数理解析研究所講究録
第 1992 巻 2016 年 151-161

151

Categorification of Coxeter groups andbraid groups
名古屋大学多元数理科学研究科水野有哉
Yuya Mizuno
Graduate School of Mathematics, Nagoya University

1. $INTRODUC’$rION
mainly a summary of [$M$ , AM], where we discuss preproof Dynkin type. Preprojective algebras first appeared inof
the work Gelfand-Ponomarev [GP]. Since then, they have been one of
the important objects in the representation theory of algebras and they also
appear in many branches of mathematics such as quantum groups.
Recently, the notion of support -tilting modules was introduced in [AIR],
as a generalization of tilting modules. Support -tilting modules have several
nice properties. For example, it is shown that there are deep connections
between -tiltihg theory, torsion theory, silting theory and cluster tilting
theory. Moreover, support -tilting modules over selfinjective algebras areuseful to provide tilting complexes. It is therefore fruitful to investigate
these remarkable modules for preprQjective algebras. To explain our results,
we give the following set-up.
Letbe a finite connected graph (without loop) with the set$\{1, \cdots, n\}$ of vertices, A the preprojective algebra of
. and
the
sided
ideal of A generated by $1-e_{i}$ , where is an idempotent of A correspondingto. We denote by , . .. , the set of ideals of of the form
for some $k\geq 0$ and , . .. ,These ideals are quite useful to study structure of categories [IR, BIRS,
AIRT, ORT]. They also play important roles in $Geiss-Leclerc-Schr\ddot{\circ}er$ ’s construction of cluster monomials of certain types of cluster algebras [GLSI,
GLS2] and Baummn-Kamnitzer-Tingley’s works of MV polytopes [BK, BKT].
One of the results in this paper is to show that elements of , . . . ,are support -tilting modules over preprojective algebras of Dynkin type,
and they are bijective to the elements of the Coxeter group.
Another aim is to study tilting complexes. It is known that derived equivalences are controlled by tilting complexes [Ric] and therefore these objects
have been extensively studied. By applying the above result, we give aclassification of tilting complexes.

This paper
jective algebl$is$$\mathfrak{W}$$\tau$$\tau$$\tau$$\tau$$\Delta$$\Delta_{0}=$$\Delta$$tworightarrow$$I_{i}$$e_{i}$$i\in\Delta_{0}$$i_{1}$$I_{i_{1}}I_{i_{2}}\cdots I_{i_{k}}$$\Lambda$$I_{n}\rangle$$\langle I_{1}$$i_{k}\in\Delta_{0}.$$\langle I_{1}$$I_{n}\rangle$$\tau$

Notation. Let $K$ be an algebraically closed field and $D$ $:=Hom_{K}$All modules are right modules. For a finite dimensional algebra , we denoteby mod A the category of finitely generated -modules.
$K\rangle.$$\Lambda$$\Lambda$

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2. PRELIMINARIES
2.1. Support -tilting modules. We recall the definition of support
tilting modules. We refer to [AIR] for the details about support -tilting
modules. Let be a finite dimensional algebra and denote the AR translation [ARS].
$\tau-$$\tau$$\tau$$\Lambda$$\tau$(a) We call $X$ in
Definition 2.1.
-rigid if $Hom_{\Lambda}(X, \tau X)=0.$$X$(b) We call in-tilting respectively, almost complete tilting)
if $X$ is -rigid and $|X|=|\Lambda|$ respectively, $|X|=|\Lambda|-1)$ , where$|X|$ denotes the number of non.isomorphic indecomposable direct
summands of $X.$(c) We call $X$ in
support -tilting if there exists an idempotent
of A such that $X$ is a -tilting-module.
(d) We call a pair $(X, P)$ of $X\in mod$ A and $P\in$ proj
-rigid if $X$ is-rigid and $Hom_{\Lambda}(P, X)=0.$(e) We call a -rigid pair $(X, P)$ a support -tilting (respectively, almost
complete support -t\’ilting) pair if $|X|+|P|=|\Lambda|$ (respectively, $|X|+$$mod \Lambda\tau$$mod \Lambda\tau$$($$\tau-$$($$\tau$$mod \Lambda$$e$$\tau$$(\Lambda/\langle e\rangle)$$\tau$$\Lambda\tau$$\tau$$\tau$$\tau$$\tau$$|P|=|\Lambda|-1)$.

We call $(X, P)$ basic if $X$ andare basic, and we say that $(X, P)$ is$X$
a direct summand ofif
is a direct summand of $X’$ andis adirect summand of $P’$ . Note that $(X, P)$ is a -rigid (respectively, support-tilting) pair for
if and only if $X$ is a -r\’igid respectively, tilting)-module, where is an idempotent of A such that add $P=$ add[AIR, Proposition 2.3]. Moreover, if $(X, P)$ and (X,are support -tilting$P’$pairs for , then we get add $P=add$ , Thus, a basic support -tilting
module $X$ determines a basic support -tilting pair $(X, P)$ uniquely andwe can identify basic support -tilting modules with basic support -tilting
pairs.
We denote by -tiltA the set of isomorphism classes of basic support-tilting -modules.
Next we recall some properties of support -tilting modules. The set of
support -tilting modules has a natural partial order as follows.
$P$

$P$

$(X\prime, P’)$$\tau$$\Lambda$$\tau$$(\Lambda/\langle e\rangle)$$($$\tau$$\tau-$$e\Lambda$$e$$P’\rangle$$\tau$$\Lambda$$\tau$$\tau$$\tau$$\tau$$s\tau$$\Lambda$$\tau$$\tau$$\tau$

Definition 2.2. [AIR, Theorem 2.18] Let
For$T,$ $T’\in s\tau$$\Lambda$

be a finite dimensional algebra.

-tilt , we write$\Lambda$$T’\geq T$

if Fac$T’\supset$

FacT. Then$\geq$

gives a partial order on$s\tau$-tilt$\Lambda.$

Then we give the following results, which play important roles in thispaper.
Definition-Theorem 2.3. [AIR, Theorem 2.28] Let be a finite dimensional algebra. Then(i) any basic almost support -tilting pair $(U, Q)$ is a direct summand ofexactly two basic support -tilting pairs $(T, P)$ and $(T’,$Moreover, we have $T>T’$ or $T<T’.$$\Lambda$$\tau$$\tau$$P$

153
Under the above setting, let $X$ be an indecomposable -module satisfying
either $T=U\oplus X$ er $P=Q\oplus X$ . We write $(T’, P’)=\mu$ く $x,0$) $(T, P)$ if $X$ is adirect summand of and $(T^{I}, P^{1})=\mu_{(0,X)}(T, P)$ if $X$ is a direct summandof $P$ , and we say that $(T’,P’)$ is a mutation of $(T, P)$ . In particular, we
say that $(T’, P’\rangle is a left$ mutat\’ion respectively, $right$ mutation) of $(T, P)$ if$T>T’$ respectively, $if T<T’)$ and write(respectively,$\mu^{+}(T, P\rangle=(T’, P By \langle i)$ , exactly one of the left mutation or right$\Lambda$

$T$

$($

$\mu^{\ovalbox{\tt\small REJECT}}(T, P)=(T’, P^{J})$$($. mutation occurs.

Now, assume that $X$ is a direct summand of and $T=U\oplus X$ . In
this case, for simplicity, we write a left mutation $T’=\mu_{\tilde{X}}(T)$ and a rightmutation $T’=\mu_{X}^{+}(T)$ .
$T$

Finally, we define the support -tilting quiver-tilt as follows.
The set of vertices is -tiltA.
Draw an arrow from to $T’$ if $T’$ is a left mutation of (i.e. $T’=$$\mathcal{H}(s\tau-$$\lrcorner r$$\Lambda)$$s\tau$$\bullet$

$T$

$T$

$\bullet$$\mu_{\tilde{X}}(T\rangle)$.

The following theorem relates } ( -tilt A) with partially orders of sr-tilt$t$$s\tau$Theorem 2,4. [AIR, Corollary 2.34] The support -tilting
is the Hasse quiver of the partially $0/dered$ set -tiltA.
$\tau$$quiver\mathcal{H}$$\Lambda.$( -tilt )$s\tau$$\Lambda$$s\tau$2.2. Preprojective algebras. In this subsection, we recall definitions andsome properties of preprojective algebras. We refer to [BBK, BGL, Ri] forbasic properties and background information.
Definition 2.5. Let
be a finite connected acyclic quiver with vertices$Q_{0}=\{1, \cdots , n\}$ . The preprojective algebra associated tois the algebra
$Q$

$Q$

$\Lambda=K\overline{Q}/\langle\sum_{a\epsilon Q_{1}}(aa^{*}-a^{*}a)\rangle$

where is the double quiver of , which is obtained from by adding foreach arrow $a:iarrow j$ inan arrow $a^{*}:iarrow j$ pointing in the opposite
direction.
$Q$$\overline{Q}$$Q$$Q_{1}$We remark that does not depend on the orientation of . Hence, for, where is aa graph , we define the preprojective algebra byquiver whose undrlying graph is . We denote byvertices ofLet be a Dynkin (ADE) graph. The preprojective algebra of is finite
dimensional and selfinjective. We denote the Nakayama permutation of
.
by
$Q$

$\Lambda$$\Delta$$Q$$\Lambda_{\Delta}=\Lambda_{Q}$$\Delta$$\Delta.$$\Delta_{0}$$\Delta$$\Delta$$\Lambda$$\iota:\Delta_{0}arrow\Delta_{0}$$(i.e. D(\Lambda e_{\iota(i\rangle})\cong e_{i}\Lambda)$

to $F$ . The
2.3. Coxeter group. Let
be a Dynkin graph of type
Coxeter groupassociated to is defined by the generatorsand relations $(s_{i}s_{j})^{m(i,j)}=1$ , where$A$$\Delta$$W_{\Delta}$$\Delta$$s_{i}(i\in\Delta_{0})$154

1$m(i,j):=$if $i=j$ ;$\{\begin{array}{ll}2 if no edge between i and j;3 if there is an edge i-j,4 if there is an edge ij\underline{4}.\end{array}$

Each element
can be written in the form
. If,
is minimal among all such expressions forthen is called the length ofand we denote by $l(w)=k$ . In this case, we calla reduced
expression of $w.$
Let be a permutation of . Then acts on an element of the Coxeter
group
by
. We
for
by
of
define the subgroup$w\in W_{\Delta}$$k$$w=s_{i_{1}}\cdots s_{i_{k}}$

$k$

$w$

$w$

$s_{i_{1}}\cdots s_{i_{k}}$$\Delta_{0}$$\iota$$W_{\Delta}$$\iota$$\iota(w):=s_{\iota(i_{1})}s_{\iota(i_{2})}\cdots s_{\iota(i_{\ell})}$$w=\mathcal{S}_{i_{1}}S_{i_{2}}\cdots \mathcal{S}_{i\ell}\in W_{Q}$$W_{\Delta}$$W_{\Delta}^{\iota}$$W_{\Delta}^{\iota}:=\{w\in W|\iota(w)=w\}.$

Then the following result is well-known.

be a Dynkin $(A,D,E)$ quiver andTheorem 2.6. Let
the Coxeter$Otherwise_{f}$group of A. Let
type
is
and
let
be
if
a quiver, respectively, given by the following type.
$\Delta$$W_{\Delta}$$\Delta’=\Delta$$\Delta$$D_{2n},$ $E_{7}$Then, for the Nakayama permutation
is isomorphic to$\iota$$\Delta’$$E_{8}.$of the preprojective algebra of$\Delta,$$W_{\Delta’}.$$W_{\Delta}^{\iota}$

3. PREPROJECTIVE ALGEBRAS AND THE COXETER GROUPS

Let A be a Dynkin graph withand the preprojective
algebra of . We denote by. We denote byfor, . .. ,the set of ideals of A which can be written as$\Delta_{0}=\{1, .

$\Delta$$\langle I_{1}$$I_{i}$.

.

$:=\Lambda(1-e_{i}\rangle\Lambda$$\Lambda$, n\}$$i\in\Delta_{0}$$I_{n}\rangle$$I_{i_{1}}I_{i_{2}}\cdots I_{i_{k}}$for some $k\geq 0$ and , .. . ,The following lemma plays a key role.
$i_{1}$

Lemma 3.1. Let

of

$T$

$T\in\langle I_{1}$$i_{k}\in\Delta_{0}.$,...,$I_{n}\rangle$:. If $I_{i}T\neq T$ , then there is a left mutation$\mu_{e_{i}T}^{-}(T)\cong I_{i}T.$

Moreover we recall the following important result.
Theorem 3.2. [IR, BIRS] There exists a bijection
given byfor any reduced expression$W_{\Delta}arrow\langle I_{1}$$w\mapsto I_{w}=I_{i_{1}}I_{i_{2}}\cdots I_{i_{k}}$, .. . ,$I_{n}\rangle$. It is$w=s_{i_{1}}\cdots s_{l_{k}’}.$

Then using Lemma 3,1 and Theorem 3.2, we obtain the following result.
Theorem 3.3. The map in Theorem 3.2 gives a bijection between the elements of the Coxeter groupand -tiltA.
$W_{\Delta}$$s\tau$155
We remark that the above ideals
non-Dynkin type [IR, BIRS].

$I_{w}$

are tilting modules in the case of

Example 3.4. (a) Let be the preprojective algebra of typecase, $H$ ( -tilt ) is given as follows.
$\Lambda$$ST$$A_{2}$. In this$\Lambda$

Here we represent modules by their radical filtrations and we write a
direct sum $X\oplus Y$ by $XY$ . For example, 1 denotes the support tilting, whereis the simple module associated with the vertex
module1.
(b) Let be the preprojective algebra of type . In this case, ( -tilt )given
isas follows.
$\tau*$$2^{}$$e_{1}\Lambda\oplus S_{1}$$\Lambda$$S_{1}$$A_{3}$$\mathcal{H}$$s\tau$$\Lambda$

156

32
321

$s\tau$Morever we study a close relationship between partial orders of-tiItA. This lemma is crucial.

Lemma 3.5. Let$w\in W_{\Delta}$

and$W_{\Delta}$

and$i\in\Delta_{0}.$$I_{i}I_{w}=I_{sw}:\subsetneq I_{w}$and we have a left mutation$I_{i}I_{w}=I_{w}\subsetneq I_{\epsilon_{i}w}$

and we have a right mutation(i) If $l(w)<l(s_{i}w)$ , then$\mu_{i}^{-}(I_{w}, P_{w})$ ,(ii) If $l(w)>l(s_{i}w\rangle$ , then$\mu_{i}^{+}(I_{w}, P_{w})$.

We denote by $\leq the$ (left) weak order ofand byquiver induced by weak order on
Then, by Lemma 3.5, we have the following result $|M$ ].
$W_{\Delta}$$\mathcal{H}(W_{\Delta},\underline{<})$

the Hasse$W_{\Delta}.$

Theorem 3.6. The bijection$W_{\Delta}arrow s\tau$morphism ofsets

partially $0/dered$-tiltA in Theorem 3.2 gives an isoand$(W_{\Delta}, \leq)$$(s\tau- tilt\Lambda, \leq)^{op}.$157
4. $SILTING-msCRE’$rENESS

In this section, we discuss some properties of silting complexes. First we
recall the notion of silting complexes.

4.1. Silting complexes. Silting complexes are a generalization of tilting
complexes, which were introduced by Keller-Vossieck [KV]. They were originally invented as a tool for studying tilting complexes. Nonetheless, silting
complexes have turned out to have deep connections with several importantcomplexes such as -structures [KY, BY].
We recall the definition of silting complexes as follows.
projA) the
Definition 4.1. Let be a finite dimensional algebra andbounded homotopy category of the finitely generated projective -modules.
$t$

$A$

$K^{b}($$A$

Let $T:=K^{b}(projA\rangle$ for simplicity.
(a) We call a complex in (or in the derived category of $mod A$ ) is$=0$
presigting (respectively, pretilting) if it satisfies
for any $i>0$ respectively, $i\neq 0)$ .
(b) We call a complex $P$ in
silting (respectively, tilting)) if it is presiiting (respectively, pretilting) and the smallest thick subcategorycontaining $P$ is
We denote by silt (respectively, tilt
the set of non-isomorphic basic
silting (respectively, tilting) complexes in
For complexes $P$ and of , we write $P\geq U$ if $Hom_{\mathcal{T}}(P, tJ[i])=0$ for
any $i>0$ . Then the relation gives a partial order on silt [AI, Theorem$P$$\mathcal{T}$$Hom\mathcal{T}(P_{\}} P[\’{i}])$$($$\mathcal{T}$$\mathcal{T}.$$A$$A\rangle$$\mathcal{T}.$$U$$\mathcal{T}$$A$$\geq$

2.11].

Moreover, a complex
is called 2-term provided it concerned in
the degree and-l. We denote by 2-silt respectively, tilt $A)$ the subset
of silt
respectively, tilt $A)$ consisting of 2-term complexes. Note that acomplex is 2-term if and only if $A\geq T\geq A[1].$Then we give the definition of silting-discrete triangulated categories asfollows.
(a) We call silting-discrete if the set $\{T\in$ silt
Definition 4.2.
$T\geq A[P]\}$ is finite for any $P>0$ . Similarly, we calltilting-discrete$T\in$$A\geq
T\geq
A[\ell]$$l>0.$
any
the
is
set {
if
tiltT}} finite for(b) For a silting complex of , we denote by 2 siltp the subset of$si1\{T$ such that-silting-finite ifwith $P\geq U\geq P[1]$ . We call
2 si
is a finite set for any silting complex of . Similarly, we
denote by 2-ti
the subset of tilt such that with $P\geq U\geq$$is$$P\in \mathcal{T}$

$A$

$O$

$A$

$2-$

$($

$($

$P$

$\mathcal{T}$$\mathcal{T}|A\geq$$\mathcal{T}$$P$$\mathcal{T}$$\mathcal{T}$$U$$\mathcal{T}$$P$$\prime t_{P}\mathcal{T}$$1t_{P}\mathcal{T}$$2$$\mathcal{T}$$U$$\mathcal{T}$$P[1].$

Moreover we recall mutation for silting complexes [AI, Theorem 2.31].
Definition 4.3. Let $P$ be a basic silting complex of and decompose it as$P=X\oplus M$ . We take a triangle$\mathcal{T}$$Xarrow^{f}M^{1}arrow Y\sim X[1]$158
with a minimal left (add $M$ )-approximation of $X$ . Then $\mu_{X}^{-}(P):=Y\oplus M$
is again a silting complex, and we call it the left mutation of with respect
to $X$ . Dually, we define the right mutation. Mutation will mean$X$
is indecomposable, then we say thateither left or right mutation. If
mutation is irreducible. In this case, we haveand there is no
silting complex satisfying $P>Q>\mu_{\overline{X}}(P)$ [AI, Theorem 2.35].
Moreover, if andare tilting complexes, then we call it the (left)
tilting mutation. In this case, if there exists no non-trivial direct summand$X’$ of $X$ such thatis tilting, then we say that tilting mutation isirreducible.
$f$

$P$

$\mu_{X}^{+}(P)$$P>\mu_{\overline{X}}(P)$

$Q$

$P$

$\mu_{X}^{-}(P)$$\mu_{X}^{-},(T)$The following theorem play a key role.
projA) .
Theorem 4.4. [AM] Let be a finite dimensional algebra and
The following are equivalent.
(a)is sitting-discrete.
(b)
is 2-silting-finite.
(c)
is a finite set for any silting complex $P$ which is given by
iterated irreducible left mutation from $A.$$A$$\mathcal{T}:=K^{b}($$\mathcal{T}$$\mathcal{T}$$2-si\ovalbox{\tt\small REJECT} t_{P}\mathcal{T}$Moreover if$A$

is selfinjective, we have the following result.

Corollary 4.5. Assume that
following are equivalent.

$A$

is selfinjective and let$\mathcal{T}:=K^{b}($projA) . The(a)is tilting-discrete.
(b)
is 2-tilting-finite.
(c) 2-siltp is a finite set for any tilting complex $P$ which is given by
iterated irreducible tilting left mutation from $A.$$\mathcal{T}$$\mathcal{T}$$\mathcal{T}$5. PREPROJECTIVE ALGEBRAS AND THE BRAID GROUPS
Using the previous results, we study tilting complexes over the preprojective algebra of Dynkin type.
First we recall the following nice correspondence,

Theorem 5.1. [AIR, Theorem 3.2] Let be a finite dimensional algebra.
There exists a bijection
tiltA-silt A.
$A$

$rightarrow 2$$s\tau-$By the above correspondence, we can give a description of 2-term silting
complexes by calculating support -tilting modules, which is much simpler
than calculations of silting complexes.
Rom now on, let be a Dynkin graph and A the preprojective algebraof . Then, as a consequence of Theorem 3.3 and 5.1, we have the following
corollary.
$\tau$$\Delta$$\Delta$

Corollary 5.2. We have a bijection$W_{\Delta}rightarrow 2$-silt$\Lambda.$

159
Thus we can parameterize 2-term silting complexes by the Coxeter group.
Moreover, we can describe 2-term tilting complexes in terms of the Coxeter
group as follows,
Proposition 5.3. Let $v:=DHom_{\Lambda}$) the Nakayama functor of and:
the Nakayama permutation of A. Then
if and only$\Lambda$$\Lambda$$\iota$$\nu(I_{w})\cong I_{w}$$\Delta_{0}arrow\Delta_{0}$if $\iota(w)=w$ . In particular, We have a bijection$W_{\Delta}^{\iota}rightarrow 2$-tilt A.

Then, by Theorem 2,6, we can understandCoxeter group.

as another type of the$W_{\Delta}^{\iota}$

Example 5.4. Let be a Dynkin graph of type
and A the preprojective
algebra of . Then the support -tilting quiver of is given as follows.
$\Delta$$A_{3}$$\Delta$$\Lambda$$\tau$

The framed modules indicate -stable modules $(i.e. I_{w}\cong\nu(I_{w}))$ , whichis equivalent to say that $\iota(w)=w.$$v$

Let $\Lambda=X\oplus Y$ . We denote byof A with respect to $X.$$\mu_{X}^{-}(A)$Proposition 5.5.
thatof A. Then we have an isomorphism$\mathcal{A}ssume$$\mu_{\vec{X}}(\Lambda)$

the irreducible tilting left mutation

is an irreducible tilting left mutation$End_{K^{b}(proj\Lambda\rangle}(\mu_{X}^{-}(\Lambda))\cong A.$

160
In particular, by Corollary 4.5,$\Lambda$

is tilting-discrete.

Consequently, we extend Proposition 5.3 and obtain the following result.

Theorem 5.6. We denote the braid group by$B_{\Delta’}rightarrow$$B_{\Delta’}$. Then we have a bijection

tilt A.

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