ππ,π =ξ1π΄π
0 1 ξ,1β€πβ€π , (1.6)
for constants π΄π, π΅πβCat a point of Bπ+1. Then, by equations (1.2) and (1.3), we can see that
ππ,πππ,π Β· Β· Β· ππ,1ππ,1=ξ0 1
1π΅πξξ0 1
1π΄πξ...ξ0 1
1π΅1ξξ0 1
1π΄1ξ
=ξ(π΄π, . . . , π΅1) ( π΄π, . . . , π΄1)
(π΅π, . . . , π΅1) (π΅π, . . . , π΄1)ξ.
(1.7)
Thus, the condition that this product belongs to πΊβ¦is equivalent to requiring the invertibility of the Euler
continuant (π΅π, . . . , π΄1)placed in the (2,2)-entry. In view of equation (1.4), we get that the diagonal
part of the Gauss decomposition is diag(Λπ π,(π΅π, . . . , π΄1)) , where
Λπ π:=(π΄π, . . . , π΅1) β ( π΄π, . . . , π΄1)(π΅π, . . . , π΄1)β1(π΅π, . . . , π΅1).(1.8)
By taking the determinant in equation (1.7), we can ο¬nd that1Λπ π=(π΄1, π΅1, . . . , π΄π, π΅π)β1. The upshot
is that we can reexpress Bπ+1and its Lie group valued moment map ππ:Bπ+1βπin terms of the space
b
Bπ+1:={π΄π, π΅πβCfor 1 β€πβ€π| (π΄1, . . . , π΅π)β 0}(1.9)
and the map
bπ:b
Bπ+1ββ (CΓ)2,{π΄π, π΅π} β¦ββ ξ(π΅π, . . . , π΄1)β1,(π΄1, . . . , π΅π)ξ.(1.10)
From this point of view, we can realise b
Bπ+1as an aο¬ne subspace2inside Rep(Ξπ,(1,1)), where we
assign π΄π, π΅πto the arrows ππ, ππ, respectively. Moreover, the subspace πβ1
π(π‘πΎ)needed to deο¬ne the wild
character variety Mπ(π‘πΎ)can be identiο¬ed with bπβ1(πΎ, πΎβ1), which can be further seen as a moduli space
of representations of a noncommutative algebra, the ο¬ssion algebra FππΎ(Ξπ); see equation (3.8) with
parameter ππΎ=(πΎ, πΎβ1). Introduced in [14], ο¬ssion algebras generalise the (deformed) multiplicative
preprojective algebras of [25], and they are linked to the generalised double aο¬ne Hecke algebras of
Etingof, Oblomkov and Rains, as explained in [27].
The importance of the discussion made so far is that we do not really need to consider the elements
{π΄π, π΅π}parametrising b
Bπ+1as numbers, but, more generally, we can use matrices π΄πβMatπ2Γπ1(C),
π΅πβMatπ1Γπ2(C)for any π1, π2β₯1. Indeed, we get in that way other examples of reduced ο¬ssion spaces
[13]; see [38]. Hence, we can work with a space of invertible representations inside Rep(Ξπ,(π1, π2)),
rather than Rep(Ξπ,(1,1)) . Therefore, it seems natural to try to understand the structures unveiled by
1Here, one uses that the constants (π΄π, π΅π)π
π=1and the Euler continuants are valued in Cand hence are commuting. The same
form for Λπ πcan be obtained for noncommuting variables π΄π, π΅π, though it requires a diο¬erent proof.
2The reader should be warned that Boalch [15] writes b
Bπ+1=Repβ(Ξπ,(1,1)) as a space of invertible representations inside
Rep(Ξπ,(1,1) ). In that work, one considers Ξπas a graph where each edge represents two arrows going in opposite directions,
that is, this is for us the double quiver Ξπso that Rep(Ξπ,(1,1)) in [15] corresponds to Rep(Ξπ,(1,1) ) for us.
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6Maxime Fairon and David FernΓ‘ndez
Boalch directly at the level of the quiver Ξπby using noncommutative quasi-Poisson geometry [41]. In
that case, we deο¬ne from Ξπthe noncommutative (2π)-th Euler continuants
(ππ, . . . , π1)β1,(π1, . . . , ππ),
which shall play the role of noncommutative moment maps. This will be the aim of this article, whose
main achievements will be presented in Β§1.3.
To gain insight into this noncommutative framework, it may be illuminating to illustrate the case
π=1. Given two complex vector spaces π1, π2of ο¬nite dimensions π1, π2, respectively, we can introduce
the Van den Bergh space (see [4,13,17,43])
BVdB :={π΄βHom(π1, π2), π΅ βHom(π2, π1) | det(Idπ2+π΄π΅)β 0}.
Van den Bergh proved in [41,42] that this space is a complex quasi-Hamiltonian manifold in the sense
of [1] for the natural action of π»=GL(π1) Γ GL(π2). Furthermore, its Lie group valued moment map
is given by
πVdB ({ π΄, π΅}) :=ξ(π΅, π΄)β1,(π΄, π΅)ξ=ξ(Idπ1+π΅ π΄)β1,Idπ2+π΄π΅ξβπ» .
Note that we easily recover (b
B2,bπ)from (BVdB, πVdB)when π1=π2=C. In the setting of multiplicative
quiver varieties, Van den Bergh crucially observed that the pair ξBVdB, πVdBξand its corresponding
2-form can be understood at the level of the quiver Ξ1. Indeed, he introduced the notion of a quasi-
bisymplectic algebra [42] as a noncommutative analogue of a quasi-Hamiltonian manifold, and he
showed that an appropriate localisation, denoted A(Ξ1), of the path algebra of the double quiver Ξ1is
endowed with such a structure. In fact, Van den Bergh ο¬rst introduced a βquasi-Poisson versionβ of this
result, when he derived in [41] that A(Ξ1)is a Hamiltonian double quasi-Poisson algebra; this is the
noncommutative version of a Hamiltonian quasi-Poisson manifold in the sense of [2]; see Deο¬nition 2.5.
The key point is that this structure is such that the second Euler continuants (π, π)β1and (π, π)become
noncommutative moment maps. Our aim is to generalise this result of Van den Bergh: We want to prove
that we can attach to each quiver Ξπan explicit Hamiltonian double quasi-Poisson algebra structure,
whose moment map is given in terms of the (2π)-th Euler continuants (π1, . . . , ππ)and (ππ, . . . , π1)β1.
1.3. Main results
We ο¬x a ο¬eld kof characteristic zero. Recall that Ξπdenotes the quiver on 2 vertices and 2πarrows
labelled {ππ, ππ|1β€πβ€π}as in Figure 1. We introduce the Boalch algebra B(Ξπ)which is obtained
from the path algebra kΞπby requiring that the (2π)-th Euler continuants (π1, . . . , ππ)and (ππ, . . . , π1)
are inverted (see Β§3.1 for the deο¬nition of Euler continuants in kΞπ). The main result of this article is
Theorem 3.4: We prove that B(Ξπ)is endowed with a double quasi-Poisson bracket {{β,β}}, explicitly
deο¬ned in equation (3.10), which is such that Ξ¦:=(π1, . . . , ππ) + (ππ, . . . , π1)β1is a multiplicative
moment map. In other words, the triple ξB(Ξπ),{{β,β}},Ξ¦ξis a Hamiltonian double quasi-Poisson
algebra. As a consequence of this result, we obtain in Corollary 3.5 that the corresponding ο¬ssion
algebra Fπ(Ξπ)deο¬ned by Boalch [15] carries a noncommutative Poisson structure, called an π»0-
Poisson structure, as introduced in [23]. Such an π»0-Poisson structure induces a Poisson bracket on the
corresponding varieties
Mk
π(π):=Rep ξFπ(Ξπ),(π1, π2)ξ//ξGL(π1) Γ GL(π2)ξ
when kis algebraically closed. If k=C,(π1, π2)=(1,1)and π=(πΎ, πΎβ1), we recover the wild
character variety Mπ(π‘πΎ)from Β§1.2. In particular, the Poisson bracket hence obtained is nondegenerate
and corresponds to Boalchβs symplectic form [15]; see Β§5.6. We will investigate this nondegeneracy
property for a general ο¬eld kin a future work.
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Forum of Mathematics, Sigma 7
It is important to observe that, if π=1, the Boalch algebra B(Ξ1)coincides with Van den Berghβs
algebra A(Ξ1)described above, and the Hamiltonian double quasi-Poisson structures are the same.
Thus, Theorem 3.4 can be regarded as a generalisation of [41, Theorem 6.5.1]. We also note that, in the
inductive proof of Theorem 3.4 carried out in Section 4, formulae (3.5) and (3.6) play a key role since
they enable us to rewrite Euler continuants in terms of lower ones. To the best of our knowledge, these
formulae are new and may be of independent interest. After that, using the correspondence between
double (quasi-Poisson) brackets and noncommutative bivectors explained in [41, Β§4.2], we exhibit in
Proposition 3.6 the bivector Pπthat gives rise to the double quasi-Poisson bracket from Theorem 3.4.
As in [42, Proposition 8.3.1], we expect that this result will be important in the future to prove that the
double quasi-Poisson bracket (3.10) is nondegenerate.
Furthermore, Boalch observed [15, Remark 5] that, at certain values (π=2), Mk
π(π)is isomorphic
to the FlaschkaβNewell surface [32]. This is an aο¬ne cubic surface endowed with a Poisson structure;
in the setting of integrable systems, it is closely related to solutions to PainlevΓ© II equation (see also
[22]). Recently, based on [33] (see also [15, Β§5]), Bertola and Tarricone [3] explicitly wrote the Poisson
bracket on the wild character variety Mπ(π‘πΎ)(with no restriction on n); note that an explicit expression
for the corresponding symplectic 2-form is well-known by experts. Whereas we originally obtained
the Hamiltonian double quasi-Poisson structure of Theorem 3.4 employing entirely noncommutative
arguments, in Β§3.4 we are able to show that it induces the Poisson bracket due to Flaschka and Newell
on Rep ξB(Ξπ),(1,1)ξ(i.e., before performing quasi-Hamiltonian reduction to end up with Mπ(π‘πΎ)).
In particular, this shows that our Theorem 3.4 should be regarded as the natural noncommutative
counterpart of the well-known commutative theory.
Finally, one of the driving ideas in quasi-Hamiltonian geometry consists of constructing interesting
moduli spaces from simple pieces by using the operation of fusion [2, Β§5]; algebraically, it endows
the category of quasi-Hamiltonian manifolds with a symmetric monoidal category structure. As we
explain in Β§2.2, Van den Bergh [41, Β§5.3] unveiled a noncommutative analogue of this method to
obtain a double quasi-Poisson bracket and a multiplicative moment map from a Hamiltonian double
quasi-Poisson algebra by identifying several idempotents. The last important result of this article is
Theorem 5.1 which states that, after further localisation at 2πβ2 Euler continuants, the structure of
Hamiltonian double quasi-Poisson algebra of B(Ξπ)unveiled in Theorem 3.4 can be obtained by fusing
the idempotents in ncopies of B(Ξ1). In the simplest case π=2, this can be interpreted as a way to get
the factorisation
(π1, π1, π2, π2)=(πβ²
1, πβ²
1)(πβ²
2, πβ²
2)
of the fourth Euler continuant in terms of a product of second ones through the substitution
πβ²
1=π1, πβ²
1=π1;πβ²
2=π2+ (π1, π1)β1π1, πβ²
2=π2.(1.11)
The proof of Theorem 5.1 is based on an explicit description of the Hamiltonian double quasi-Poisson
algebra structure obtained after the change of coordinates (5.16) that generalises equation (1.11) for any
πβ₯2. We also note that this result relies on the curious identity of Lemma 3.3, which commutes the
adjacent arrows in a product of Euler continuants.
Layout of the article
In Section 2, we recall the basics of Van den Berghβs noncommutative quasi-Poisson geometry [41].
We introduce the important notions of double quasi-Poisson brackets and Hamiltonian double quasi-
Poisson algebras, as well as the method of fusion. In this context, we also present an example associated
with the quiver Ξ1by Van den Bergh [41,42], which is generalised in the rest of this work. In Section
3, we state as part of Theorem 3.4 that we can obtain a Hamiltonian double quasi-Poisson algebra
B(Ξπ)by localisation of the path algebra kΞπof the double of the quiver Ξπ. This structure is such
that its noncommutative moment map is given in terms of Euler continuants. We then write down the
corresponding noncommutative bivector Pπin Proposition 3.6, and we link our result to the Flaschkaβ
https://doi.org/10.1017/fms.2022.76 Published online by Cambridge University Press
8Maxime Fairon and David FernΓ‘ndez
Newell Poisson bracket [33] in Β§3.4. The proof of Theorem 3.4 is the subject of Section 4. Finally, in
Section 5we exhibit a factorisation of (a localisation of) the algebra B(Ξπ)in terms of ncopies of
B(Ξ1)through the method of fusion.
2. Noncommutative quasi-Poisson geometry
2.1. Hamiltonian double quasi-Poisson algebras
Hereafter, we follow [41,24,29].
2.1.1. Double derivations
We ο¬x a ο¬nitely generated associative unital algebra Aover a ο¬eld kof characteristic zero, and we use
the unadorned notations β=βk, Hom =Homkfor brevity. The opposite algebra and the enveloping
algebra of Awill be denoted π΄op and π΄e:=π΄βπ΄op, respectively. We shall identify the category of
A-bimodules and the category of (left) π΄e-modules. Note that the underlying A-bimodule of Acarries
two A-bimodule structures with the same underlying vector space π΄βπ΄, namely the outer and the inner
A-bimodule structures, respectively denoted by (π΄βπ΄)out and (π΄βπ΄)inn; they are given by
π1(πβ²βπβ²β²)π2:=(π1πβ²) β (πβ²β²π2)on (π΄βπ΄)out ;
π1β (πβ²βπβ²β² ) β π2:=(πβ²π2) β (π1πβ²β²)on (π΄βπ΄)inn ,
for all πβ², πβ²β², π1, π2βπ΄.
Given another unital associative k-algebra B,Ais called a B-algebra if there exists a unit preserving
k-algebra morphism π΅βπ΄