Strictly wild algebras with radical square zero. By. YANG HAN. Abstract. It is proved that a wild algebra with radical square zero is strictly wild if and only if it has ...
Arch. Math. 76 (2001) 95 ± 99 0003-889X/01/020095-05 $ 2.50/0 Birkhäuser Verlag, Basel, 2001
Archiv der Mathematik
Strictly wild algebras with radical square zero By YANG HAN Abstract. It is proved that a wild algebra with radical square zero is strictly wild if and only if it has a wild hereditary algebra as its factor algebra, which, on the one hand supports one conjecture in [3], on the other hand can be used to construct many nonstrictly wild algebras.
Throughout the paper k denotes a fixed algebraically closed field. By an algebra we mean an associative finite-dimensional k-algebra with an identity, which we shall assume (without loss of generality) to be basic and connected. By a module over an algebra A we mean a left A-module of finite k-dimension. Let A be an algebra and Q
Q0 ; Q1 be the Gabriel quiver of A, where Q0 is the set of vertices of Q and Q1 is the set of arrows of Q. Then we can write A kQ=I where I is an admissible ideal of path algebra kQ (See [2]). We denote by modA the category of finite-dimensional left A-modules. From now on we always denote by Kn
n ^ 3 the quiver with two vertices 1 and 2 and n arrows starting at 1 and ending at 2. Recall that A kQ=I is called an algebra with radical square zero if the composition of any two non-trivial paths in Q is in I, i.e. I k2 Q, where k2 Q denotes the ideal of kQ generated by the paths of length 2. A finite-dimensional algebra A is called wild if there is a finitely generated A-khx; yiÿbimodule M which is free as a right khx; yiÿmodule and such that the N functor M khx;yi ÿ from modkhx; yi to modA preserves indecomposability and isomorphism N classes. We say that A is strictly wild if in addition the functor M khx;yi ÿ is full. It is conjectured in [7] that all wild algebras are controlled wild. In [3], the author gave a covering criterion for a wild algebra to be controlled wild. According to that criterion, the problem ± when a wild algebra is controlled wild ± is transformed into two parts: one is when the wildness can be lifted to its covering, which has been studied for a long time; the other is when a wild algebra is strictly wild. About the second part, the author gave a conjecture ± an algebra is strictly wild if and only if it has a wild tilted algebra as its factor algebra. Indeed, combining the main results of [5] and [9], it follows that a tilted algebra is wild if and only if it has a wild concealed factor algebra. Therefore the author s conjecture can be reformulated as follows: Conjecture. An algebra is strictly wild if and only if it has a wild concealed factor algebra. In this note, we prove that the conjecture above is true for wild algebras with radical square zero. The main result is the following theorem. Mathematics Subject Classification (2000): 16G20, 16G60.
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Theorem. A wild algebra A with radical square zero is strictly wild if and only if A has a wild hereditary algebra as its factor algebra. The theorem above can also be used to construct many non-strictly wild algebras, which is the other motivation to do this work. We say that an algebra A is minimal strictly wild (resp. minimal wild) if A is strictly wild (resp. wild), but no proper factor algebra has this property. Denote by
modAf the full subcategory of modA consisting of the zero module and all faithful A-modules. Lemma 1. Let A be a minimal strictly wild algebra. Then
modAf is strictly wild. P r o o f. First of all, there is a fully faithful exact functor f : modkK3 ÿ! modkhx; yi, which is defined by sending
V1 ; V2 ; a; b; g to 0
2
0 1 0
0 0 0 0
3 2
0
0
0
0
0
0
0
0
0
0 0
31
C 0 07 7C 7C C 1 0 0 0 0 07 7C 7C s 1 0 0 0 0 7C; 7C C 0 d 1 0 0 07 7C 7C 0 0 0 b 1 0 0 5A 0 0 0 0 0 g0 1 0 0 0 0 0 1 0 ; ; d where the entries of two matrices are 2 2 matrices and s 0 1 0 0 0 0 0 0 0 0 a0 ; b0 and g0 in the case of dimk V1 dimk V2 j 0, and the a 0 b 0 g 0 entries of two matrices are 1 1 matrices and s 0; d 1 (resp. s 1; d 0), and a0 b0 g0 0, in the case of V1 0 (resp. V2 0). Moreover, there is also a fully faithful exact functor g : modkhx; yi ÿ! modkK3 which is defined by sending
V; x; y to
V; V; 1; x; y: Since A is strictly wild, there exists a fully faithful exact functor h : modkK3 ÿ! modA. Let I be the ideal of A consisting of all the annihilators of the A-module h
P1 where P1 is the projective kK3 -module corresponding to vertex 1. We claim that I is zero. Indeed, h
P1 and h
S2 , where S2 is the simple kK3 -module corresponding to vertex 2, belong to the full subcategory modA=I of modA. Since modA=I is closed under cokernels and direct sums, the image Im h fCoker
h
S2 a ÿ!h
P1 b h
S2 c ja; b; c 2 Ng, is contained in modA=I, too. Since A is minimal strictly wild, I 0 follows. Therefore h
P1 is a faithful A-module. Finally, it is not difficult to see that h
P1 is a submodule of both hgf
S1 and hgf
S2 . Hence hgf
S1 and hgf
S2 are faithful. Therefore, hgf defines a strictly wild h functor from modkK3 to
modAf . B 60 B 6 B 6 B 60 B 6 B 7 6 B
V1 V2 ; 6 0 B 6 B 60 B 6 B 6 @ 40 0
0 0 0 0
7 1 07 6 6 7 6 0 7 07 6 6a 7 6 0 7; 6 0 7 6 6 07 7 60 7 6 0 0 0 15 4 0 0 0 0 0 0
1 0 0 1 0 0 0 0
0 0 1 0
0 0 0 1
Lemma 2. Let A be a radical square zero algebra such that modA contains a faithful brick M. Then A is hereditary. P r o o f. Let Q be the Gabriel quiver of A. Since A is a radical square zero algebra, A is hereditary if and only if Q has sink-source orientation. Suppose that A is not hereditary.
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Then Q contains a vertex z which is neither a sink nor a source. We denote by a1 ; a2 ; . . . ; an all the arrows which start from z and by b1 ; b2 ; . . . ; bm all the arrows which end at z. Here some of these ak s and bl s may coincide. m P ImM
bj . Since A We decompose the vector space M
z into M1 M2 such that M2 j1
is an algebra with radical square zero, we have M2 7 Ker M
ai for every i. Hence M
ai and h i 0 M
bj can be written as M
ai 0 and respectively. For any endomorphism M
bj 11 12 with ij 2 Homk
Mj ; Mi . If 2 EndA
M, we can also write z as the matrix 21 22 M1 j 0 and M2 j 0, then it is easy to see that 0 defined by 8 if x j z x 1, this contradicts to M being a brick. Therefore we have M1 0 or M2 0. If M1 0, then M
ai 0 for every i, and if M2 0, then M
bj 0 for every j. In any case we have that M is not faithful, which is a contradiction. h P r o o f o f Th e o r e m . Let B be a minimal strictly wild factor algebra of A. By lemma 1 we know that (mod Bf is strictly wild. Thus there is a faithful B-module M which is a brick. It follows from Lemma 2 that B is wild hereditary. h Note that each minimal (in the sense of [4]) wild hereditary algebra with radical square e e e e e e e e 1; A e 3; A e 5; A e 7; D e 4; D e 5; D e 6; zero corresponds to just one of the quivers K3 ; T11111 ; A e e e e e e 6; E e 7; E e 8 with sink-source orientation. We have altogether 28 minimal (in the e 8; E e 7; D D sense of [4]) wild hereditary algebras with radical square zero. Let Q be a quiver. The quiver Z QZ 2 is defined by:
Q2 0 : Q0 f1; 2g; for any v ! w 2 Q1, there is an arrow
v; 1 !
w; 2 Z in
Q2 1 . It is well-known that A kQ=k2 Q is stably equivalent to kQZ 2 (See for instance (See for instance [6]). Thus every [1]) and A is of the same representation type as kQZ 2 (minimal) wild algebra with radical square zero can be obtained from a (minimal (in the sense of [4])) wild hereditary algebra by identifying i different sources with i different sinks in turn for some i 2 N and adding zero-relation to the composition of any two arrows. In this way we may classify all the minimal wild algebras with radical square zero as those in the following table, where n denotes the number of vertices and X
i denotes the quiver obtained from X by identifying i sources and i sinks in turn. For each minimal wild algebra with radical square zero, its Gabriel quiver is of some type displayed in the table. The following corollary implies that all minimal wild algebras with radical square zero of type X
i for i ^ 1 in the table below are non-strictly wild. Corollary. A minimal wild algebra A with radical square zero is strictly wild if and only if A is hereditary. In the following example we construct non-strictly wild algebra with radical square zero which is of n simple modules for each non-zero natural number n.
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Types of Gabriel Quiver
1
1
K3 e e
1 A 1
2 e e
2 A
3
3
4 5
e e
4 E 6
e e
3 A 5
e e
3 D 5
e e
2 D 4
e e
1 A 3 e e3 A
e e
5 E 8
e e
4 A 7
e e
4 D 7
e e
4 E 7
e e
3 D 6
e e
3 E 6
e e
2 A 5
e e
2 D 5
T11111
1
e e
1 D 4
T11111
e e4 D
6
e e
4 D 8
e e
4 E 8
e e
3 A 7
e e
3 D 7
e e
3 E 7
e e
2 D 6
e e
2 E 6
e e
1 A 5
e e
1 D 5
7
e e
3 D 8
e e
3 E 8
e e
2 A 7
e e
2 D 7
e e
2 E 7
e e
1 D 6
e e
1 E 6
e e5 A
e e5 D
8
e e
2 D 8
e e
2 E 8
e e
1 A 7
e e
1 D 7
e e
1 E 7
e e6 D
e e6 E
9
e e
1 D 8
e e
1 E 8
e e7 A
e e7 D
e e7 E
10
e e8 D
e e8 E
K3
e e1 A
E x a m p l e. Let A be the algebra with radical square zero with Gabriel quiver Q as follows (the orientation of the last arrow is uniquely determined by n is odd or even):
1
2
3
4
nÿ1
n
e e It is easy to see that A is wild for any n 2 N and n ^ 4 since kQZ 2 has kD5 as its factor algebra and A is not strictly wild since A does not have a wild hereditary algebra as its factor. By our theorem we know that above table and the example give at least one non-strictly wild algebra with n simple modules for each n 2 N and n ^ 1. A c k n o w l e d g e m e n t s. This work is a part of the author s doctoral thesis which was done under the patient guidance of Professor Claus Michael Ringel. The author is indebted to him for his advice, many stimulating discussions and hospitality. He thanks Volkswagen Stiftung for the financial support. He thanks also the referee for his valuable suggestions. References [1] M. AUSLANDER and I. REITEN, Stable equivalence of artin algebras. LNM 353, 8 ± 71, BerlinHeidelberg-New York 1973. [2] P. GABRIEL, Auslander-Reiten sequences and representation-finite algebras. In: Representation Theory I, LNM 831, 1 ± 71, Berlin-Heidelberg-New York 1980. [3] Y. HAN, Controlled wild algebras. Submitted. [4] O. KERNER, Preprojective components of wild tilted algebras. Manuscripta Math. 61, 429 ± 445 (1988). [5] O. KERNER, Tilting wild algebras. J. London Math. Soc. 39, 29 ± 47 (1989). [6] H. KRAUSE, Stable equivalence preserves representation type. Comm. Math. Helv. 72, 266 ± 284 (1997).
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[7] C. M. RINGEL, The development of the representation theory of finite-dimensional algebras 1968 ± 1975. Representation theory and algebraic geometry, 89 ± 115. London. Math. Soc. Lecture Note Ser. 238, Cambridge 1997. [8] C. M. RINGEL, Representations of K-species and bimodules. J. Algebra 41, 269 ± 302 (1976). [9] H. STRAUSS, On the perpendicular category of a partial tilting module. J. Algebra 144, 43 ± 66 (1991). Eingegangen am 10. 9. 1999 Anschrift des Autors: Yang Han Institute of Systems Science Chinese Academy of Sciences Beijing 100080 P. R. China
