Clustered Hyperbolic Categories

0.27 MB·

by  Ibrahim Saleh

Clustered Hyperbolic Categories
Ibrahim Saleh
7 Mathematics Department
1 University of Wisconsin-Marathon
0
2 Wausau, WI 54401

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Abstract
a
J
We introduce a class of categories, called clustered hyperbolic cate-

2
gories, which are generated by equivalent categories of representations

] of some Weyl cluster algebras. Every preseed p gives rise to a categori-
T
cal preseed P which generates a clustered hyperbolic category through
R
categorical mutations. A “categorification” of Weyl cluster algebra is
.
h introduced in the sense of defining a map F from the clustered hyper-
p
t
a bolic category C(P) to the Weyl cluster algebra H(p) where image of
m
F generates H(p).
p
[

Mathematics Subject Classification (2000): 16T05, 06B15, 17B37.
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v
3
Keywords: Representations Theory, Generalized Weyl Algebras, Cluster
2
4 Algebras.
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.
1 1 Introduction
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7
1 Cluster algebras were introduced by S. Fomin and A. Zelevinsky in [10, 11,
:
v 12, 19, 2]. A cluster algebra is a commutative algebra with a distinguished
Xi set of generators called cluster variables and particular type of relations called
r mutations. A quantum version was introduced in [3] and [7, 8, 9].
a
Generalized Weyl algebras were first introduced by V. Bavula in [1] and
separately as hyperbolic algebras by A. Rosenberg in [16]. Their motivation
was to find a ring theoretical frame work to study the representations theory
of some important “small algebras” such as the first Heisenberg algebra, Weyl
algebras and the universal enveloping algebra of the Lie algebra sl(2). In [16],
Rosenberg introduced hyperbolic categories which arebasically generalizations
of the categories of representations of generalized Weyl algebras.
In [18], we introduced Weyl cluster algebras which is a non-commutative
algebra generated by a cluster structure that is formed by mutations from
(possibly infinite) many copies of generalized Weyl algebras.
Several attempts have been done to introduce “categorifications” for clus-
ter algebras, taking into account the different ways of defining the notion of
categorification. In [13, 14], cluster algebras of certain finite types were re-
alized as Grothendick rings of categories of representations of some quantum
2 Ibrahim Saleh
affine algebras. Another type of categorification of cluster algebras was intro-
duced in [5], which is Caldero-Chapoton map. In [4], cluster category C(Q)
was introduced for any finite quiver Q with neither loops nor two cycles. The
Caldero-Chapoton map X is a map from C(Q) to the ring of Laurent poly-
T
nomials over Z in the initial cluster variables associated to Q. It sends certain
indecomposable objects in C(Q) to cluster variables such that its image gener-
ates the cluster algebra A(Q).
In this paper we provide a similar type of categorification for Weyl cluster
algebra. We introduce a categorical version of Weyl preseed called categorical
preseeds and a “functorial” version of preseeds mutations called categorical
mutations, see Definitions 3.5 and Definition 3.9 respectively. Every categori-
calpreseedgeneratesanambient hyperboliccategory, calledmutation category,
which is generated by equivalent copies of categories of representations of Weyl
cluster algebras. Every mutation category contains a clustered hyperbolic cat-
egory as a subcategory which we use as the categories of categorification for
Weyl cluster algebras. That is, we define a map from clustered hyperbolic cat-
egory to a Weyl cluster algebra of the sam rank such that its image generates
the Weyl cluster algebra.
A technique of identifying clustered hyperbolic categories as subcategories
of mutation categories, is provided through combinatorial tools introduced in
this paper, namely zigzag presentations. Which is a presentation that encodes
the relations between the expressions of the skew Laurent objects, which are
dual to cluster variables. In the following we summarize the main statements
of this article.
Theorem 1.1. Every mutation category contains a clustered hyperbolic cate-
gory as a subcategory.
Every Weyl preseed p gives rise to a categorical preseed P which is used to
generate a mutation category H(P).
Theorem 1.2. The clustered hyperbolic category C(P) is generated by (possibly
infinitely many) equivalent hyperbolic categories; each one of these hyperbolic
categories is equivalent to the category H(p)-mod, where H(p) is the Weyl
cluster algebra generated by p.
Theorem 1.3. There is a map from the category C(P) to the hyperbolic cluster
algebra H(p) such that its image generates H(p).
The paper is organized as follows. Section 2 is devoted for basic definitions
of Weyl cluster algebras. In Section 3, we introduce the notion of categorical
preseeds and their mutations. Examples and properties of categorical preseeds
are also given. In the same section we introduce, hyperbolic objects and the
zigzag presentations and some of their properties are given in Lemma 3.15,
Generalized Cluster Structure on Hyperbolic Categories 3
which takes us to clustered hyperbolic categories definition. We finish Section
3 with proving the relation between the clustered hyperbolic category and the
category of representations of its associated Weyl cluster algebra. In Theo-
rem 3.22, we introduce a map from the clustered hyperbolic category to the
associated Weyl cluster algebra.
Throughout the paper, K is a field of zero characteristic and the notation
[1,k] stands for the set {1,...,k}. All our categories are small, Obj.A stands
for the set of all objects of the category A and Mor. (M,M′) denotes all
A
morphisms in the category A from the object M to the object M′. Let R
be an associative ring with a non-trivial center Z(R). Then, the group of all
automorphisms of R over the filed of zero characteristic K will be denoted by
Aut. (R). The functor id is the identical functor of the category A.
K A
2 Weyl cluster algebras
2.1 Generalized Weyl algebras
Definition 2.1 (GeneralizedWeylalgebra(1,16, 17)). LetRbeanassociative
ring with ξ = {ξ ,...,ξ } be a fixed set of elements of the center of R and
1 n
θ = {θ ,...,θ } be a set of ring automorphisms such that θ (ξ ) = ξ for
1 n i j j
all i 6= j. The generalized Weyl algebra of degree n, denoted by R {θ,ξ},
n
is defined to be the ring extension of R generated by the 2n indeterminates
x ,...,x ,y ,...,y modulo the commutation relations
1 n 1 n
x r = θ (r)x and ry = y θ (r), ∀i ∈ [1,n], ∀r ∈ R, (2.1)
i i i i i i
x y = ξ , y x = θ−1(ξ ), ∀i ∈ [1,n], (2.2)
i i i i i i
and
x y = y x , x x = x x and y y = y y , ∀i 6= j ∈ [1,n]. (2.3)
i j j i i j j i i j j i
We warn the reader that x y 6= y x in general.
i i i i
Example 2.2 (6, 1, 16, 17). Let A be the nth Weyl algebra generated by
n
x ,...,x ,y ...,y over K subject to the relations
1 n 1 n
x y −y x = 1, and x x = x x , y y = y y for i 6= j, ∀i,j ∈ [1,n]. (2.4)
i i i i i j j i i j j i
Let ξ = x y ,ε = y x and R be the ring of polynomials K[ξ ,...,ξ ] and
i i i i i i 1 n
θ : R → R, induced by ξ 7→ ξ + 1,ξ 7→ ξ ,j 6= i, for all i,j ∈ [1,n]. It is
i i i j j
known that A is isomorphic to the generalized Weyl algebra R {θ,ξ}.
n n
Example 2.3 (16, 17). The coordinate algebra A(SL (2,k)) of algebraic quan-
q
tum group SL (2,k) is the K-algebra generated by x,y,u, and v subject to
q
the following relations
qux = xu, qvx = xv, qyu = uy, qyv = vy, uv = vu, q ∈ K∗ (2.5)
4 Ibrahim Saleh
xy = quv+1, and yx = q−1uv +1. (2.6)
A(SL (2,k)) is isomorphic to the generalized Weyl algebra R(ξ,θ,1), where R
q
is the algebra of polynomials K[u,v]; ξ = 1+q−1uv and θ is an automorphism
of R, defined by θ(f(u,v)) = f(qu,qv) for any polynomial f(u,v).
2.2 Weyl cluster algebras
Every thing in this subsection can be found in [18], we start with introducing
a simpler version of the definition of preseeds.
Definition 2.4 (Preseeds). 1. Let P be a finitely generated (free) abelian
group, written multiplicatively, with set of generators
n
F = F where F = {f ,...,f }for some natural numbersm ,i ∈ [1,n].
[ i i i1 imi i
i=1
(2.7)
Let R = K[P] be the group ring of P over K. The set of algebraically
independent indeterminates t ,··· ,t is called a cluster if they satisfy
1 n
the following two conditions
(a)
t t = t t and t f = f t , for everyi 6= j ∈ [1,n], for all r ∈ [1,m ];
i j j i i jr jr i j
(2.8)
(b) The division ring of fractions D = R(t ,··· ,t ) is an Ore domain.
n 1 n
Notethat: Forevery i ∈ [1,n] theelement t doesnot necessary commute
i
with elements from the set F . In such case {t ,...,t } will be called a
i 1 n
cluster in D .
n
2. The triple p = (X,θ,ξ) is called a Weyl preseed of rank n in D if we
n
have the following
(a) X = {x ,...,x } is a cluster in D ;
1 n n
(b) θ = {θ ,...,θ } be a set of n automorphisms of R such that
1 n
x±1f = θ±1(f)x±1, ∀f ∈ F ,∀i ∈ [1,n]; (2.9)
k i i i
(c) ξ = {ξ ,...,ξ } be a subset of R such that for every i ∈ [1,n],ξ is
1 n i
a monomial in the elements of F . The set ξ will be called the set
i
of exchange monomials of p.
Generalized Cluster Structure on Hyperbolic Categories 5
In the following, we will omit the word Weyl from the expression Weyl
preseeds, and all preseeds are of rank n unless stated otherwise. Also for
simplicity we will use D for the division ring of fractions instead of D .
n
Definition 2.5 (Preseeds mutations). Let p = (X,θ,ξ) be a preseed in D. For
eachk ∈ [1,n],twonewtriplesµR(p) = (µR(X),θ,ξ)andµL(p) = (µL(X),θ,ξ)
k k
can be obtained from p as follows
• (Right mutation)
ξ x−1, i = k;
µR(x ) = i i (2.10)
k i ® xi, i 6= k.
• (Left mutation)
x−1ξ , i = k;
µL(x ) = i i (2.11)
k i ® xi, i 6= k.
Proposition 2.6 (18). Let p = (X,θ,ξ) be a preseed in D. Then the following
are true
1. For any sequence of right mutations (respectively left) µRµR ...µR, we
i1 i2 iq
have µRµR ...µR(p) (respectively µLµL ...µL(p)) is again a preseed;
i1 i2 iq i1 i2 iq
2. For every k ∈ [1,n],
µRµL(p) = µLµR(p) = p. (2.12)
k k k k
Definition 2.7 (Cluster sets and exchange graphs). 1. Let p be a preseed
in D. An element y ∈ D is said to be a cluster variable of p if y is
an element in some cluster Y where s = (Y,θ,ξ), is obtained from p by
applying some sequence of (right or left) mutations. The set of all cluster
variables of p is called the cluster set of p and is denoted by X(p). The
elements of the cluster of p are called initial cluster variables.
2. The exchange graph of a preseed p, denoted by G(p), is the n-regular
graph whose vertices are labeled by the preseeds that can be obtained
from p by applying some sequence of right or left mutations and whose
edges correspond to mutations. Two adjacent preseeds in G can be
obtained from each other by applying right and left mutations µR and
k
µL for some k ∈ [1,n].
k
Example 2.8. Let p be the rank 1 preseed ({x },θ ,ξ ) with F = {f }, ξ is
1 1 1 1 1 1
some monomial in f , θ is an R-automorphism, where R = K[fn,n ∈ Z] and
1 1 1
D = R(x ). Applying mutation at x produces the following cluster variables
1 1
6 Ibrahim Saleh
µL
x ⇒k x−1ξ
1 1 1
µL
⇒k (ξ )−1x ξ
1 1 1
µL
⇒k (ξ )−1x−1(ξ )2
1 1 1
µL
⇒k (ξ )−2x (ξ )+2
1 1 1
...
µL
⇒k (ξ )−kx−1(ξ )k+1
1 1 1
µL
⇒k (ξ )−(k+1)x (ξ 1)k+1
1 1 1
... ,
and
µR
x ⇒k ξ x−1
1 1 1
µR
⇒k ξ x (ξ )−1
1 1 1
µR
⇒k (ξ )2x−1(ξ )−1
1 1 1
µR
⇒k (ξ )2x (ξ )−2
1 1 1
...
µR
⇒k (ξ )k+1x−1(ξ )−k
1 1 1
µR
⇒k (ξ )k+1x (ξ )−(k+1)
1 1 1
... .
So we have the infinite cluster set
X(p) = {x ,(ξ )k+1x−1(ξ )−k,(ξ )kx (ξ )−k,(ξ )−kx−1(ξ )k+1,(ξ )−kx−1(ξ )k,k ∈ Z}.
1 1 1 1 1 1 1 1 1 1 1 1 1
In the following example we will see that every generalized Weyl algebra
gives rise to a preseed.
Example 2.9. Let R(θ,ξ,n) be a generalized Weyl algebra. Consider the triple
p = (Y,ξ,θ), where F = {f ;f = y x },i ∈ [1,n], Y = {y ,...,y }. From the
i i i i i 1 n
properties of the R-automorphisms θ = (θ ,...,θ ) given in Equations (2.1)
1 n
and (2.2) one can see that θ satisfies Equation (2.9) for each i ∈ [1,n] which
i
makes p a preseed in D, where D is the division ring of rational functions
in y ,...,y over the ring R = K[P]. In particular, in the case of the nth
1 n
Generalized Cluster Structure on Hyperbolic Categories 7
Weyl algebra A , the ring R is ring of polynomials K[ξ ,...,ξ ]. One can see
n 1 n
that the ambient division ring D of rational functions is an Ore domain. For
information about Ore domains we refer to [13, 3].
Example 2.10. Recall the coordinate algebra A(SL (2,k)) of the algebraic
q
quantum group SL (2,k), Example 2.3. Let F = {qu,v}. Consider the rank
q 1
1 preseed p = ({x},{θ},{ζ}), where θ : R → R given by θ(f(u,v)) = f(qu,qv)
and ζ = quv + 1. One can see that p is a preseed in the division ring
D = R[P](x). The cluster set of p is given by
X(p) = {x,ζjxζ−j,ζj+1x−1ζ−j−1,j ∈ N} {y,ζjyζ−j,ζj+1y−1ζ−j−1,j ∈ N}
[
(2.13)
Definition 2.11 (Weyl cluster algebras). Let p = (X,ξ,θ) be a preseed in
D. The Weyl cluster algebra H(p) is defined to be the R-subalgebra of D
generated by the cluster set X(p).
The following remark and theorem shed some light on the structure of the
Weyl cluster algebra H(p ). Remark 2.12 and first part of Theorem 2.13 can
n
be phrased as following: The Weyl cluster algebra H(p ) is generated by R
n n
and many (could be infinite) isomorphic copies of generalized Weyl algebras,
each vertex in the exchange graph of p gives rise to two copies of them.
n
Remark and Definition 2.12. Let p = (X,θ,ξ) be a Weyl preseed and R =
K[ξ ,...,ξ ] be the ring of polynomials in ξ ,...,ξ where ξ ,i = 1,...,n are
1 n 1 n i
as defined in Example 2.9. Then p gives rise to two copies of generalized Weyl
algebras of rank n, as follows
(a) HR(p )istheringextensionofRgeneratedbyµR(x ),...,µR(x ),x ,...,x .
n 1 1 n n 1 n
(b) HL(p )istheringextensionofRgeneratedbyx ,...,x ,µL(x ),...,µL(x ).
n 1 n 1 1 n n
(c) In particular, if p = (X,ξ,θ) is the preseed given in Example 2.9, then
eachofHR(p)andHL(p)areisomorphictoR(θ,ξ,n)asgeneralizedWeyl
algebras.
Theorem 2.13. Let p = (X,ξ,θ) be a Weyl preseed in D. Then the following
are true
1. Right and left mutations on p induce isomorphisms between the gen-
eralized Weyl algebras HR(p) and HR(µR(p)) (respectively HL(p) and
k
HL(µL(p))).
k
2. The Weyl cluster algebra H(p) is a subring of the (non-commutative)
ring of Laurent polynomials in the initial exchange cluster variables with
coefficients from ring of polynomials R[θ±1(ξ−1),...,θ±1(ξ−1)].
1 1 n n
8 Ibrahim Saleh
3. The Weyl cluster algebra H(p) is finitely generated and is isomorphic to
each of HR(s) and HL(s) for every preseed s mutationally equivalent to
p.
Proof. Proof of Parts (1) and (2) are provided in [18].
To prove Part (3), we only need to prove that the generators of the algebra
H(p) are also elements in the algebras HR(s) and HL(s). The algebra H(p) is
generated by the set of all cluster variables that are obtained from the initial
cluster variables x ,...,x . We will show that every cluster variable generated
1 n
from x ,k = 1,...,n by applying some sequence of mutations is already an
k
element of HR(s) and HL(s). From the proof of Part (3) in Theorem 4.12
in [18], a cluster variable y that obtained from x by applying a sequence of
k
mutations (right or left) of length l, can be written in the form
ξl+21x−1ξ−(l+21−1) or ξ−(l+21−1)x−1ξl+21, if l is an odd number;
y =  k k k k k k
 l −l −l l
ξ2x ξ 2 or ξ 2x ξ2, if l is an even number.
k k k k k k

(2.14)
Assume that the sequence of mutations that creates s from p contains m
copiesofµR (respecttoµL),thenthesetofgeneratorsofthehyperbolicalgebra
k k
HR(s) (respect to HL(s)) contains elements of the form ξm2+1x−1ξ−(m2+1−1) and
k k k
ξm2 x ξ−m2 (respect to ξ−(m2+1−1)x−1ξm2+1 and ξ−m2 x ξm2 ). One can see that
k k k k k k k k k
whether m > l or m < l, y can be obtained from the generators of HR(s)
(respect to HL(s)) by multiplying them from left and right by ξ±q for some
k
natural number q.
Example 2.14 (Weyl cluster algebra associated to first Weyl algebra). Recall
the nth Weyl algebra given and the associated preseed given in Example 2.9.
Let A be the first Weyl algebra and consider the preseed p = ({y},{f},θ}).
1 1
Here R = K[P], where P is the cyclic group generated by f = yx. Then We
have the following exchange graph
• G(p )
1
...oo R //y−·3 oo R // y−·2 oo R //y−·1 oo R // y0·=y oo R // y·1 oo R //y·2 oo R // y·3 oo R //...,
L L L L L L L L
(2.15)
(here oo · is left mutation and · R // is right mutation). Which can be
L
encoded by the following equations
y y = y y +1, for k ∈ Z. (2.16)
2k 2k±1 2k±1 2k
Generalized Cluster Structure on Hyperbolic Categories 9
The Weyl cluster algebra H(p (y)) is the R-subalgebra of D generated by
1
the set of cluster variables {y ,k ∈ Z}. Relations (2.15) can be interpreted
k
as follows, each double heads arrow in G(p ) corresponds to a copy of first
1
Weyl algebra, denoted by Ak = Khy ,y i,k ∈ Z and right (respectively
1 k k+1
left) mutations define isomorphisms between the adjacent copies, given by
T : Ak → Ak+1, y 7→ y for k ∈ Z (respectively to the inverses of
k 1 1 k k+1
T ,k ∈ Z).
k
3 Hyperbolic cluster category
3.1 Hyperbolic category
Definition 3.1 (Hyperbolic category (15, 17)). Let A be an additive category
with a set of n-auto-equivalences Θ = {Θ ,...,Θ } and another set of n-
1 n
endomorphisms ξ = {ξ ,...,ξ } of the identical functor of A. Consider the
1 n
endomorphism ε of the identical functor of A given by
i
ε := Θ−1(ξ ), i ∈ [1,n]. (3.1)
i i
The hyperbolic category of rank n on A is denoted by A {Θ,ξ} and is defined
n
as follows: Objects are triples (γ,M,η) where M is an object in A and γ and
η are two sets of A-morphisms such that γ = {γ ,...,γ }, η = {η ,...,η }
1 n 1 n
where
γ : M −→ Θ (M) and η : Θ (M) −→ M, i ∈ [1,n]
i i i i
given by
η ◦γ = ξ and γ ◦η = ε , ∀i ∈ [1,n]. (3.2)
i i i,M i i i,Θi(M)
A morphism from the object (γ,M,η) to the object (γ′,M′,η′) is the set
f = {f ,...,f }wheref isinMor. (M,M′),i ∈ [1,n], suchthatthefollowing
1 n i A
diagram is commutative
M γi // Θ (M) ηi //M (3.3)
i
fi Θi(fi) fi
(cid:15)(cid:15) γ′ (cid:15)(cid:15) η′ (cid:15)(cid:15)
M′ i //Θ (M′) i //M′.
i
Remark 3.2. For every object M in A, the object (γ,M,η) in the hyperbolic
category A {Θ,ξ}, has unique sets of morphisms γ and η if and only if the
n
identity morphism id in A is also a morphism in A {Θ,ξ}. In particular, in
M n
such casetheobject M inAwill appearexactly onceas(−,M,−)inA {Θ,ξ}.
n
Example 3.3. Let R {θ,ξ} be a generalized Weyl algebra with indeterminants
1
x and y. Let A be the full subcategory of the category R−mod with objects
10 Ibrahim Saleh
are R {θ,ξ}-modules, forgetting about the actions of x and y. The category
1
R {θ,ξ}-modules is equivalent to a hyperbolic category A {Θ,ξ} where Θ :
1 1
A −→ A is induced by the R-automorphism θ and for M ∈ objets of A,
ξ : M −→ M is given by ξ (m) := ξm. Objects of A {Θ,ξ} are the triples
M M 1
(x,M,y) where M is an object in A, x : M −→ θ(M) given by x(m) = xm
and y : θ(M) −→ M given by y(m) = ym.
Example 3.4 (Trivial hyperbolic category). Let R {ξ,θ} be a generalized Weyl
n
algebra. Consider the additive category B with only one object which is the
ring R and Hom (R,R) is the underlying vector space of R {ξ,θ}. One can
B n
form a hyperbolic category with one object which is (x,R,y), where x =
{x ,··· ,x } and y = {y ,··· ,y }.
1 n 1 n
3.2 Categorical preseeds
Let A be an additive category.
Definition 3.5 (Categorical preseeds). A categorical preseed of rank n in A
is the data P = (Θ,ξ,AhΘi) where
1. Θ = {Θ ,...,Θ } is a set of n auto-equivalences in A;
1 n
2. ξ = {ξ ,...,ξ } is set of n endomorphisms of the identical functor of A;
1 n
3. AhΘi is the following category:
Objects arepairs (M,f), where M is anobject inA andf = {f ,...,f }
1 n
is a set of n invertible elements of Mor. (Θ(M),M).
A
Morphismsfrom(M,f)to(M′,f′)inAhΘiarethen-tuplesh = {h ,...,h }
1 n
where h ∈ Mor. (M,M′),i ∈ [1,n] such that the following diagram is
i A
commutative
Θ (M) fi //M (3.4)
i
Θi(hi) hi
(cid:15)(cid:15) f′ (cid:15)(cid:15)
Θ (M′) i //M′.
i
Furthermore, if the set {Θ ;i ∈ [1,n]} is a commutative set of functors
i
and for every object M in A, the set {ξ ;i ∈ [1,n]} is a set of commu-
i,M
tative morphisms, then P is called a hyperbolic categorical preseed or for
simplicity HCPS.
Remark 3.6. 1. Let P = (Θ,ξ,AhΘi) be a categorical preseed. Since ξ is a
set of endomorphisms of id , then for every object (M,f) in AhΘi we
A
have
f ◦ξ = ξ ◦f ; ∀i,j ,...,j ∈ [1,n]. (3.5)
i i,Θi(Θjk...Θj1(M)) i,Θjk...Θj1(M) i k 1