Description of the derived categories of tubular algebras
in terms of dimension vectors
Hongbo Lv, Zhongmei Wang
School of Mathematical Science, University of Jinan, Jinan 250022, P.R.China
School of Control Science and Engineering, Shandong University, Jinan 250061, P.R.China
Abstract In this paper, we give a description of the derived category of a tubular algebra by calculating the dimension
vectors of the objects in it.
Keywords:derived categories; tubular algebra; dimension vectors
1. Introduction
Let /
/
be a basic connected algebra over an algebraically
closed field k. We denote by mod/
/
the category of all
finitely generated right /
/
-modules and by ind /
/
a full
subcategory of mod /
/
containing exactly one
representative of each isomorphism class of
indecomposable /
/
-modules. For a /
/
-module M, we
denote the dimension vector by dim M. The bounded
derived category of mod /
/
is denoted by Db(
/
/
). We
denote the Grothendieck group of /
/
by K0(/
/
), Auslander-
Reiten translation by W
W
, the Cartan matrix by
C
/
/
. Let

/
/
be the repetitive algebra of /
/
, mod

/
/
the stable module
category. When the global dimension of /
/
is finite,
C
/
/
is
invertible by [1], and Db( /
/
) is equivalent to mod

/
/
as
triangulated categories by [2].
By [1], a tubular extension A of a tame-concealed
algebra
of extension type T = (2, 2, 2, 2), (3, 3, 3), (4, 4, 2) or (6, 3,
2)
is called a tubular algebra. For example, the canonical
tubular algebras of T (2, 2, 2, 2) is determined by the
following quiver with relations.
By [1], global dimension of a tubular algebra A is 2, then
Db(A) is equivalent to mod
l
A
. And tubular algebras of the
same extension type are tilt-cotilt equivalent, see [3]. Then
we only consider the derived categories of canonical tubular
algebras, whose structures are given in [4].
 ()
b
r
rQ
DA T


where (1) for anyrෛ  Q, 7 r is the standard stable P1(k)-
tubular family of type T;
(2) for any r ෛQ, 7 r is separating
s
sr
T

 from u
ruT 
 .
Based on the results above, we give a description of the
derived category of a canonical tubular algebra by
calculating the dimension vectors of the objects in it.
2. Description of The Derived
Categories of Tubular Algebras In
Terms Of Dimension Vectors
In this section, let A be a canonical tubular algebra of
type T.
Definition 1.1. ([1]) Let n be the rank of Grothendieck
group K0(A),
A
C
the Cartan matrix of A. Then
(1) The Coxeter matrix A
) is defined by
T
AA
CC



(2) The quadratic form A F in
n
] is defined by
1
1
() ()
2
TT
AAA
CC FD DD


for any D
in
n
]
.
(3) Let 0,hh
fbe the positive generators of rad A
F . For an
A෥module M, define
0
(dim )
() (dim )
lM
indexMlM
f

where
00
(dim )(dim ),
(dim )(dim ).
TT
A
TT
A
lMhC M
lMhC M


ff
In particular, for any
00
rad , ,
Arhr h
DFD
ff
 
where
0
0
,. Then, index().
r
rr r D
f
f
 ]
Open Journal of Applied Sciences
Supplement：2012 world Congress on Engineering and Technology
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(4) rad
A
DF  is called a real (respectively, imaginary)
root,
if
1
1
()() 1
2
TT
AAA
CC FD DD


(respectively, = 0).
It is well known that there exists a ”minimal ” imaginary
root G
such that rad
.
A
FG ]
Now we recall some results in [4]. Let /
be a finite
dimensional k෥algebra and
l
/ the repetitive algebra.
Denote
by
l
()P/
the subgroup of
l
0
()K/
generated by the
dimension vectors of indecomposable projective
l
/
෥
modules.
Lemma 1.2.
ll
00
()() ().KKP/ //
Definition 1.3. Let
l
00
:() ()KK S
/
/o / be the
projective morphism. Define
l
0
dim: mod()K
//o /
where for any
l
/ -module X, dim(dim ).XX S
/
/
Lemma 1.4. Let / )be the Coxeter matrix of
/
, W

the
Auslander-Reiten translation of
l
/ . Then
dim(dim ).XX W
/
/
)

Note that if
/
has finite global dimension, we have a
triangulated equivalence:
l
:()mod
b
D
K
/o / . For an
object
(), define dim(1)dim.
bii
i
XDX X/ 
¦
<<
Then we have
Lemma 1.5. dimdim ().XX K
/
<<
By representation theory of Auslander-Reiten quivers in
[5]
and the results above, we have a method to describing the
derived category of a canonical tubular algebra in terms of
dimension vectors.
Theorem 1.6. Let A be a canonical tubular algebra of
type
T, the rank of 0()KA
be n. Then
(1) Let

G be the minimal imaginary root in
l
0()KA
corresponding the P1(k)- tubular family r T, and let

dim ().
A
GG Then G
is determined by () 0.
A
FG
(2) Let X be an object in the bottom of a tube of rank r in
r
T. Then dimAX
is determined by the following:
1
1
1
(dim )dim ()(dim )1
2
()
dim(dim )(dim ).
AA A
TT
AAA
AA A
r
AA
XXCCX
XX X
F
G


 
°
®
°))
¯"
Proof. (1) By [4], 00rad ,
A
rhr h GF
ff
and thus
() 0.
A
FG Since
0
index( ),
r
r
G
f
_
where
0,,rr f] and 0
(,) 1,rr
f
it suffices to calculating
0 and .rr f
(2) Directly from Lemma 1.4 and 1.5.
Example 1.7. Now let A be a canonical tubular algebra of
type T(2, 2, 2, 2). The Cartan matrix and Coxeter matrix are
as following:
111112 111112
01000101 0001
0010010 01001
,.
0001010 00101
000011 00001 1
000001000001
AA
C

§·§ ·
¨¸¨ ¸

¨¸¨ ¸
¨¸¨ ¸

)
¨¸¨ ¸

¨¸¨ ¸
¨¸¨ ¸

¨¸¨ ¸
¨¸¨ ¸
©¹© ¹
For each object
l
mod ,XA
denote
12 6
dim( ,,,),
AXxx x "
Then
2
516
2
(dim)() .
2
A
Ai
i
xx
Xx F


¦
(1)Description of the minimal imaginary root G .
Case 1. If
01(mod2)rr f
{
100000
(2,,,,,2).rr rr rr rr rr G
fffff

Case 2. If 00(mod 2)rr
f
{
0000
20
(,,,,,).
2222
rrrrrrrr
rr G
ffff
f

(2) Description of dimAX where X is an object in the
bottom of a tube of rank 2.
If 10 ,that is 1(mod2),rr GG
f
{
(ѽ) in Theorem
1.6 should be as follows:
65 5
2
1616
12 2
5
60
2
5
160
2
5
1
2
21
22
22
iii
ii i
i
i
i
i
i
i
xxxxxxx
xx r
xxx rr
xxr
f
f

°
°
°
°
°
®
° 
°
°
°
°
¯
¦¦ ¦
¦
¦
¦
Then,
2
50
2
()1.
2
i
i
rr
xf


¦
case 1. We have four different tubes of rank 2.
(i)
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00
0
00
11
dim(, , ,
22
11
,,)
22
Arr rr
Xr
rrrr r
ff
ff
f
 
 
00
0
00
11
dim(,,,
22
11
,,)
22
A
rr rr
Xr
rrrr r
W
ff
ff
f
 
 

(ii)
00
0
00
11
dim( ,,,
22
11
,,)
22
Arr rr
Xr
rrrr r
ff
ff
f
 
 
00
0
00
11
dim(,,,
22
11
,,)
22
A
rr rr
Xr
rrrr r
W
ff
ff
f
 
 

(iii)
00
0
00
11
dim( ,,,
22
11
,,)
22
Arr rr
Xr
rrrr r
ff
ff
f
 
 
00
0
00
11
dim(,,,
22
11
,,)
22
A
rr rr
Xr
rrrr r
W
ff
ff
f
 
 

(iv)
00
0
00
11
dim(1,,,
22
11
,,1)
22
A
rr rr
Xr
rrrr r
ff
ff
f
 

 
00
0
00
11
dim(1,, ,
22
11
,,1)
22
A
rr rr
Xr
rrrr r
W
ff
ff
f
 

 

If
2
, GG
(ѽ) in Theorem 1.6 should be as follows:
65 5
2
1616
12 2
5
60
2
50
16
2
5
1
2
21
2
2
2
iii
ii i
i
i
i
i
i
i
xxxxxxx
xxr
rr
xxx
xxr
f
f

°
°
°
°
°
®
°
°
°
°
°
¯
¦¦ ¦
¦
¦
¦
Then,
2
50
2
()1.
4
i
i
rr
xf


¦
case 2. When r0+r1 ิ2 (mod 4), we have four different
tubes of rank 2.
(i)
00 0
00
122
dim( ,,,
24 4
22 1
,,)
442
A
rrrrr
X
rr rr r
ff
ff f

 
00 0
00
122
dim( ,,,
24 4
22 1
,,)
442
Arrr rr
X
rr rr r
W
ff
ff
f
 
 

(ii)
000
00
122
dim( ,,,
24 4
22
1
,,)
442
A
rrr rr
X
rr rr r
ff
ff
f
 
 
00 0
00
122
dim(,,,
24 4
22 1
,,)
442
A
rrr rr
X
rr rr r
W
ff
ff f
 
 

(iii)
000
00
122
dim(,,,
24 4
22 1
,,)
442
A
rrr rr
X
rr rr r
ff
ff f
 
 
00 0
00
122
dim(,,,
24 4
22
1
,,)
442
Arrr rr
X
rr rr r
W
ff
ff
f
 
 

(iv)
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000
00
122
dim(,,,
24 4
22 1
,,)
442
A
rrr rr
X
rr rr r
ff
ff f
 
 
00 0
00
122
dim(,,,
24 4
22 1
,,)
442
A
rrr rr
X
rr rr r
W
ff
ff f
 
 

case 3. When r0+r1 ิ0 (mod 4), we have four different
tubes of rank 2.
(i)
00 0
00
1
dim(,,,
24 4
41
,,)
442
A
rrrrr
X
rrrrr
ff
ff f
 
 
00 0
00
1
dim(,,,
244
41
,,)
442
A
rrrrr
X
rrrrr
W
ff
ff f
 



(ii)
00 0
00
1
dim(,,,
24 4
41
,,)
442
Arrrrr
X
rr rr r
ff
ff f
 
 
00 0
00
1
dim(,,,
244
41
,,)
442
A
rrrrr
X
rr rr
r
W
ff
ff
f
 
 

(iii)
00 0
00
14
dim( ,,,
24 4
1
,,)
442
A
rrrrr
X
rrrr
r
ff
ff f
 

00 0
00
14
dim( ,,,
24 4
1
,,)
442
A
rrrrr
X
rrrr
r
W
ff
ff f
 


(iv)
00 0
00
14
dim( ,,,
24 4
1
,,)
442
A
rrr rr
X
rrrr
r
ff
ff
f
 

000
00
14
dim( ,,,
24 4
1
,,)
442
A
rrr rr
X
rrrr r
W
ff
ff f
 


3. Acknowledgment
The authors would like to thank the referee for his or her
valuable suggestions and comments. The first-named author
thanks NSF of China (Grant No. 11126300) and of
Shandong Province (Grant No. ZR2011AL015 ) for support.
REFERENCES
[1] C.M.Ringel, Tame algebras and integral quadratic forms,
Lecture Notes in Math. 1099. Springer Verlag, 1984.
[2] D.Happel, Triangulated categories in the representation
theory of finite dimensional algebras, Lecture Notes series
119. Cambridge Univ. Press, 1988.
[3] D.Happel, On the derived category of a finite-
dimensional algebra. Comment. Math. Helv. 62(1987),
339_389.
[4] D.Happel, C.M.Ringel, The derived category of a
tubular
algebra. LNM1273, Berlin-Heidelbelrg-NewYork:
Springer-Verlag, 1986:156_180.
[5] W.Crawley-Boevey, Lectures on Representations of
Quivers. Preprint.
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