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Euler continuants in noncommutative quasi-Poisson geometry

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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 P1\mathbb {P}^1 by 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 Ξ“n\Gamma _n on two vertices and n equioriented 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 Ξ“n\Gamma _n . We prove that the Poisson structure carried by these varieties is induced, via the Kontsevich–Rosenberg principle, by an explicit Hamiltonian double quasi-Poisson algebra defined at the level of the quiver Ξ“n\Gamma _n such 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 Ξ“1\Gamma _1 by Van den Bergh. Moreover, using the method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to Ξ“n\Gamma _n admits a factorisation in terms of n copies of the algebra attached to Ξ“1\Gamma _1 .
Forum of Mathematics, Sigma (2022), Vol. 10:e88 1–54
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 Classification: Primary – 16G20, 17B63; Secondary – 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 Γ𝑛on 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 Γ𝑛. We prove that the Poissonstructure car ried by these varieties is induced, via the Kontsevich–Rosenberg
principle, by an explicit Hamiltonian double quasi-Poisson algebra defined at the level of the quiver Γ𝑛such 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 Ξ“1by Van den Bergh. Moreover, using the
method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to Γ𝑛admits a factorisation
in terms of ncopies of the algebra attached to Ξ“1.
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
Commons Attribution-NonCommercial-ShareAlike licence (https://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial
re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly
cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
https://doi.org/10.1017/fms.2022.76 Published online by Cambridge University Press
2Maxime Fairon and David FernΓ‘ndez
2.3 𝐻0-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 Flaschka–Newell 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
𝑛..................................... 39
5.3 Alternative description of the algebra Bloc
𝑛........................ 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=β€˜π‘₯1. . . π‘₯π‘˜β€™ of π‘˜β‰₯1 indeterminates, which are not necessarily invertible or commuting.
We define the k-th Euler continuant (polynomial), denoted (π‘₯1, . . . , π‘₯π‘˜), by starting with the product
π‘₯1Β· Β· Β· π‘₯π‘˜and then taking the sum of all the distinct substrings of Sobtained by removing adjacent pairs
π‘₯β„“π‘₯β„“+1in all possible ways, with +1 assigned to the empty substring. The first 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 defined by the following recurrence:
(βˆ…) =1,(π‘₯1)=π‘₯1,(π‘₯1, . . . , π‘₯π‘˜)=(π‘₯1, . . . , π‘₯π‘˜βˆ’1)π‘₯π‘˜+ (π‘₯1, . . . , π‘₯π‘˜βˆ’2)if π‘˜β‰₯2.(1.1)
Originally, Euler introduced this class of polynomials to grasp the numerators and denominators of
continued fractions (hence the name β€˜continuants’). 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 Euclid’s 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 polygonsβ€”see [38], where a connection to Loday’s free duplicial
https://doi.org/10.1017/fms.2022.76 Published online by Cambridge University Press
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 defining equations are closely linked to Euler continuants
[21]. Another appearance of Euler continuants occurs when we consider a matrix version of the Stern–
Brocot tree [34], which is a classical construction of all nonnegative fractions whose numerators and
denominators are coprime. If we let
𝐿=ξ˜’1 1
0 1 ξ˜“, 𝑅 =ξ˜’1 0
1 1 ξ˜“,
each node in the matrix Stern–Brocot tree can be represented as a sequence
𝑅𝑐0𝐿𝑐1𝑅𝑐2𝐿𝑐3Β·Β·Β·π‘…π‘π‘›βˆ’2πΏπ‘π‘›βˆ’1,where 𝑐0, π‘π‘›βˆ’1β‰₯0, 𝑐2, . . . , π‘π‘›βˆ’2β‰₯1,
and nis even. Then it is not difficult to prove that the four entries of the obtained matrix can be rewritten
in terms of Euler continuants. To show this, it suffices to note that
ξ˜’1𝛼
0 1 ξ˜“=ξ˜’0 1
1 0 ξ˜“ξ˜’0 1
1π›Όξ˜“,ξ˜’1 0
𝛼1ξ˜“=ξ˜’0 1
1π›Όξ˜“ξ˜’0 1
1 0 ξ˜“,(1.2)
for any constant 𝛼and then use that, given π‘˜β‰₯2 and constants π‘₯𝑖for 1 β‰€π‘–β‰€π‘˜, we have
ξ˜’0 1
1π‘₯π‘˜ξ˜“ξ˜’0 1
1π‘₯π‘˜βˆ’1ξ˜“Β· Β· Β· ξ˜’0 1
1π‘₯1ξ˜“=ξ˜’(π‘₯π‘˜βˆ’1, . . . , π‘₯2) (π‘₯π‘˜βˆ’1, . . . , 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 effective way to define Euler continuants uses the determinant
of a tridiagonal matrix. For such a matrix, the only nonzero entries are given by the indeterminates π‘₯𝑖
along the main diagonal, βˆ’1 on the first diagonal below it and +1 on the first 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 Coxeter’s frieze patterns,
Ptolemy–PlΓΌcker relations and cluster algebras, or the discrete Sturm–Liouville, 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
Bergh’s noncommutative quasi-Poisson geometryβ€”see 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]–[16]. Second, the application of the Kontsevich–Rosenberg 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 fix 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 defined 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 Atiyah–Bott 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 finite-dimensional
multiplicative symplectic quotients of smooth affine 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 fission and fusion. Whereas the former consists in
breaking the structure group into some relevant subgroupsβ€”see [10,13]β€”the latter provides a way to
glue pieces of surfaces together out of building blocks given by conjugacy classes, pairs of pants and the
important fission spaces [13]. In the rest of this subsection, we will come across one of these building
blocks: The (reduced) fission space B𝑛+1; see equation (1.5).
In [15], Boalch examined an interesting class of wild character varieties arising as specific 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
Eguchi–Hanson spaces. If we denote by Γ𝑛the quiver with two vertices {1,2}and narrows π‘Žπ‘–: 1 β†’2
(whose double Γ𝑛has nextra arrows 𝑏𝑖: 2 β†’1; see Figure 1), Calabi’s varieties can be described as
(Nakajima/additive) quiver varieties attached to Γ𝑛. They are obtained by Hamiltonian reduction from
the moduli space of representations of Γ𝑛with dimension vector 𝑑=(1,1). This prompted Boalch [15]
to make the crucial observation that Sibuya’s varieties, as multiplicative analogues of Calabi’s varieties,
could also be defined using the quiver Γ𝑛. This point of view naturally leads to Euler continuants as we
explain now.
If 𝐺:=GL2(C), let π‘ˆ+(resp. π‘ˆβˆ’) be the subgroup of unipotent upper (resp. lower) triangular
matrices, Tbe the maximal diagonal torus of Gand we let 𝐺◦=π‘ˆ+π‘‡π‘ˆβˆ’denote the subspace of
matrices in Gadmitting a Gauss decomposition. Recall that, for a matrix 𝑀=(𝑀𝑖 𝑗 ) ∈ 𝐺, we have that
π‘€βˆˆπΊβ—¦is equivalent to the condition 𝑀22 β‰ 0, in which case we can write that
𝑀=ξ˜’1𝑀12 π‘€βˆ’1
22
0 1 ξ˜“ξ˜’π‘€11 βˆ’π‘€12π‘€βˆ’1
22 𝑀21 0
0𝑀22 ξ˜“ξ˜’ 1 0
π‘€βˆ’1
22 𝑀21 1ξ˜“.(1.4)
Following Boalch [15] from now on (see also Paluba’s thesis [38] and even more generally [9,13]), we
introduce for any 𝑛β‰₯1 the reduced fission space B𝑛+1βŠ‚ (π‘ˆβˆ’Γ—π‘ˆ+)𝑛, which is the smooth complex
variety defined by
B𝑛+1:=ξ˜ˆπ‘†π‘,𝑖 βˆˆπ‘ˆβˆ’, 𝑆 π‘Ž, 𝑖 βˆˆπ‘ˆ+for 1 ≀𝑖≀𝑛|𝑆𝑏, 𝑛 π‘†π‘Ž, 𝑛 ···𝑆𝑏, 1π‘†π‘Ž,1βˆˆπΊβ—¦ξ˜‰.(1.5)
1β€’ β€’ 2
.
.
.
.
.
.
π‘Ž1
π‘Žπ‘›
𝑏1
𝑏𝑛
Figure 1. The quiver Γ𝑛with 2𝑛arrows is the double of Γ𝑛, which is only formed of the n arrows
π‘Žπ‘–: 1 β†’2.
https://doi.org/10.1017/fms.2022.76 Published online by Cambridge University Press
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 B𝑛+1by simultaneous conjugation, which preserves a 2-form, and the inverse πœ‡π‘‡
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, πœ‡π‘‡takes value in π‘‡βˆ©SL2(C). This implies that, if we fix the T-valued moment map πœ‡π‘‡to
𝑑𝛾:=diag(𝛾, π›Ύβˆ’1) ∈ π‘‡βˆ©SL2(C)for generic π›ΎβˆˆCΓ—, the corresponding geometric invariant theory
(GIT) quotient M𝑛(𝑑𝛾)=πœ‡βˆ’1
𝑇(𝑑𝛾)//𝑇admits a symplectic form. The space M𝑛(𝑑𝛾)hence obtained is
an example of the Sibuya spaces mentioned earlier.
To relate the construction of M𝑛(𝑑𝛾)to quivers and Euler continuants, we let
𝑆𝑏,𝑖 =ξ˜’1 0
𝐡𝑖1ξ˜“, π‘†π‘Ž,𝑖 =ξ˜’1𝐴𝑖
0 1 ξ˜“,1≀𝑖≀𝑛 , (1.6)
for constants 𝐴𝑖, π΅π‘–βˆˆCat a point of B𝑛+1. Then, by equations (1.2) and (1.3), we can see that
𝑆𝑏,π‘›π‘†π‘Ž,𝑛 Β· Β· Β· 𝑆𝑏,1π‘†π‘Ž,1=ξ˜’0 1
1π΅π‘›ξ˜“ξ˜’0 1
1π΄π‘›ξ˜“...ξ˜’0 1
1𝐡1ξ˜“ξ˜’0 1
1𝐴1ξ˜“
=ξ˜’(𝐴𝑛, . . . , 𝐡1) ( 𝐴𝑛, . . . , 𝐴1)
(𝐡𝑛, . . . , 𝐡1) (𝐡𝑛, . . . , 𝐴1)ξ˜“.
(1.7)
Thus, the condition that this product belongs to 𝐺◦is equivalent to requiring the invertibility of the Euler
continuant (𝐡𝑛, . . . , 𝐴1)placed in the (2,2)-entry. In view of equation (1.4), we get that the diagonal
part of the Gauss decomposition is diag(Λœπ‘ π‘›,(𝐡𝑛, . . . , 𝐴1)) , where
Λœπ‘ π‘›:=(𝐴𝑛, . . . , 𝐡1) βˆ’ ( 𝐴𝑛, . . . , 𝐴1)(𝐡𝑛, . . . , 𝐴1)βˆ’1(𝐡𝑛, . . . , 𝐡1).(1.8)
By taking the determinant in equation (1.7), we can find that1Λœπ‘ π‘›=(𝐴1, 𝐡1, . . . , 𝐴𝑛, 𝐡𝑛)βˆ’1. The upshot
is that we can reexpress B𝑛+1and its Lie group valued moment map πœ‡π‘‡:B𝑛+1→𝑇in terms of the space
b
B𝑛+1:={𝐴𝑖, π΅π‘–βˆˆCfor 1 ≀𝑖≀𝑛| (𝐴1, . . . , 𝐡𝑛)β‰ 0}(1.9)
and the map
bπœ‡:b
B𝑛+1βˆ’β†’ (CΓ—)2,{𝐴𝑖, 𝐡𝑖} β†¦βˆ’β†’ (𝐡𝑛, . . . , 𝐴1)βˆ’1,(𝐴1, . . . , 𝐡𝑛)ξ˜‘.(1.10)
From this point of view, we can realise b
B𝑛+1as an affine subspace2inside Rep(Γ𝑛,(1,1)), where we
assign 𝐴𝑖, 𝐡𝑖to the arrows π‘Žπ‘–, 𝑏𝑖, respectively. Moreover, the subspace πœ‡βˆ’1
𝑇(𝑑𝛾)needed to define the wild
character variety M𝑛(𝑑𝛾)can be identified with bπœ‡βˆ’1(𝛾, π›Ύβˆ’1), which can be further seen as a moduli space
of representations of a noncommutative algebra, the fission algebra Fπ‘žπ›Ύ(Γ𝑛); see equation (3.8) with
parameter π‘žπ›Ύ=(𝛾, π›Ύβˆ’1). Introduced in [14], fission algebras generalise the (deformed) multiplicative
preprojective algebras of [25], and they are linked to the generalised double affine 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
{𝐴𝑖, 𝐡𝑖}parametrising b
B𝑛+1as numbers, but, more generally, we can use matrices π΄π‘–βˆˆMat𝑑2×𝑑1(C),
π΅π‘–βˆˆMat𝑑1×𝑑2(C)for any 𝑑1, 𝑑2β‰₯1. Indeed, we get in that way other examples of reduced fission spaces
[13]; see [38]. Hence, we can work with a space of invertible representations inside Rep(Γ𝑛,(𝑑1, 𝑑2)),
rather than Rep(Γ𝑛,(1,1)) . Therefore, it seems natural to try to understand the structures unveiled by
1Here, one uses that the constants (𝐴𝑖, 𝐡𝑖)𝑛
𝑖=1and the Euler continuants are valued in Cand hence are commuting. The same
form for Λœπ‘ π‘›can be obtained for noncommuting variables 𝐴𝑖, 𝐡𝑖, though it requires a different proof.
2The reader should be warned that Boalch [15] writes b
B𝑛+1=Repβˆ—(Γ𝑛,(1,1)) as a space of invertible representations inside
Rep(Γ𝑛,(1,1) ). In that work, one considers Γ𝑛as a graph where each edge represents two arrows going in opposite directions,
that is, this is for us the double quiver Γ𝑛so that Rep(Γ𝑛,(1,1)) in [15] corresponds to Rep(Γ𝑛,(1,1) ) for us.
https://doi.org/10.1017/fms.2022.76 Published online by Cambridge University Press
6Maxime Fairon and David FernΓ‘ndez
Boalch directly at the level of the quiver Γ𝑛by using noncommutative quasi-Poisson geometry [41]. In
that case, we define from Γ𝑛the noncommutative (2𝑛)-th Euler continuants
(𝑏𝑛, . . . , π‘Ž1)βˆ’1,(π‘Ž1, . . . , 𝑏𝑛),
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
𝑛=1. Given two complex vector spaces 𝑉1, 𝑉2of finite dimensions 𝑑1, 𝑑2, respectively, we can introduce
the Van den Bergh space (see [4,13,17,43])
BVdB :={𝐴∈Hom(𝑉1, 𝑉2), 𝐡 ∈Hom(𝑉2, 𝑉1) | det(Id𝑉2+𝐴𝐡)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 𝐻=GL(𝑉1) Γ— GL(𝑉2). Furthermore, its Lie group valued moment map
is given by
πœ‡VdB ({ 𝐴, 𝐡}) :=(𝐡, 𝐴)βˆ’1,(𝐴, 𝐡)ξ˜‘=(Id𝑉1+𝐡 𝐴)βˆ’1,Id𝑉2+π΄π΅ξ˜‘βˆˆπ» .
Note that we easily recover (b
B2,bπœ‡)from (BVdB, πœ‡VdB)when 𝑉1=𝑉2=C. In the setting of multiplicative
quiver varieties, Van den Bergh crucially observed that the pair ξ˜€BVdB, πœ‡VdBand its corresponding
2-form can be understood at the level of the quiver Ξ“1. 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(Ξ“1), of the path algebra of the double quiver Ξ“1is
endowed with such a structure. In fact, Van den Bergh first introduced a β€˜quasi-Poisson version’ of this
result, when he derived in [41] that A(Ξ“1)is a Hamiltonian double quasi-Poisson algebra; this is the
noncommutative version of a Hamiltonian quasi-Poisson manifold in the sense of [2]; see Definition 2.5.
The key point is that this structure is such that the second Euler continuants (𝑏, π‘Ž)βˆ’1and (π‘Ž, 𝑏)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 Γ𝑛an explicit Hamiltonian double quasi-Poisson algebra structure,
whose moment map is given in terms of the (2𝑛)-th Euler continuants (π‘Ž1, . . . , 𝑏𝑛)and (𝑏𝑛, . . . , π‘Ž1)βˆ’1.
1.3. Main results
We fix a field kof characteristic zero. Recall that Γ𝑛denotes the quiver on 2 vertices and 2𝑛arrows
labelled {π‘Žπ‘–, 𝑏𝑖|1≀𝑖≀𝑛}as in Figure 1. We introduce the Boalch algebra B(Γ𝑛)which is obtained
from the path algebra kΓ𝑛by requiring that the (2𝑛)-th Euler continuants (π‘Ž1, . . . , 𝑏𝑛)and (𝑏𝑛, . . . , π‘Ž1)
are inverted (see Β§3.1 for the definition of Euler continuants in kΓ𝑛). The main result of this article is
Theorem 3.4: We prove that B(Γ𝑛)is endowed with a double quasi-Poisson bracket {{βˆ’,βˆ’}}, explicitly
defined in equation (3.10), which is such that Ξ¦:=(π‘Ž1, . . . , 𝑏𝑛) + (𝑏𝑛, . . . , π‘Ž1)βˆ’1is a multiplicative
moment map. In other words, the triple ξ˜€B(Γ𝑛),{{βˆ’,βˆ’}},Φis a Hamiltonian double quasi-Poisson
algebra. As a consequence of this result, we obtain in Corollary 3.5 that the corresponding fission
algebra Fπ‘ž(Γ𝑛)defined by Boalch [15] carries a noncommutative Poisson structure, called an 𝐻0-
Poisson structure, as introduced in [23]. Such an 𝐻0-Poisson structure induces a Poisson bracket on the
corresponding varieties
Mk
𝑛(π‘ž):=Rep ξ˜€Fπ‘ž(Γ𝑛),(𝑑1, 𝑑2)//ξ˜€GL(𝑑1) Γ— GL(𝑑2)
when kis algebraically closed. If k=C,(𝑑1, 𝑑2)=(1,1)and π‘ž=(𝛾, π›Ύβˆ’1), we recover the wild
character variety M𝑛(𝑑𝛾)from Β§1.2. In particular, the Poisson bracket hence obtained is nondegenerate
and corresponds to Boalch’s symplectic form [15]; see Β§5.6. We will investigate this nondegeneracy
property for a general field kin a future work.
https://doi.org/10.1017/fms.2022.76 Published online by Cambridge University Press
Forum of Mathematics, Sigma 7
It is important to observe that, if 𝑛=1, the Boalch algebra B(Ξ“1)coincides with Van den Bergh’s
algebra A(Ξ“1)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 P𝑛that 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 (𝑛=2), Mk
𝑛(π‘ž)is isomorphic
to the Flaschka–Newell surface [32]. This is an affine 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 M𝑛(𝑑𝛾)(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 ξ˜€B(Γ𝑛),(1,1)(i.e., before performing quasi-Hamiltonian reduction to end up with M𝑛(𝑑𝛾)).
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 2π‘›βˆ’2 Euler continuants, the structure of
Hamiltonian double quasi-Poisson algebra of B(Γ𝑛)unveiled in Theorem 3.4 can be obtained by fusing
the idempotents in ncopies of B(Ξ“1). In the simplest case 𝑛=2, this can be interpreted as a way to get
the factorisation
(π‘Ž1, 𝑏1, π‘Ž2, 𝑏2)=(π‘Žβ€²
1, 𝑏′
1)(π‘Žβ€²
2, 𝑏′
2)
of the fourth Euler continuant in terms of a product of second ones through the substitution
π‘Žβ€²
1=π‘Ž1, 𝑏′
1=𝑏1;π‘Žβ€²
2=π‘Ž2+ (π‘Ž1, 𝑏1)βˆ’1π‘Ž1, 𝑏′
2=𝑏2.(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
𝑛β‰₯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 Bergh’s 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 Ξ“1by 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(Γ𝑛)by localisation of the path algebra kΓ𝑛of the double of the quiver Γ𝑛. This structure is such
that its noncommutative moment map is given in terms of Euler continuants. We then write down the
corresponding noncommutative bivector P𝑛in Proposition 3.6, and we link our result to the Flaschka–
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(Γ𝑛)in terms of ncopies of
B(Ξ“1)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 fix a finitely generated associative unital algebra Aover a field kof characteristic zero, and we use
the unadorned notations βŠ—=βŠ—k, Hom =Homkfor brevity. The opposite algebra and the enveloping
algebra of Awill be denoted 𝐴op and 𝐴e:=π΄βŠ—π΄op, respectively. We shall identify the category of
A-bimodules and the category of (left) 𝐴e-modules. Note that the underlying A-bimodule of Acarries
two A-bimodule structures with the same underlying vector space π΄βŠ—π΄, namely the outer and the inner
A-bimodule structures, respectively denoted by (π΄βŠ—π΄)out and (π΄βŠ—π΄)inn; they are given by
π‘Ž1(π‘Žβ€²βŠ—π‘Žβ€²β€²)π‘Ž2:=(π‘Ž1π‘Žβ€²) βŠ— (π‘Žβ€²β€²π‘Ž2)on (π΄βŠ—π΄)out ;
π‘Ž1βˆ— (π‘Žβ€²βŠ—π‘Žβ€²β€² ) βˆ— π‘Ž2:=(π‘Žβ€²π‘Ž2) βŠ— (π‘Ž1π‘Žβ€²β€²)on (π΄βŠ—π΄)inn ,
for all π‘Žβ€², π‘Žβ€²β€², π‘Ž1, π‘Ž2∈𝐴.
Given another unital associative k-algebra B,Ais called a B-algebra if there exists a unit preserving
k-algebra morphism 𝐡→𝐴