Representations of quivers on abelian categories and monads on

21 Pages·2009·0.28 MB·English

Representations of quivers on abelian
categories and monads on projective varieties
Marcos Jardim & Daniela Moura Prata
IMECC - UNICAMP
Departamento de Matem´atica
Caixa Postal 6065
13083-970 Campinas-SP, Brazil
November 16, 2009
Abstract
We consider representations of quivers in arbitrary categories and
twisted representations of quivers in arbitrary tensor categories. We
show that if 𝒜 is an abelian category, then the category of representa-
tions of a quiver in 𝒜 is also abelian, and that the category of twisted
linear representations of a quiver is equivalent to the category of lin-
ear (untwisted) representations of a different quiver. We conclude by
discussing how representations of quivers arise naturally in certain
important problems concerning monads and sheaves on projective va-
rieties.
1 Introduction
Quivers are a valuable tool in the theory of finite dimensional associative
algebras and their representations [3]. Moreover, linear representations of
quivers are a beautiful subject in itself, with many interesting connections
with other areas. Recently, many authors have considered representations
of quivers in arbitrary categories, see in particular [11], motivated by the
relevance of such concept in algebraic geometry and mathematical physics.
More precisely, representations of quivers in the category of vector bundles
or coherent sheaves on a projective variety, called quiver bundles or quiver
sheaves, unify many of the vector bundles with extra structure which have
been previously considered in the literature (e.g. Higgs bundles, coherent
1
systems, holomorphic triples, etc), see [2, 11]. Several recent papers also
consider quiver bundles in connection with string theory, see for instance
[7, 19].
In this paper, we consider representations of quivers in arbitrary cate-
gories and twisted representations of quivers in arbitrary tensor categories,
aswellassomeapplicationsoftheseconceptstothestudyofcertainproblems
concerning vector bundles over projective varieties.
First, we prove that if 𝒜 is an additive (abelian) category, then the cat-
egory of representations of a quiver in 𝒜 is also additive (abelian), showing
that the category of representations often inherits some of the properties of
the original category. We also discuss how functors between categories in-
duce functors between the respective categories of representations, and show
that such induced functors also inherit properties from the original ones.
Next, we consider twisted representations of quivers in arbitrary tensor
categories, and show that the category of twisted linear representations of a
quiver is equivalent to the category of linear (untwisted) representations of
a different quiver.
Finally, we turn to one of the original motivations behind this project by
discussing the theory of monads on a projective variety 𝑋 from a categorical
point of view, and showing that these can be regarded as representations of a
quiver with relation in the category of vector bundles over 𝑋. We then focus
on a particular class of sheaves on projective space, so-called linear sheaves,
and show that the category of such sheaves is equivalent to a subcategory of
the category of twisted linear representations of a quiver with relations. We
look at examples in which geometric properties of sheaves are translated into
algebraic properties of the corresponding twisted representations, and vice
versa. This leads to a possibly (and hopefully) fruitful application of repre-
sentation theoretical methods to the study of vector bundles over projective
varieties.
Acknowledgments. MJ is partially supported by the CNPq grant num-
ber 305464/2007-8 and the FAPESP grant number 2005/04558-0. DMP’s re-
search is supported by the FAPESP grant number 2007/07469-3. We thank
Vyacheslav Futorny, Adriano Moura, Fla´vio Coelho, Plamen Koshlukov and
EduardoMarcosfortheirvaluablecommentsinvariousstagesofthisproject.
2 Representations of Quivers
Recall that a quiver 𝑄 consists of a pair (𝑄 ,𝑄 ) where 𝑄 is the set of
0 1 0
vertices and 𝑄 is the set of arrows, and we have two maps 𝑡,ℎ : 𝑄 → 𝑄
1 1 0
2
named tail and head. We only consider finite quivers, i.e. the sets 𝑄 and
0
𝑄 are finite. A path in the quiver is a sequence of arrows
1
𝑝 = 𝑎 𝑎 ⋅⋅⋅𝑎
1 2 𝑛
such that ℎ(𝑎 ) = 𝑡(𝑎 ), 𝑖 = 1,⋅⋅⋅ ,𝑛−1.
𝑖+1 𝑖
𝑎𝑛 (cid:47)(cid:47) 𝑎𝑛−1(cid:47)(cid:47)⋅⋅⋅ 𝑎2 (cid:47)(cid:47) 𝑎1 (cid:47)(cid:47)
We say that the path starts in 𝑡(𝑎 ) and ends in ℎ(𝑎 ). The path algebra
𝑛 1
𝑘𝑄 is the associative algebra generated by all paths of 𝑄 with the product
given by concatenation of paths.
A relation 𝑅 in 𝑄 is a 𝑘-linear combination of paths 𝑝 ∈ 𝑘𝑄 such
𝑖
that 𝑡(𝑝 ) = 𝑡(𝑝 ) and ℎ(𝑝 ) = ℎ(𝑝 ), 𝑖,𝑗 = 1,...,𝑛. Given relations 𝑅 =
𝑖 𝑗 𝑖 𝑗 𝑗
∑𝑛𝑗 𝜆𝑗𝑝𝑗, 𝑗 = 1,...,𝑚, 𝑝𝑗 ∈ 𝑘𝑄, the path algebra with relations is given by
𝑖=1 𝑖 𝑖 𝑖
the quotient 𝑘𝑄/𝐼, where 𝐼 is the ideal of 𝑘𝑄 generated by the relations 𝑅 ,
𝑗
𝑗 = 1,...,𝑚.
2.1 Representations of quivers in additive and abelian
categories
So let 𝒜 be a category and 𝑄 a quiver. Our first goal is to consider represen-
tations of quivers in arbitrary categories. In what follows, we will omit the
usual categorical definitions like additive, abelian and tensor categories, and
functors between them; we refer to the definitions found in [10, 16].
Definition. A representation of 𝑄 in 𝒜 consists of
∙ a collection of objects {𝑉 },𝑖 ∈ 𝑄 ;
𝑖 0
∙ a collection of morphisms {𝜑 },𝑎 ∈ 𝑄 , where 𝜑 ∈ Hom (𝑉 ,𝑉 ).
𝑎 1 𝑎 𝒜 𝑡(𝑎) ℎ(𝑎)
The representations of 𝑄 in 𝒜 is a category that we denote by Rep(𝑄,𝒜).
It is interesting to observe that the category of representations Rep(𝑄,𝒜)
inherits some of the properties of the original category 𝒜; in particular, one
has the following result.
Theorem 1. If 𝒜 is additive (abelian) then Rep(𝑄,𝒜) is also additive
(abelian).
Proof. Suppose 𝒜 is additive. To prove that Rep(𝑄,𝒜) is additive we have
to check that Rep(𝑄,𝒜) satisfies some conditions, see [10].
3
(𝑖) Rep(𝑄,𝒜) has a zero object: just take 𝑉 = 0 ∈ Obj(𝒜), ∀𝑖 ∈ 𝑄 , and
𝑖 0
𝜑 = 0 ∈ Hom (0,0), ∀𝑎 ∈ 𝑄 .
𝑎 𝒜 1
(𝑖𝑖) If 𝑋,𝑌,𝑍 ∈ Obj(ℛ) then Hom (𝑋,𝑌) is an abelian group and
ℛ
Hom (𝑋,𝑌)×Hom (𝑌,𝑍) → Hom (𝑋,𝑍)
ℛ ℛ ℛ
is bi-additive.
Let 𝑋 = (𝑉,𝜙), 𝑌 = (𝑊,𝜓) and 𝑍 = (𝑈,𝜂) be objects of ℛ =
Rep(𝑄,𝒜). The result follows because Hom (𝑉 ,𝑊 ) is an abelian
𝒜 𝑖 𝑖
group and the pairing
Hom (𝑉 ,𝑊 )×Hom (𝑊 ,𝑈 ) → Hom (𝑉 ,𝑈 )
𝒜 𝑖 𝑖 𝒜 𝑖 𝑖 𝒜 𝑖 𝑖
is bi-additive for all 𝑖 ∈ 𝑄 .
0
(𝑖𝑖𝑖) Rep(𝑄,𝒜) has coproducts.
Let (𝑉,𝜙) and (𝑊,𝜓) be objects of Rep(𝑄,𝒜). We want to define
the direct sum of (𝑉,𝜙) and (𝑊,𝜓). As 𝒜 is additive, for all 𝑖 ∈ 𝑄 ,
0
𝑈 = 𝑉 ⊕ 𝑊 is an object of 𝒜 with the maps 𝑙 : 𝑉 → 𝑈 and
𝑖 𝑖 𝑖 𝑖 𝑖 𝑖
ℎ : 𝑊 → 𝑈 .
𝑖 𝑖 𝑖
By a property of coproducts, there exists an unique morphism
𝜂 : 𝑈 → 𝑈
𝑎 𝑡(𝑎) ℎ(𝑎)
such that the diagram
𝑉 𝜙𝑎 (cid:47)(cid:47)𝑉
𝑡(𝑎) ℎ(𝑎)
𝑙 𝑙
𝑡(𝑎) ℎ(𝑎)
(cid:15)(cid:15) (cid:15)(cid:15)
𝑈 𝜂𝑎 (cid:47)(cid:47)𝑈
𝑡((cid:79)(cid:79) 𝑎) ℎ(cid:79)(cid:79)(𝑎)
ℎ ℎ
𝑡(𝑎) ℎ(𝑎)
𝑊 𝜓𝑎 (cid:47)(cid:47)𝑊
𝑡(𝑎) ℎ(𝑎)
commutes. Then (𝑈,𝜂) with 𝑈 = 𝑉 ⊕𝑊 for 𝑖 ∈ 𝑄 and 𝜂 : 𝑈 →
𝑖 𝑖 𝑖 0 𝑎 𝑡(𝑎)
𝑈 for each 𝑎 ∈ 𝑄 is the direct sum of (𝑉,𝜙) and (𝑊,𝜓) with the
ℎ(𝑎) 1
morphisms
𝐿 = {𝑙 } : (𝑉,𝜙) → (𝑈,𝜂) and 𝐻 = {ℎ } : (𝑊,𝜓) → (𝑈,𝜂).
𝑖 𝑖∈𝑄0 𝑖 𝑖∈𝑄0
Therefore Rep(𝑄,𝒜) is an additive category.
4
If 𝒜 is abelian, we have to verify other four conditions.
1. Every morphism has kernel and cokernel.
Let 𝑓 : (𝑉,𝜙) → (𝑊,𝜓) be a morphism. We want to define ker𝑓. As
𝒜 is abelian, for each 𝑖 ∈ 𝑄 , 𝑓 : 𝑉 → 𝑊 has a kernel (𝑉′,𝜇 ). By
0 𝑖 𝑖 𝑖 𝑖 𝑖
a property of kernel, for each 𝑎 ∈ 𝑄 there is an unique morphism
1
𝜙′ : 𝑉′ → 𝑉′ such that the diagram commutes.
𝑎 𝑡(𝑎) ℎ(𝑎)
𝑉′ 𝜙′𝑎 (cid:47)(cid:47)𝑉′
𝑡(𝑎) ℎ(𝑎)
𝜇 𝜇
𝑡(𝑎) ℎ(𝑎)
(cid:15)(cid:15) (cid:15)(cid:15)
𝑉 𝜙𝑎 (cid:47)(cid:47)𝑉
𝑡(𝑎) ℎ(𝑎)
𝑓 𝑓
𝑡(𝑎) ℎ(𝑎)
(cid:15)(cid:15) (cid:15)(cid:15)
𝑊 𝜓𝑎 (cid:47)(cid:47)𝑊
𝑡(𝑎) ℎ(𝑎)
If we set
𝜇 = {𝜇 } : (𝑉′,𝜙′) → (𝑉,𝜙)
𝑖 𝑖∈𝑄0
where 𝑉′ = {𝑉′} and 𝜙′ = {𝜙′} , one can see that ker𝑓 =
𝑖 𝑖∈𝑄0 𝑎 𝑎∈𝑄1
((𝑉′,𝜙′),𝜇) is in fact kernel of 𝑓.
Similarly one can see that 𝑓 has cokernel.
2. Every monomorphism is the kernel of its cokernel.
Let 𝑓 : (𝑉,𝜙) → (𝑊,𝜓) be a monomorphism. Then 𝑓 : 𝑉 → 𝑊
𝑖 𝑖 𝑖
is a monomorphism for each 𝑖 ∈ 𝑄 . As 𝒜 is an abelian category, if
0
(𝑊′,𝑐 ) = coker𝑓 then (𝑉 ,𝑓 ) = ker𝑐 , ∀𝑖 ∈ 𝑄 and there is an unique
𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 0
𝜓′ : 𝑊′ → 𝑊′ such that 𝑐 𝜓 = 𝜓′𝑐 . We have ((𝑊′,𝜓′),𝑐) =
𝑎 𝑡(𝑎) ℎ(𝑎) ℎ(𝑎) 𝑎 𝑎 𝑡(𝑎)
coker𝑓 and therefore ((𝑉,𝜙),𝑓) = ker𝑐.
3. Every epimorphism is cokernel of its kernel.
Let 𝑓 : (𝑉,𝜙) → (𝑊,𝜓) be an epimorphism. Then for each 𝑖 ∈ 𝑄
0
𝑓 : 𝑉 → 𝑊 is an epimorphism. As 𝒜 is abelian if (𝑉′,𝑘 ) = ker𝑓
𝑖 𝑖 𝑖 𝑖 𝑖 𝑖
there is an unique 𝜙′ : 𝑉′ → 𝑉′ , 𝑎 ∈ 𝑄 , such that 𝜙 𝑘 = 𝑘 𝜙′
𝑎 𝑡(𝑎) ℎ(𝑎) 1 𝑎 𝑡(𝑎) ℎ(𝑎) 𝑎
and ((𝑉′,𝜙′),𝑘) = ker𝑓. We also have (𝑊 ,𝑓 ) = coker𝑘 therefore
𝑖 𝑖 𝑖
((𝑊,𝜓),𝑓) = coker𝑘 and follows the result.
4. Every morphism can be written as composition of epimorphism and
monomorphism.
Let 𝑓 : (𝑉,𝜙) → (𝑊,𝜓) be a morphism. Since 𝒜 is abelian for each
𝑖 ∈ 𝑄 , 𝑓 = ℎ 𝑔 , where 𝑔 : 𝑉 → 𝑊′ is epimorphism and ℎ : 𝑊′ → 𝑊
0 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖
5
is monomorphism, with 𝑊′ ∈ Obj(𝒜). By a lemma for each 𝑎 ∈ 𝑄
𝑖 1
there is an unique
𝜓′ : 𝑊′ → 𝑊′
𝑎 𝑡(𝑎) ℎ(𝑎)
such that
𝑉 𝜙𝑎 (cid:47)(cid:47)𝑉
𝑡(𝑎) ℎ(𝑎)
𝑔 𝑔
𝑡(𝑎) ℎ(𝑎)
(cid:15)(cid:15) (cid:15)(cid:15)
𝑊′ 𝜓𝑎′ (cid:47)(cid:47)𝑊′
𝑡(𝑎) ℎ(𝑎)
ℎ ℎ
𝑡(𝑎) ℎ(𝑎)
(cid:15)(cid:15) (cid:15)(cid:15)
𝑊 𝜓𝑎 (cid:47)(cid:47)𝑊
𝑡(𝑎) ℎ(𝑎)
commutes. Then if we take the representation (𝑊′,𝜙′) and the mor-
phisms ℎ = {ℎ } : (𝑊′,𝜓′) → (𝑊,𝜓) and 𝑔 : {𝑔 } (𝑉,𝜙) →
𝑖 𝑖∈𝑄0 𝑖 𝑖∈𝑄0
(𝑊′,𝜓′) follows that 𝑔 is epimorphism, ℎ is monomorphism and 𝑓 =
ℎ∘𝑔.
Remark 2. We believe that other standard categorical properties of 𝒜, like
the existence of sufficiently many projectives and injectives, will also be in-
herited by the category of representations Rep(𝑄,𝒜).
If 𝒜 is an additive category, one can also consider representations of
quivers with relations in 𝒜.
Definition. Let 𝒜 be an additive category, 𝑄 a quiver and 𝑅 = {𝑅 ,...,𝑅 }
1 𝑚
be a set of relations in 𝑄, where 𝑅 = ∑𝑛𝑖 𝜆𝑖𝑝𝑖, 𝑝𝑖 = 𝑎𝑖 ⋅⋅⋅𝑎𝑖 . A re-
presentation (𝑉,𝜙) in 𝒜 satisfies th𝑖e relati𝑗o=n1s 𝑗if𝑗∑𝑛𝑗𝑖 𝜆𝑖𝜙𝑗1 = 0𝑗𝑙𝑗for each
𝑗=1 𝑗 𝑝𝑖
𝑗
𝑖 = 1,⋅⋅⋅ ,𝑚, where 𝜙 = 𝜙 ⋅⋅⋅𝜙 .
𝑝𝑖 𝑎𝑖 𝑎𝑖
𝑗 𝑗1 𝑗𝑙𝑗
WedenotebyRep((𝑄,𝑅),𝒜)thecategoryofrepresentationsofthequiver
with relations. It is not difficult to see that Rep((𝑄,𝑅),𝒜) is a full additive
subcategory of Rep(𝑄,𝒜).
2.2 Induced functors
Let 𝒜 and ℬ be categories, 𝑄 a quiver and F : 𝒜 → ℬ a functor. We can
define a functor F : Rep(𝑄,𝒜) → Rep(𝑄,ℬ) in the following way:
𝑄
∙ given an object (𝑉,𝜙) ∈ Rep(𝑄,𝒜), F ((𝑉,𝜙)) = (𝑊,𝜓) where 𝑊 =
𝑄 𝑖
F(𝑉 ) ∈ Obj(ℬ),𝑖 ∈ 𝑄 , and𝜓 = F(𝜙 ) ∈ Hom (𝑊 ,𝑊 ),𝑎 ∈ 𝑄 ;
𝑖 0 𝑎 𝑎 ℬ 𝑡(𝑎) ℎ(𝑎) 1
6
∙ given a morphism 𝑓 : (𝑉,𝜙) → (𝑊,𝜓), 𝑓 = {𝑓 },𝑓 ∈ Hom (𝑉 ,𝑊 ) we
𝑖 𝑖 𝒜 𝑖 𝑖
define F (𝑓) = 𝑔, where 𝑔 = F(𝑓 ) ∈ Hom (F(𝑉 ),F(𝑊 )),𝑖 ∈ 𝑄 .
𝑄 𝑖 𝑖 ℬ 𝑖 𝑖 0
It turns out that the induced functor F also inherits properties of the
𝑄
functor original F, as in the following Proposition.
Proposition 3. Let F : 𝒜 → ℬ be a functor.
∙ If F is an equivalence of categories, then so is the induced functor F
𝑄
for any quiver 𝑄;
∙ If 𝒜 and ℬ are abelian categories and F is an exact functor, then so is
the induced functor F for any quiver 𝑄.
𝑄
Proof. First, suppose F is an equivalence of categories. To show that F is
𝑄
equivalence of categories we must see that F is fully faithful and essentially
𝑄
surjective, see [10, p. 71].
Let 𝑋 = (𝑉,𝜙) and 𝑌 = (𝑊,𝜓) be objects of Rep(𝑄,𝒜), and take 𝑓,𝑔 ∈
Hom(𝑋,𝑌) such that F (𝑓) = F (𝑔), with
𝑄 𝑄
F : Hom(𝑋,𝑌) → Hom(F(𝑋),F(𝑌)).
𝑄
Then F(𝑓 ) = F(𝑔 ),∀𝑖 ∈ 𝑄 , 𝑓 ,𝑔 ∈ Hom(𝑉 ,𝑊 ). Since F is faithful,
𝑖 𝑖 0 𝑖 𝑖 𝑖 𝑖
𝑓 = 𝑔 ,∀𝑖 ∈ 𝑄 then 𝑓 = 𝑔 therefore F is also faithful.
𝑖 𝑖 0 𝑄
If 𝑔 ∈ Hom(𝐹 (𝑋),𝐹 (𝑌)) where 𝑔 = {𝑔 } , 𝑔 ∈ Hom(𝐹(𝑉 ),𝐹(𝑊 )),
𝑄 𝑄 𝑖 𝑖∈𝑄0 𝑖 𝑖 𝑖
as F is full, for each 𝑖 ∈ 𝑄 there is a morphism 𝑔 ∈ Hom(𝑉 ,𝑊 ) such that
0 𝑖 𝑖 𝑖
F(𝑔 ) = 𝑔 . Thus F (𝑔) = 𝑔 where 𝑔 = {𝑔 } , hence F is full. We have
𝑖 𝑖 𝑄 𝑖 𝑖∈𝑄0 𝑄
shown that if F is fully faithful, then so is F .
𝑄
Let 𝑌 = (𝑊,𝜓) ∈ Obj(Rep(𝑄,ℬ)). As F is essentially surjective, there
is an isomorphism 𝜆 : 𝑊 → F(𝑉 ),∀𝑖 ∈ 𝑄 , 𝑉 ∈ Obj(𝒜). For each 𝑎 ∈ 𝑄
𝑖 𝑖 𝑖 0 𝑖 1
consider the map 𝜙′ = 𝜆 𝜓 𝜆−1 . Then 𝜙′ ∈ Hom(F(𝑉 ),F(𝑉 )) and
𝑎 ℎ(𝑎) 𝑎 𝑡(𝑎) 𝑎 𝑡(𝑎) ℎ(𝑎)
since F is full, there is a map 𝜙 ∈ Hom(𝑉 ,𝑉 ) such that F(𝜙 ) =
𝑎 𝑡(𝑎) ℎ(𝑎) 𝑎
𝜙′. Let 𝑋 = (𝑉,𝜙) where 𝑉 = {𝑉 } and 𝜙 = {𝜙 } . Then 𝑋 ∈
𝑎 𝑖 𝑖∈𝑄0 𝑎 𝑎∈𝑄1
Obj(Rep(𝑄,𝒜))andF (𝑋) ≃ 𝑌 withisomorphism𝜆andsoF isessentially
𝑄 𝑄
surjective, therefore F is equivalence of categories.
𝑄
Now suppose that F is an exact functor and let 𝑋 = (𝑉,𝜙), 𝑌 = (𝑊,𝜓),
𝑍 = (𝑈,𝜆) be objects of Rep(𝑄,𝒜), 𝑓 ∈ Hom(𝑋,𝑌) and 𝑔 ∈ Hom(𝑌,𝑍)
such that
(cid:47)(cid:47) 𝑓 (cid:47)(cid:47) 𝑔 (cid:47)(cid:47) (cid:47)(cid:47)
0 𝑋 𝑌 𝑍 0
is a short exact sequence. Since F is an exact functor one can easily check
that
0 (cid:47)(cid:47)F (𝑋)F𝑄(𝑓)(cid:47)(cid:47)F (𝑌) F𝑄(𝑔)(cid:47)(cid:47)F (𝑍) (cid:47)(cid:47)0
𝑄 𝑄 𝑄
is a short exact sequence and therefore F is an exact functor.
𝑄
7
3 Twisted linear representations of quivers
The concept of twisted representations of quivers was first introduced by
Gothen and King [11, p. 88], motivated by certain problems involving vector
bundles with extra structures, like Higgs bundles and holomorphic triples.
Here, we will define twisted representations in an arbitrary tensor category,
and, for representations in the category of vector spaces over a field, we
will relate the category of twisted representations of 𝑄 with the category of
˜
representations of a different quiver 𝑄.
3.1 Twisted representations of quivers
Definition. Let 𝑄 be a quiver and 𝒜 a tensor category. Fix a collection
𝑀 = {𝑀 } of objects of 𝒜. A right 𝑀-twisted representation of 𝑄
𝑎 𝑎∈𝑄1
consists of
∙ a collection of objects {𝑉 ∣ 𝑖 ∈ 𝑄 };
𝑖 0
∙ a collection of morphisms {𝜑 : 𝑉 → 𝑉 ⊗𝑀 ∣ 𝑎 ∈ 𝑄 }.
𝑎 𝑡(𝑎) ℎ(𝑎) 𝑎 1
We also denote the representation by (𝑉,𝜙). Alternatively, one could also
consider the morphisms {𝜑 : 𝑀 ⊗ 𝑉 → 𝑉 ∣ 𝑎 ∈ 𝑄 }, leading to the
𝑎 𝑎 𝑡(𝑎) ℎ(𝑎) 1
notion of a left 𝑀−twisted representation of 𝑄.
A morphism between two twisted representations (𝑉,𝜙) and (𝑊,𝜓) in 𝒜,
is a collection of morphisms 𝑓 : 𝑉 → 𝑊 , 𝑖 ∈ 𝑄 , such that the following
𝑖 𝑖 𝑖 0
diagram commutes for each 𝑎 ∈ 𝑄 .
1
𝑉 𝜙𝑎 (cid:47)(cid:47)𝑉 ⊗𝑀
𝑡(𝑎) ℎ(𝑎) 𝑎
𝑓𝑡(𝑎) 𝑓ℎ(𝑎)⊗1𝑀𝑎
(cid:15)(cid:15) (cid:15)(cid:15)
𝑊 (cid:47)(cid:47)𝑊 ⊗𝑀
𝑡(𝑎) ℎ(𝑎) 𝑎
𝜓𝑎
that is,
(𝑓 ⊗1 )∘𝜙 = 𝜓 ∘𝑓 , ∀𝑎 ∈ 𝑄 .
ℎ(𝑎) 𝑀𝑎 𝑎 𝑎 𝑡(𝑎) 1
We have then a new category of right 𝑀−twisted representations of 𝑄,
denoted by Rep (𝑄,𝒜).
𝑀
Suppose 𝑄 has relations 𝑅 = {𝑅 ,⋅⋅⋅ ,𝑅 }, where 𝑅 = ∑𝑛𝑖 𝜆𝑖𝑝𝑖,
1 𝑚 𝑖 𝑗=1 𝑗 𝑗
and 𝑝𝑖 = 𝑎𝑖 ⋅⋅⋅𝑎𝑖 . Let 𝑀 = {𝑀 }, 𝑎 ∈ 𝑄 , be a collection of objects
𝑗 𝑗1 𝑗𝑙𝑗 𝑎 1
of 𝒜, 𝑀 = 𝑀 ⊗ 𝑀 ⊗ ... ⊗ 𝑀 , 𝑗 = 1,...𝑛 ,𝑖 = 1,...,𝑚 and
𝑝𝑖 𝑎𝑖 𝑎𝑖 𝑎𝑖 𝑖
𝑗 𝑗1 𝑗2 𝑗𝑙𝑗
8
˜ ⊗
𝑀 = 𝑀 . Given a twisted representation (𝑉,𝜙) we have induced
𝑎∈𝑄1 𝑎
maps
˜
𝜙 : 𝑉 → 𝑉 ⊗𝑀
𝑝𝑖 𝑡(𝑝𝑖) ℎ(𝑝𝑖) 𝑝𝑖
𝑗 𝑗 𝑗 𝑗
where
˜
𝜙 = (𝜙 ⊗1 ⊗...⊗1 )∘(𝜙 ⊗1 ⊗...⊗1 )∘...
𝑝𝑖𝑗 𝑎𝑖𝑗1 𝑀𝑎𝑖𝑗2 𝑀𝑎𝑖𝑗𝑙𝑗 𝑎𝑖𝑗2 𝑀𝑎𝑖𝑗3 𝑀𝑎𝑖𝑗𝑙𝑗
...∘(𝜙 ⊗1 )∘𝜙 .
𝑎𝑖𝑗𝑙𝑗−1 𝑀𝑎𝑖𝑗𝑙𝑗 𝑎𝑖𝑗𝑙𝑗
˜ ˜
If 𝑓 : 𝑀 → 𝑀 is the inclusion map and 𝜙 = (1 ⊗𝑓 )∘𝜙 , we have
𝑝𝑖𝑗 𝑝𝑖𝑗 𝑝𝑖𝑗 𝑉ℎ(𝑝𝑖𝑗) 𝑝𝑖𝑗 𝑝𝑖𝑗
˜
𝜙 : 𝑉 → 𝑉 ⊗𝑀,
𝑝𝑖 𝑡(𝑝𝑖) ℎ(𝑝𝑖)
𝑗 𝑗 𝑗
for 𝑗 = 1,...,𝑛
𝑖
A twisted representation (𝑉,𝜙) satisfies the relations if 𝜆𝑖𝜙 + ... +
1 𝑝𝑖
1
𝜆𝑖 𝜙 = 0, for 𝑖 = 1,⋅⋅⋅ ,𝑚. We have again a new category, the category
𝑛𝑖 𝑝𝑖𝑛𝑖
of 𝑀−twisted representations of the quiver 𝑄 with relations.
3.2 Twisted representations of quivers
Let 𝑄 be a quiver, 𝒜 be the category of finite dimensional vector spaces
over a field 𝑘 and let 𝑀 = {𝑀 },𝑎 ∈ 𝑄 , be a collection of objects of 𝒜.
𝑎 1
Let Rep (𝑄) denote the category of 𝑀−twisted representations of 𝑄 in the
𝑀
category of 𝑘-vector spaces. The main result of this section is to relate, as
we mentioned before, the categories of twisted 𝑘-linear representations of a
quiver with a category of 𝑘-linear representations of another quiver.
Theorem 4. The category Rep (𝑄) of 𝑀−twisted 𝑘-linear representations
𝑀
˜ ˜
of 𝑄 is equivalent to the category Rep(𝑄) of 𝑘-linear representations of 𝑄,
˜
where 𝑄 is obtained from 𝑄 in the following way;
˜
∙ the set of vertices is the same, that is, 𝑄 = 𝑄 ;
0 0
˜
∙ for each arrow 𝑎 ∈ 𝑄 , 𝑄 possesses 𝑚 = dim𝑀 arrows, 𝑎 ,⋅⋅⋅ ,𝑎 ,
1 1 𝑎 1 𝑚
such that 𝑡˜(𝑎 ) = 𝑡(𝑎) and ℎ˜(𝑎 ) = ℎ(𝑎), where 𝑡˜, ℎ˜ are the tail and
𝑗 𝑗
˜
head maps of 𝑄.
˜
Proof. We will define a functor from Rep (𝑄) to Rep(𝑄) and show that it
𝑀
is an equivalence of categories. First, consider the case of the quiver 𝐴 ,
2
consisting of two vertices and one arrow between them:
9
1 𝑎 (cid:47)(cid:47)2.
The proof for the general (finite) quiver will follow from performing the
construction below for each individual arrow of the quiver.
Fix 𝑀 a 𝑘−vector space, dim𝑀 = 𝑚. We show that 𝒞 = Rep (𝐴 ) is
𝑎 𝑎 𝑀 2
equivalent to 𝒟 = Rep(𝐴˜), where 𝐴˜ is the quiver consisting of two vertices
2 2
and 𝑚-arrows (𝑎 ,...,𝑎 ) between them, all pointing in the same direction.
1 𝑚
Let
({𝑉,𝑊},{𝜙 : 𝑉 → 𝑊 ⊗𝑀 }) be a 𝑀-twisted representation of 𝐴 . Let
𝑎 2
{ℎ ,...,ℎ } be a basis for 𝑀 . Since Hom(𝑉,𝑊 ⊗𝑀 ) ≃ Hom(𝑉,𝑊)⊗𝑀 ,
1 𝑚 𝑎 𝑎 𝑎
for any 𝜙 ∈ Hom(𝑉,𝑊 ⊗𝑀 ), there are 𝜙 ,⋅⋅⋅ ,𝜙 ∈ Hom(𝑉,𝑊) such that
𝑎 1 𝑚
𝑚
∑
𝜙 = 𝜙 ⊗ℎ . (1)
𝑖 𝑖
𝑖=1
˜
Fix an order in the set of arrows between two vertices of the quiver 𝐴
2
and consider the functor
˜
F : Rep (𝑄) → Rep(𝑄)
𝑀
such that
∙ for each object 𝑋 = ({𝑉 ,𝑉 },𝜙) ∈ Rep (𝐴 )
𝑡(𝑎) ℎ(𝑎) 𝑀 2
˜
F(𝑋) = F({𝑉 ,𝑉 },𝜙) = ({𝑉 ,𝑉 },𝜙)
𝑡(𝑎) ℎ(𝑎) 𝑡(𝑎) 𝑡(𝑎)
˜
where 𝜙 = {𝜙 ,...,𝜙 } are obtained as in (1) and 𝜙 is associated to
1 𝑚 𝑛
˜
the 𝑛−th arrow of 𝐴 ;
2
∙ for each morphism 𝑓 = {𝑓 ,𝑓 } between 𝑋 = ({𝑉 ,𝑉 },𝜙) and
𝑡(𝑎) ℎ(𝑎) 𝑡(𝑎) ℎ(𝑎)
𝑌 = ({𝑊 ,𝑊 },𝜓) we define F(𝑓) = 𝑓.
𝑡(𝑎) ℎ(𝑎)
It is easy to see that 𝑓 = {𝑓 ,𝑓 } is a morphism between the repre-
𝑡(𝑎) ℎ(𝑎)
sentations ({𝑉 ,𝑉 },{𝜙 ,...,𝜙 }) and ({𝑊 ,𝑊 },{𝜓 ,...𝜓 }), so
𝑡(𝑎) ℎ(𝑎) 1 𝑚 𝑡(𝑎) ℎ(𝑎) 1 𝑚
F is a functor. Now we must show that F is an equivalence of categories.
Let us show that F is fully faithful functor and essentially surjective.
1. F : Hom (𝑋,𝑌) → Hom (F(𝑋),F(𝑌)) is injective and surjective.
𝒞 𝒟
Clearly F is injective. Let 𝑔 ∈ Hom (F(𝑋),F(𝑌)), 𝑔 = {𝑔 } such
𝒟 𝑖 𝑖∈𝑄0
that
𝜓 ∘𝑔 = 𝑔 ∘𝜙 , 𝑖 = 1,...,𝑚.
𝑖 𝑡(𝑎) ℎ(𝑎) 𝑖
Is easy to see that 𝜓𝑔 = (𝑔 ⊗ 1 )𝜙. Then 𝑓 = {𝑔 } ∈
𝑡(𝑎) ℎ(𝑎) 𝑀𝑎 𝑖 𝑖∈𝑄0
Hom (𝑋,𝑌) is such that F(𝑓) = 𝑔. Therefore F is fully faithful.
𝒞
10