The Feigin Tetrahedron

0.55 MB·

by  Dylan Rupel

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 024, 30 pages
The Feigin Tetrahedron(cid:63)
Dylan RUPEL
Department of Mathematics, Northeastern University, Boston, MA 02115, USA
E-mail: [email protected]
URL: http://www.northeastern.edu/drupel/
Received September 11, 2014, in final form March 03, 2015; Published online March 19, 2015
http://dx.doi.org/10.3842/SIGMA.2015.024
5
Abstract. The first goal of this note is to extend the well-known Feigin homomorphisms
1
takingquantumgroupstoquantumpolynomialalgebras. Moreprecisely,wedefinegenerali-
0
2 zedFeiginhomomorphismsfromaquantumshufflealgebratoquantumpolynomialalgebras
whichextendtheclassicalFeiginhomomorphismsalongtheembeddingofthequantumgroup
r
a intosaidquantumshufflealgebra. InarecentworkofBerensteinandtheauthor,analogous
M extensions of Feigin homomorphisms from the dual Hall–Ringel algebra of a valued quiver
to quantum polynomial algebras were defined. To relate these constructions, we establish

9
ahomomorphism,dubbedthequantum shuffle character,fromthedualHall–Ringelalgebra
1
tothequantumshufflealgebrawhichrelatesthegeneralizedFeiginhomomorphisms. These

] constructions can be compactly described by a commuting tetrahedron of maps beginning
T
with the quantum group and terminating in a quantum polynomial algebra. The second
R goal in this project is to better understand the dual canonical basis conjecture for skew-
. symmetrizable quantum cluster algebras. In the symmetrizable types it is known that dual
h
t canonicalbasiselementsneednothavepositivemultiplicativestructureconstants,whilethis
a
is still suspected to hold for skew-symmetrizable quantum cluster algebras. We propose an
m
alternate conjecture for the symmetrizable types: the cluster monomials should correspond
[
to irreducible characters of a KLR algebra. Indeed, the main conjecture of this note would

2 establish this “KLR conjecture” for acyclic skew-symmetrizable quantum cluster algebras:
v that is, we conjecture that the images of rigid representations under the quantum shuffle
8
character give irreducible characters for KLR algebras. We sketch a proof in the symmetric
3
case giving an alternative to the proof of Kimura–Qin that all non-initial cluster variables
3
4 in an acyclic skew-symmetric quantum cluster algebra are contained in the dual canonical
. basis. With these results in mind we interpret the cluster mutations directly in terms of the
1
representation theory of the KLR algebra.
0
4
Key words: cluster algebra; Hall algebra; quantum group; quiver Hecke algebra; KLR alge-
1
: bra; dual canonical basis; Feigin homomorphism; categorification
v
i 2010 Mathematics Subject Classification: 13F60; 16G20; 17B37; 20G42
X
r
a
1 Introduction
In the struggle to understand the quantum groups and their duals, the quantized enveloping al-
gebras, onesearchesforconcreterealizationsofthesealgebras, eitherbygeneratorsandrelations
or via embeddings, or more generally simply homomorphisms, into “nicer” algebras. This note
grew out of an attempt to understand three well-known and well-studied examples, namely the
embedding into a (dual) Hall–Ringel algebra, the embedding into a quantum shuffle algebra,
and the Feigin homomorphisms to quantum polynomial algebras.
The embeddings of the (positive part of) quantized Kac–Moody algebras into Hall–Ringel
algebras originated with the incredible insights of Ringel, for the finite-types in [29] and fully
(cid:63)This paper is a contribution to the Special Issue on New Directions in Lie Theory. The full collection is
available at http://www.emis.de/journals/SIGMA/LieTheory2014.html
2 D. Rupel
realized in [30]. This is merely the tip of a fantastic geometric iceberg discovered by Lusztig [25]
that has lead to a deep understanding of certain “canonical” bases in the quantized Kac–Moody
algebras.
A careful study of multiplicative properties of these bases helped motivate Fomin and Zele-
vinsky in the definition of cluster algebras. In the hopes of gaining a combinatorial grasp of the
structure of the canonical basis they provided the definition of a recursively, combinatorially
defined algebra with a deep conjecture in mind: certain “cluster monomials” appearing in
this construction should be identifiable with (dual) canonical basis vectors. This motivating
conjecture has been established so far in a very restricted set of cases by Lampe [20, 21] and
Kimura–Qin [16]. One aim of this note is to shed some additional light on this “dual canonical
basis conjecture”.
ThequantizedKac–Moodyalgebrasaremostcompactlydescribedbygeneratorsandrelations
or better as a certain quotient of a free algebra with an unconventional (dare I say twisted)
multiplication defined on its tensor square. Dualizing this construction leads to an embedding
of the quantum group into the quantum shuffle algebra and thus, via an isomorphism, an
embedding of the quantized Kac–Moody algebra. This is the tip of yet another (not entirely
unrelated) iceberg, this one categorical [14, 15, 32, 33], which provides another description of
these positively amazing bases of the quantum group and quantized Kac–Moody algebra.
Our third major tool for studying quantized Kac–Moody algebras are certain algebra homo-
morphisms to quantum polynomial rings suggested by B. Feigin during a talk at RIMS in 1992
(thus we refer to these as Feigin homomorphisms). There are actually infinitely many Feigin
homomorphisms acting on a given quantized Kac–Moody algebra, one corresponding to each
finite sequence of simple roots. For finite-types, we get an embedding whenever the sequence
corresponds to a reduced word for the longest element of the corresponding Weyl group. In
general a Feigin homomorphism has a very large kernel, a feature which we hope will shed some
light on the relationship of KLR characters to quantum cluster algebras. The Feigin homo-
morphisms, as Feigin predicted, turn out to be the essential tool [2, 10, 12] for studying the
skew-field of fractions of the quantized Kac–Moody algebras.
The relationship between the first and third approach was studied by Berenstein and the
author in [3] where Feigin homomorphisms were extended to the dual Hall–Ringel algebra for
representations of an acyclic valued quiver. Applications to quantum cluster algebras were
a central result in [3] where the images of rigid objects are identified with quantum cluster
characters describing non-initial cluster variables of a corresponding acyclic quantum cluster
algebra [28, 34, 35]. This also provided the essential tool for understanding a certain “twist
automorphism” of the quantum cluster algebra, it would be interesting to use the construc-
tions/conjectures of this note to interpret the twist in terms of the representation theory of
KLR algebras.
The main result of the current work is to complete the above tetrahedron. We begin in
Section 6 by extending Feigin homomorphisms to the quantum shuffle algebra. Section 7 intro-
duces a homomorphism, the quantum shuffle character, from the dual Hall–Ringel algebra to
the quantum shuffle algebra making all of the relevant triangles commute.
Asecondarygoalofthisprojectistounderstandthedualcanonicalbasisconjectureforquan-
tum cluster algebras. The generalized Feign homomorphism of [3] produces non-initial cluster
monomials (of a certain acyclic quantum cluster algebra) as the images of rigid objects. Tracing
the other path through the tetrahedron gives elements of the quantum shuffle algebra which,
according to the dual canonical basis conjecture, should be expected to identify (in symmetric
types) with dual canonical basis elements, i.e. with irreducible characters of Khovanov–Lauda–
Rouquier (quiver Hecke) algebras [14, 15, 32, 33] living in the quantum shuffle algebra. In
Section 8 we recall the necessary background material to precisely formulate this conjecture as
well as sketch a proof for symmetric types giving an alternative to the proof by Kimura and
The Feigin Tetrahedron 3
Qin using graded quiver varieties. Assuming the conjecture, we formulate a mutation opera-
tion for certain clusters defined using the representation theory of KLR algebras. We finish by
presenting several conjectures about how this should be done in general.
The results of this work are undoubtedly closely related to several other works in this area
though the precise connections are still a little bit of a mystery, perhaps the other construc-
tions provide the categorified/geometric perspective of our constructions. We mention here
only a few main examples that should be investigated further. The geometric construction of
KLR algebras by Varagnolo–Vasserot [37] and Webster [38] motivate our approach to the dual
canonical basis conjecture in acyclic skew-symmetric types, more precisely they have given a ge-
ometric construction of certain faithful modules over KLR algebra using flag varieties of quiver
representations. The other works we mention are mainly focused on understanding the precise
relationship of the representation theory of KLR algebras to an alternative categorification of
quantum groups using quantum affine algebras [13]. A central ingredient in establishing a di-
rect link was the paper [9] where the authors find an isomorphism between between a deformed
Grothendieck ring associated to the quantum affine algebra and the derived Hall algebra of
a quiver [36]. Tracing through these isomorphisms one should find a direct relationship between
the derived Hall algebra and the KLR algebra, perhaps a Feigin-type homomorphism is lurking
in the background.
The structure of the paper is as follows: Section 2 sets up certain combinatorial and nota-
tional conventions that we have found most useful in these investigations. Section 3 recalls the
quantizedKac–MoodyalgebraandbackgroundonclassicalFeiginhomomorphisms. InSection4
we introduce quantum cluster algebras. Section 5 defines valued quivers and their Hall–Ringel
algebras and recalls the results of [28, 34, 35] and [3] relating these to quantum cluster alge-
bras. Sections 6–8 have been discussed above. Section 9 discusses ideas for generalizations and
proposes several conjectures on irreducible KLR modules.
2 Combinatorial conventions and notations
Fix an indeterminate q. We define the q-numbers, q-factorials, and q-binomials by
(n) = (n) = 1+q+···+qn−1,
q
(cid:18) (cid:19)
n (n)!
(n)! = (n)·(n−1)···(2)·(1), = .
k (k)!·(n−k)!
q
In defining our generalized Feigin homomorphism Ψ we will make particular use of the bar-
i √
invariant versions of the quantum numbers, for convenience we introduce v = q:
[n] = [n] = v−n+1+v−n+3+···+vn−3+vn−1,
q
(cid:20) (cid:21)
n [n]!
[n]! = [n]·[n−1]···[2]·[1], = .
k [k]!·[n−k]!
q
The q-binomials satisfy several useful analogues of classical binomial identities. We collect these
and sketch their proofs in the following.
Lemma 2.1.
1. The q-binomial coefficients satisfy the following identities:
(a) Pascal identities: (cid:0)n(cid:1) = (cid:0)n−1(cid:1) +qk(cid:0)n−1(cid:1) = qn−k(cid:0)n−1(cid:1) +(cid:0)n−1(cid:1) ;
k q k−1 q k q k−1 q k q
n
(b) row identity: for n > 0, (cid:80)(−1)kq−nk+12k(k+1)(cid:0)n(cid:1) = 0;
k q
k=0
4 D. Rupel
(c) subspace identity: (cid:0)m+n(cid:1) = (cid:80) qr(n−s)(cid:0)m(cid:1) (cid:0)n(cid:1) .
k q r q s q
r+s=k
2. The bar-invariant q-binomial coefficients satisfy the following identities:
(a) Pascal identities: (cid:2)n(cid:3) = vn−k(cid:2)n−1(cid:3) +v−k(cid:2)n−1(cid:3) = vk−n(cid:2)n−1(cid:3) +vk(cid:2)n−1(cid:3) ;
k q k−1 q k q k−1 q k q
n
(b) row identity: for n > 0, (cid:80)(−1)kvk−nk(cid:2)n(cid:3) = 0;
k q
k=0
(c) subspace identity: (cid:2)m+n(cid:3) = (cid:80) vr(n−s)−s(m−r)(cid:2)m(cid:3) (cid:2)n(cid:3) .
k q r q s q
r+s=k
Proof. We prove the identities in (1). The Pascal identities in (a) follow from the definition
and the equalities (n) = (k) +qk(n−k) = qn−k(k) +(n−k) . For (b) we simply apply the
q q q q q
second Pascal identity to get a telescopic summation:
n (cid:18) (cid:19)
(cid:88)(−1)kq−nk+12k(k+1) n
k
k=0 q
n (cid:18) (cid:19) n−1 (cid:18) (cid:19)
= (cid:88)(−1)kq−n(k−1)+12(k−1)k n−1 +(cid:88)(−1)kq−nk+12k(k+1) n−1 = 0.
k−1 k
k=1 q k=0 q
Tosee(c)weconsiderafinitefieldFwithq elements. WriteGr (Fn)forthesetofk-dimensional
k
subspacesofFn. ThenumberofpointsinGr (Fn)isgivenby(cid:0)n(cid:1) . FixadecompositionFm+n =
k k q
Fm⊕Fn so that any subspace of Fm+n can be decomposed as a direct sum of a subspace of Fm
andasubspaceofFn. ThisgivesrisetoasurjectivemapGr (Fm+n) →→ (cid:70) Gr (Fm)×Gr (Fn)
k r s
r+s=k
with fiber over any point of Gr (Fm)×Gr (Fn) an affine space of dimension r(n−s). Now
r s
counting points completes the proof when q is a power of a prime. But this is an equality of
polynomials for infinitely many values and thus the polynomials must be equal for every q.
Theidentitiesin(2)easilyfollowfromthosein(1)usingtherelation(cid:2)n(cid:3) = v−k(n−k)(cid:0)n(cid:1) . (cid:4)
k q k q
We choose a notation for symmetric groups which is most convenient for describing the
structure constants of the multiplication in the quantum shuffle algebra. For each positive
integer t let Σ denote the symmetric group on t letters, thought of as the set of all orderings
t
σ = (σ ,...,σ ) of the set (1,...,t). We write σ·τ = (σ ,...,σ ) for the multiplication in Σ .
1 t τ1 τt t
Let σ−1 denote the position of k in σ, i.e the number i so that σ = k.
k i
For positive integers r and s define the shuffle subgroup Σ ⊂ Σ consisting of all shuffles
r,s r+s
of the sequences (1,...,r) and (r+1,...,r+s), that is
Σ = (cid:8)σ ∈ Σ : σ−1 < ··· < σ−1,σ−1 < ··· < σ−1 (cid:9).
r,s r+s 1 r r+1 r+s
By convention we take Σ = Σ = {1} to be the trivial group.
0,0 0
3 Quantum groups and Feigin homomorphisms
Inthissectionweintroducethemainobjectsunderlyingthepresentinvestigations: thequantum
groups associated to a symmetrizable Cartan matrix and the Feigin homomorphisms mapping
them to quantum polynomial algebras.
Fix an index set I. Let A = (a ) be an I × I Cartan matrix with symmetrizing matrix
ij
D = diag(d), where d = (d : i ∈ I) is an I-tuple of positive integers. More explicitly, the
i
entries of A have the following properties:
• a = 2 for all i;
ii
The Feigin Tetrahedron 5
• a ≤ 0 for i (cid:54)= j;
ij
• d a = d a for all i,j ∈ I.
i ij j ji
Let Q denote the root lattice of a Kac–Moody root system Φ associated to A. Denoting by
{α } the simple roots of Φ, Q is the free abelian group with basis {α }. The pair (A,d)
i i∈I i
determines a symmetric bilinear form (·,·) : Q×Q → Z given on generators of Q by (α ,α ) =
i j
d a for i,j ∈ I. Let W denote the Weyl group associated to Φ with generators the simple
i ij
reflections s (i ∈ I). Write α∨ = α /d . The action of a simple generator s on a root α is
i i i i i
given by
s (α) = α−(α∨,α)α .
i i i
Notice that (·,·) is equivariant with respect to the action of W, i.e. (α,β) = (s α,s β) for all
i i
α,β ∈ Q and i ∈ I.
√
Fix an indeterminate q and write v = q. Define A = Z[v±1]. It will be convenient for i ∈ I
v
to abbreviate (n) = (n) and define (n)!, (cid:0)n(cid:1) similarly. Similar abbreviations will be used for
i qdi i k i
the bar-invariant versions. Associated to the Cartan matrix A with choice of symmetrizers d we
have the (integral form of the) quantized coordinate ring A [N] of the standard upper unipotent
v
subgroup N of the Kac–Moody group G. The Q-graded algebra A [N] is generated by formal
v
variables x[r] (i ∈ I, r ∈ Z ) with |x[r]| = −rα subject to the quantum Serre relations:
i ≥0 i i
1−aij
(cid:88) (−1)rx[r]x x[1−aij−r] = 0 for any i (cid:54)= j ∈ I,
i j i
r=0
where by definition [r]!x[r] = xr. The algebra A [N] is actually a twisted bialgebra where each
i i i v
generator is primitive, i.e. the comultiplication is given by ∆(x ) = x ⊗1+1⊗x for each i ∈ I,
i i i
and where the multiplication on A [N]⊗A [N] is twisted by the grading, i.e. (x⊗y)(x(cid:48)⊗y(cid:48)) =
v v
v(|x(cid:48)|,|y|)·xx(cid:48)⊗yy(cid:48).
[r]
The graded dual of A [N] is the quantized enveloping algebra U generated by E (i ∈ I,
v v i
r ∈ Z ) also subject to the quantum Serre relations:
≥0
1−aij
(cid:88) (−1)rE[r]E E[1−aij−r] = 0 for any i (cid:54)= j ∈ I,
i j i
r=0
where by definition [r]!E[r] = Er. The algebra U is also Q-graded by the rule |E[r]| = rα for
i i i v i i
i ∈ I.
A powerful tool for studying certain distinguished elements of A [N] are the twisted deriva-
v
tions θ ,θ∗ : A [N] → A [N] which satisfy θ (x ) = θ∗(x ) = δ and
i i v v i j i j ij
θ (xy) = v(αi,|y|)θ (x)y+xθ (y), θ∗(xy) = θ∗(x)y+v(αi,|x|)xθ∗(y).
i i i i i i
It is well known that the maps ϕ,ϕ∗ : U → End(A [N]) given by E (cid:55)→ θ or E (cid:55)→ θ∗ define
v v i i i i
respectively left and right actions of U on A [N].
v v
Consider a word i = (i ,...,i ) ∈ Im and define the m-dimensional quantum polynomial
1 m
ring
Pi = Z[v±12](cid:104)t1,...,tm : t(cid:96)tk = v(αik,αi(cid:96))tkt(cid:96) for k < (cid:96)(cid:105).
For a = (a ,...,a ) ∈ Zm define the bar-invariant basis element ta ∈ P by
1 m i
ta = v21k(cid:80)<(cid:96)aka(cid:96)(αik,αi(cid:96))ta1···tam.
1 m
6 D. Rupel
Theorem 3.1 ([2]). For i ∈ Im the map Ψ : A [N] → P , given on generators by Ψ (x ) =
i v i i j
(cid:80)
t for j ∈ I, defines an algebra homomorphism.
k
k:i =j
k
We will call Ψ the Feigin homomorphism of type i. These homomorphisms were proposed
i
by Boris Feigin (hence the name) as a tool for studying the skew-field of fractions of A [N]. The
v
Feigin homomorphisms were extensively studied in [2, 10, 12] where many important properties
were discovered. In particular we emphasize the following result.
Theorem3.2([2]). Supposei∈Im isareducedwordforaWeylgroupelementw = s ···s ∈W.
i1 im
Then the kernel K := kerΨ does not depend on the choice of reduced word i for w. Moreover,
w i
K is equal to the structural ideal of the quantized coordinate ring of the closure of the unipotent
w
cell Nw := N ∩B wB , where B is the standard negative Borel subgroup of G, i.e. under the
− − −
specialization v (cid:55)→ 1 the algebra A [N]/K becomes the (integral form of) the coordinate ring of
v w
Nw.
The ring A [Nw] has a natural choice of “coefficients” which become monomials under Ψ
v i
(see [3, Section 6] for more details), in particular they provide an Ore set by which we may
localize to obtain a quantization of the coordinate ring of Nw. In particular, this shows that the
Feigin homomorphism Ψ allows one to replace the complicated algebra A [Nw] by a relatively
i v
simplesubalgebraofquantumLaurentpolynomials. ItturnsoutthatΨ alsoprovidesapowerful
i
tool for revealing quantum cluster algebra structures (see the next section for details).
4 Quantum cluster algebras
A quantum cluster algebra is a certain type of non-commutative algebra recursively defined
from some initial combinatorial data. In this section we recall the combinatorial construction of
quantum cluster algebras from an initial seed.
LetQbeanacyclicquiverwithvertexsetI andrecallthesymmetrizingmatrixD = diag(d),
where d = (d : i ∈ I). Write n = n for the number of arrows connecting vertices i,j ∈ I
i ij ji
in Q and note that all such arrows point in the same direction. We assume further that there
are only finitely many arrows whose source or target is the vertex i ∈ I. From the pair (Q,d),
called a valued quiver, we define an adjacency matrix B = B = (b ) by the rule
Q ij

n d /gcd(d ,d ) if i → j in Q,
 ij j i j

b = −n d /gcd(d ,d ) if j → i in Q,
ij ij j i j

0 if i = j.
Notice that the matrix B is skew-symmetrizable with skew-symmetrizing matrix D, i.e. d b =
i ij
−d b for all i,j ∈ I. To connect with the previous section we will assume that the matrix B
j ji
is related to the Cartan matrix A by a = −|b | for i (cid:54)= j.
ij ij
Consider an index set J ⊃ I and a J ×I matrix B(cid:101) = (bij) with principal I×I submatrix B.
We say that a skew-symmetric J ×J matrix Λ = (λij) is compatible with B(cid:101) if
(cid:88)
λ b = δ d for all j ∈ J and k ∈ I.
ij ik jk k
i∈J
Associated to Λ we have the quantum torus T generated over Z[v±1] by quasi-commuting
Λ,q
cluster variables X = (X : i ∈ J) subject to the relations
i
X X = qλijX X for i,j ∈ J.
i j j i
We will call the pair Σ0 = (X,B(cid:101)) a quantum seed whenever Λ is compatible with B(cid:101), the
collection X is called the cluster and B(cid:101) is called the exchange matrix.
The Feigin Tetrahedron 7
Remark 4.1. It is often customary to include the commutation matrix Λ in the data of a quan-
tum seed, however the quantum cluster X “knows” its quasi-commutation so this is slightly
redundant and we omit it.
Inordertodefinethemutationofquantumseedsweneedtointroducemorenotation. Choose
a total order < on J. Then for a ∈ ZJ we may define the bar-invariant monomial Xa ∈ T by
Λ,q
Xa = vi(cid:80)<jaiajλji(cid:89)(cid:126) Xai,
i
i∈J
(cid:81)(cid:126)
where denotes the product in increasing order. Then for k ∈ I we are now ready to define
the mutation in direction k, µkΣ = (µkX,µkB(cid:101)), as follows:
• µ X = X\{X }∪{X(cid:48)} for X(cid:48) given by
k k k k
X(cid:48) = Xbk+−αk +Xbk−−αk,
k
where bk,bk ∈ ZJ are the unique vectors satisfying bk −bk = bk is the kth column
+ − ≥0 + −
of B(cid:101);
• µkB(cid:101) = (b(cid:48)ij), where
(cid:40)
−b if i = k or j = k,
b(cid:48) = ij
ij
b +[b ] b +b [−b ] otherwise.
ij ik + kj ik kj +
Note that the compatibility of B˜ and Λ ensures that µ X is again a quantum cluster. Precise
k
formulas for the commutation matrix of µ X are known [4], however we will not need them.
k
Now consider a rooted, labelled I-regular tree T with root vertex t , where the edges em-
0
anating from each vertex are labelled by distinct elements of I. We assign quantum seeds
Σt = (Xt,B(cid:101)t) to the vertices t ∈ T so that Σt0 = Σ0 and for t kt(cid:48) in T the seeds Σt and Σt(cid:48)
are related by the mutation in direction k. For any two (possibly distant) vertices t,t(cid:48) ∈ T we
say that the seeds Σ and Σ are mutation equivalent. A fundamental result in the theory of
t t(cid:48)
quantum cluster algebras is the following.
Theorem 4.2 ([4, quantum Laurent phenomenon]). For any vertex t ∈ T all cluster variables
of the cluster X are contained in T .
t Λ,q
The quantum cluster algebra is then the Z[v±1][X±1 : j ∈ J\I]-subalgebra of T generated
j Λ,q
by all cluster variables from all seeds mutation equivalent to Σ :
0
Aq(X,B(cid:101)) = Z[v±1][Xj±1 : j ∈ J \I][Xi;t : t ∈ T,i ∈ I] ⊂ TΛ,q.
One of the main problems in the theory of quantum cluster algebras is to explicitly describe the
Laurent expansions of all cluster variables. In general this is still an open and active area of
research. Under the current assumption of acyclicity some progress has been made on this prob-
lem using the representation theory of valued quivers; we present these results at the beginning
of the next section.
5 Representations of valued quivers
and their Hall–Ringel algebras
Fix a finite field F and an algebraic closure F¯. For each positive integer d write F for the
d
degree d extension of F in F¯. Note that with these conventions we may intersect F and F to
d d(cid:48)
8 D. Rupel
get that F is their largest common subfield. We now define representations of (Q,d) by
gcd(d,d(cid:48))
assigning an F -vector space to each vertex i ∈ I and an F -linear map to each arrow
di gcd(di,dj)
from vertex i to vertex j. The finite-dimensional representations of (Q,d) form a finitary,
hereditary, Abelian category rep (Q,d) where kernels and cokernels are taken vertex-wise, we
F
refer the reader to [34] for more details.
Recall that we work under the assumption that the quiver Q contains no oriented cycles.
Then the simple representations of (Q,d) may be labeled as S (i ∈ I) where the only non-zero
i
vector space is F at vertex i. The Grothendieck group K(Q,d) of rep (Q,d) has a Z-basis
di F
given by the classes α = [S ] (i ∈ I) of the irreducible representations S , in this way we
i i i
identify the root lattice Q with K(Q,d). Write [V] for the isomorphism class of a representation
V ∈ rep (Q,d) and write |V| for its dimension vector, i.e. the class of V in K(Q,d). Define the
F
height ht(V) of a representation V as the length of any composition series for V.
We define the Euler–Ringel form (cid:104)·,·(cid:105) : K(Q,d)×K(Q,d) → Z on generators by
(cid:40)
d if i = j,
i
(cid:104)α ,α (cid:105) =
i j
−[d b ] if i (cid:54)= j.
i ij +
The identification of Q and K(Q,d) allows to transfer the bilinear form (·,·) to K(Q,d). It
then follows from the definitions that (·,·) is the symmetrization of the Euler–Ringel form,
i.e.(α ,α ) = (cid:104)α ,α (cid:105)+(cid:104)α ,α (cid:105)foralli,j ∈ I. Recallthatwewriteα∨ = α /d . Foradimension
i j i j j i i i i
vector v ∈ K(Q,d) define ∗v ∈ K(Q,d) by
(cid:88)
∗v = (cid:104)α∨,v(cid:105)α .
i i
i∈I
In [34] we proposed the following definition and established the additive categorification of
Theorem 5.2 below in a restricted set of cases.
Definition 5.1. For any representation V ∈ rep (Q,d) we associate the quantum cluster char-
F
acter X defined by
V
XV = (cid:88) |F|−21(cid:104)e,|V|−e(cid:105)·|Gre(V)|·X−B(cid:101)e−∗|V| ∈ TΛ,|F|,
e∈K(Q,d)
where Gr (V) denotes the set of subrepresentations of V with dimension vector e.
e
A representation V is rigid if Ext1(V,V) = 0. Our main result from [35] is the following. We
also refer the reader to [28] for an analogous result in the equally valued case, i.e. when d = 1
i
for all i ∈ I.
Theorem 5.2 ([28, 35]). Every non-initial cluster variable of A|F|(X,B(cid:101)) is of the form XV for
some indecomposable rigid representation V of (Q,d).
The multiplication formulas for quantum cluster characters involved in the proof of Theo-
rem 5.2 suggest a relationship with dual Hall–Ringel algebras. In the work [3] we studied with
Arkady Berenstein the nature of this relationship, a review of these results are presented in the
next section.
5.1 Generalized Feigin homomorphisms
Define the Hall–Ringel bialgebra H(Q,d) = (cid:76)C[V] to be the free K(Q,d)-graded C-vector
space spanned by the isomorphism classes of representations of (Q,d) with grading |[V]| = |V|
given by dimension vector. The structure constants of the multiplication and comultiplication
The Feigin Tetrahedron 9
count certain extensions between representations. More precisely, for representations U, V,
and W, we define the Hall number as the cardinality of the following set:
FV = {R ⊂ V : R ∼= W and V/R ∼= U}.
UW
For convenience we abbreviate v = (cid:112)|F|.
Theorem 5.3 ([29]). The map µ : H(Q,d) ⊗ H(Q,d) → H(Q,d) (written multiplicatively)
given on basis elements by
(cid:88)
[U][W] = v(cid:104)|U|,|W|(cid:105)|FV |·[V]
UW
[V]
defines an associative multiplication on H(Q,d).
The tensor square H(Q,d)⊗H(Q,d) will be considered as an algebra via the twisted multi-
plication defined by
([U]⊗[W])([U(cid:48)]⊗[W(cid:48)]) = v(|W|,|U(cid:48)|)[U][U(cid:48)]⊗[W][W(cid:48)].
Theorem 5.4 ([7]). The map ∆ : H(Q,d) → H(Q,d)⊗H(Q,d) given on basis elements by
∆([V]) = (cid:88) v(cid:104)|U|,|W|(cid:105)|Aut(U)||Aut(W)| ·(cid:12)(cid:12)FV (cid:12)(cid:12)·[U]⊗[W]
|Aut(V)| UW
[U],[W]
defines a coassociative comultiplication on H(Q,d) which is compatible with the multiplication
on H(Q,d) and the twisted multiplication on H(Q,d)⊗H(Q,d).
The graded dual Hall–Ringel bialgebra H∗(Q,d) has a basis of delta functions δ with
[V]
grading |δ | = −|V|. It will be convenient to consider a slight rescaling [V]∗ of the standard
[V]
dual basis given by
n
[V]∗ = v−12(cid:104)V,V(cid:105)+12i(cid:80)=1diviδ[V].
For a representation U ∈ rep (Q,d) define the linear maps θ ,θ∗ : H∗(Q,d) → H∗(Q,d) by
F U U
θ (δ )([W]) = δ ([W][U]) and θ∗(δ )([W]) = δ ([U][W]) (5.1)
U [V] [V] U [V] [V]
for each V,W ∈ rep (Q,d). More precisely, we may expand θ (δ ) and θ∗(δ ) in the basis
F U [V] U [V]
of delta functions as follows:
θU(δ[V]) = (cid:88)v(cid:104)|W|,|U|(cid:105)(cid:12)(cid:12)FWV U(cid:12)(cid:12)·δ[W], θU∗(δ[V]) = (cid:88)v(cid:104)|U|,|W|(cid:105)(cid:12)(cid:12)FUVW(cid:12)(cid:12)·δ[W].
[W] [W]
The following result asserts that for any i ∈ I the linear maps θ and θ∗ are twisted
Si Si
derivations on H∗(Q,d).
Lemma 5.5. For any simple representation S of (Q,d) and U,W ∈ rep (Q,d) we have:
i F
θ (δ δ ) = v(αi,|W|)θ (δ )δ +δ θ (δ ),
Si [U] [W] Si [U] [W] [U] Si [W]
θ∗ (δ δ ) = θ∗ (δ )δ +v(αi,|U|)δ θ∗ (δ ).
Si [U] [W] Si [U] [W] [U] Si [W]
10 D. Rupel
Proof. These easily follow from the description (5.1) of θ and θ∗ . Indeed, for any V ∈
Si Si
rep (Q,d) we have
F
(cid:0) (cid:1)
θ (δ δ )([V]) = (δ δ )([V][S ]) = (δ ⊗δ ) ∆([V][S ])
Si [U] [W] [U] [W] i [U] [W] i
(cid:0) (cid:1)
= (δ ⊗δ ) ∆([V])∆([S ])
[U] [W] i
(cid:0) (cid:1) (cid:0) (cid:1)
= (δ ⊗δ ) ∆([V])([S ]⊗1) +(δ ⊗δ ) ∆([V])(1⊗[S ]) .
[U] [W] i [U] [W] i
But notice that the twisted multiplication on H(Q,d)⊗H(Q,d) gives
(δ ⊗δ )(cid:0)∆([V])([S ]⊗1)(cid:1) = v(αi,|W|)(cid:0)θ (δ )⊗δ (cid:1)(cid:0)∆([V])(cid:1)
[U] [W] i S1 [U] [W]
and that
(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)
(δ ⊗δ ) ∆([V])(1⊗[S ]) = δ ⊗θ (δ ) ∆([V]) .
[U] [W] i [U] S1 [W]
Combining these observations completes the proof of the first identity:
θ (δ δ )([V]) = v(αi,|W|)(cid:0)θ (δ )⊗δ (cid:1)(cid:0)∆([V])(cid:1)+(cid:0)δ ⊗θ (δ )(cid:1)(cid:0)∆([V])(cid:1)
Si [U] [W] S1 [U] [W] [U] S1 [W]
= (cid:0)v(αi,|W|)θ (δ )δ +δ θ (δ )(cid:1)([V]).
Si [U] [W] [U] Si [W]
The second identity follows by a similar calculation. (cid:4)
Remark 5.6. In general, one cannot hope for θ and θ∗ to be derivations. Under certain
U U
restrictions one can show that they are higher order differential operators on H∗(Q,d), though
we will not use this here and thus leave the details for another day.
For each representation V ∈ rep (Q,d) we define the quantum i-character
F
XV,i = (cid:88) v−k(cid:80)<(cid:96)aka(cid:96)(cid:104)αi(cid:96),αik(cid:105)·|Fi,a(V)|·ta ∈ Pi,
a∈Zm
≥0
where F (V) denotes the set of all flags of V of type (i,a), i.e.
i,a
F (V) = {0 = V ⊂ V ⊂ ··· ⊂ V ⊂ V = V : V /V ∼= Sak, 1 ≤ k ≤ m}.
i,a m m−1 1 0 k−1 k i
k
Theorem 5.7 ([3]). For any sequence i the assignment [V]∗ (cid:55)→ X defines an algebra homo-
V,i
morphism Ψ(cid:101)i : H∗(Q,d) → Pi.
It is a classical result of Ringel [29] that the composition subalgebra of H∗(Q,d) generated
over A = Z[v±1] by the classes [S ]∗ of simple representations is isomorphic to A [N]. Since
v j v
Ψ˜([Sj]∗) = (cid:80) tk we see that the restriction of Ψ(cid:101)i to Av[N] recovers the Feigin homomor-
k:i =j
k
phism Ψi introduced in Section 3. Thus we call the maps Ψ(cid:101)i generalized Feigin homomorphisms.
A certain choice of generalized Feigin homomorphism realizes the connection between acyclic
quantum cluster characters and dual Hall–Ringel algebras alluded to above.
Theorem 5.8 ([3]). Let i be twice a complete source adapted sequence for Q. Then there
exists a compatible pair (B(cid:101)i,Λi) so that Pi ∼= TΛi,q and Ψ(cid:101)i([V]∗) = XV for any representation
V ∈ rep (Q,d).
F
Combining Theorems 5.2 and 5.8 we see that non-initial cluster variables can be recovered
as images of rigid representations under the generalized Feigin homomorphism.
Theorem 5.9 ([3]). Let c be any Coxeter element of W and i a reduced word for c. The
0
quantized coordinate ring A [Nc2] admits the structure of an acyclic quantum cluster algebra
v
associated to the compatible pair (B(cid:101)i,Λi) for i = (i0,i0).
Together these results motivate our approach to the dual canonical basis conjecture, we will
make this precise in future sections.