INJECTIVE AND PROJECTIVE PROPERTIES OF REPRESENTATIONS OF QUIVERS WITH n EDGES. Sangwon Park

Transcription

1 Korean J. Mat. 16 (2008), No. 3, pp INJECTIVE AND PROJECTIVE PROPERTIES OF REPRESENTATIONS OF QUIVERS WITH n EDGES Sanwon Park Abstract. We define injective and projective representations of quivers wit two vertices wit n arrows. In te representation of quivers we denote n edes between two vertices as and n maps as f 1 f n, and E E E (n times) as n E. We sow tat if E is an injective left R-module, ten n E p1 pn E is an injective representation of Q = were p i (a 1, a 2,, a n ) = a i, i {1, 2,, n}. Dually we sow tat if M 2 is an injective representation of a quiver Q = ten and M 2 are injective left R-modules. We also sow tat if P is a projective left R-module, ten P i1 in n P is a projective representation of Q = were i k is te kt injection. And if M 2 is an projective representation of a quiver Q = ten and M 2 are projective left R-modules. 1. Introduction A quiver is just a directed rap wit vertices and edes (arrows) ([1]). We may consider many different types of quivers. We allow multiple edes and multiple arrows, and edes oin from a vertex back to te same vertex. Oriinally a representation of quiver assined a vector space to eac vertex - and a linear map to eac ede (or arrow) - wit te linear map oin from te vector space assined to te initial vertex Received July 2, Revised July 16, Matematics Subject Classification: Primary 16E30; Secondary 13C11, 16D80. Key words and prases: injective module, injective representation, quiver. Tis paper was supported by Don-A University Researc fund, in 2006.

2 324 Sanwon Park to te one assined to te terminal vertex. For example, a representation of te quiver Q = is V 1 f V 2, V 1 and V 2 are vector spaces and f is a linear map (morpism). Ten we can define a morpism of two representations of te same quiver i.e., iven a quiver Q =, we f can define two representations V 1 V 2 and W 1 W 2. Now we can define a morpism between tese two representations. A morpism of V 1 f V 2 to W 1 W 2 is iven by a commutative diaram V 1 f V 2 S 1 W 1 W 2 wit s 1, s 2 linear maps. In ([3]) a omotopy of quiver was developed and in ([2]) cyclic quiver rin was studies. Recently, te teory of projective representations was developed in ([4]) and te teory of injective representation was studied in ([5]). Definition 1.1. ([7]) A left R-module E is said to be injective if iven any injective linear map σ : M M and any linear map : M E, tere is a linear map : M E suc tat σ =. Tat is S 2 0 M σ M can always be completed to a commutative diaram. E Definition 1.2. ([7]) A left R-module P is said to be projective if iven any surjective linear map σ : M M and any linear map : P M, tere is a linear map : P M suc tat σ =. Tat is P M σ M 0 can always be completed to a commutative diaram.

3 Quivers of n edes 325 Let G = G 1 G 2 G i G n be a direct product of roups. Te projection map π i : G G i were π i ( 1, 2,, i,, n ) = i is a omomorpism for eac i = 1, 2,, n. Tis follows immediately from te fact tat te binary operation of G coincides in te it component wit te binary operation in G i. Let φ i : G i G 1 G 2 G i G n be iven by φ i ( i ) = (e 1, e 2,, i,, e n ) were i G i and e j is te identity of G j. Tis is an injection map. Let F = {X i i I} be an indexed family of left R-modules X i and denote P = i I X i te cartesian product of F. Define an element of P as a function f : I i I X i suc tat f(i) X i for every i I. Define (P, +) by (f + )(i) = f(i) + (i) X i for every i I, and 0(i) = 0 X i and ( f)(i) = [f(i)]. Ten easily (P, +) is an abelian roup. Define µ : R P P by (rf)(i) = r[f(i)] for every i I. Ten easily P is a left R-module. We say P as te direct product of F over R. Consider S te subset of P suc tat f(i) = 0 except only finite i I. Ten easily S is a submodule of P. We say S as direct sum of F and is denoted by S = i I X i. Remark 1. p j s : i I X i X j is called te natural projection of S = i I X i. So we ave morpisms X j d j i I X i i i I X i p k X k p k i d j : X j X k is trivial if j k, and is identity if j = k. Remark 2. Te natural injection d j : X j i I X i is a monomorpism and te natural projection p j : i I X i X j is an epimorpism. Notation : In te representation of quivers we denote n arrows between two vertices as and n maps as f 1 f n and E E E (n times) as n E. 2. Injective representation of a quiver Q = wit n edes We define injective representation of a quiver wit two vertices and multiple arrows. And consider teir various injective representations as left R-modules.

4 326 Sanwon Park Definition 2.1. A representation M1 M 2 of a quiver Q = is called an injective representation if for any representation N1 1 n N 2 wit a subrepresentation and morpisms S 2 1 S1 S2 n S1 S 1 S 2 S 1 S 2 k M 2 tere exist H Hom R (N 1, ) and K Hom R (N 2, M 2 ) suc tat te followin diaram N 1 1 n N 2 H K M 2 commutes and H S1 = K S2 = k. In oter words, every diaram of representations (0 0) (S 1 S 2 ) (N 1 1 n N 2 ) k M 2 ) can be completed to a commutative diaram as follows : (0 0) (S 1 S 2 ) (N 1 1 n N 2 ) H K M 2 ) Teorem 2.2. If E is an injective left R-module, ten E 0 is an injective representation of Q =.

5 Quivers of n edes 327 Proof. Let, M 2 be left R-modules, S 1 be a submodule of, S 2 be a submodule of M 2 and : S 1 E be an R-linear map. Consider te followin diaram (0 0) (S 1 S 2 ) 1 n M 2 ) (E 0) Ten since E is an injective left R-module, we can complete te followin commutative diaram by. 0 S 1 E Ten 0((m)) = 0 = 0( 1 (m)), 0((m)) = 0 = 0( 2 (m)),, 0((m)) = 0 = 0( n (m)). Tus, we can complete te followin diaram 1 n M 2 (E 0) as a commutative diaram by 0 : M 2 0. Terefore, we can complete te diaram 0 (0 0) (S 1 S 2 ) 1 n M 2 ) 0 (E 0) as a commutative diaram. Hence, E 0 is an injective representation. Teorem 2.3. If E is an injective left R-module, ten n E p 1 p n E is an injective representation of Q = were p i (a 1, a 2,, a n ) = a i, i {1, 2,, n}.

6 328 Sanwon Park Proof. Let, M 2 be a left R-module, S 1 be a submodule of, S 2 be a submodule of M 2 and : S 2 E be an R-linear map. Consider te followin diaram (0 0) (S 1 S 2 ) M 2 ) ( n E p 1 p n E) Ten since E is an injective left R-module, we can consider te followin commutative diaram 0 S 2 M 2 E by. Define H : n E by H(m) = ((f 1 (m)), (f 2 (m)),, (f n (m))). Ten p i (H(m)) = (f i (m)), i = 1, 2,, n and we can complete te followin diaram M 2 H ( n E p 1 p n E) as a commutative diaram. Hence, we can complete te diaram (0 0) (S 1 S 2 ) M 2 ) H ( n E p 1 p n E) is an injective rep- as a commutative diaram. Terefore n E p 1 p n E resentation. Teorem 2.4. If M 2 is an injective representation of a quiver Q = ten and M 2 are injective left R-modules.

7 Quivers of n edes 329 Proof. First we sow tat M 2 is an injective left R-module. Let S be a submodule of N and : S M 2 be an R-linear map and we consider te followin diaram 0 S N M 2 Ten since M1 M 2 is an injective representation, tere exist : N M 2 wic completes te followin (0 0) (0 S) (0 N) M 2 ) as a commutative diaram. Tus, : N M 2 completes te above diaram as a commutative diaram. Terefore, M 2 is an injective left R-module. Let : S be an R-linear map and we consider te followin diaram 0 S N Consider te followin diaram (0 0) (S i 1 i n n S) (N j 1 j n n N) G M 2 ) were G((s 1, s 2,, s n )) = n k=1 f k((s k )), and i k (s) is te kt injection.

8 330 Sanwon Park Ten since M1 M 2 is an injective representation, tere exist : N and α : n E M 2 suc tat te followin diaram N j 1 j n n N α M 2 ) as a commutative diaram. Tus, : N completes te followin diaram 0 S N as a commutative diaram. Terefore, is an injective left R-module. 3. Projective representation of a quiver Q = wit n edes Definition 3.1. A representation P1 P 2 of a quiver Q = is called a projective representation if every diaram of representations (P 1 P 2 ) 1 n M 2 ) (N 1 1 n N 2 ) (0 0) can be completed to a commutative diaram as follows : (P 1 P 2 ) H K 1 n M 2 ) (N 1 1 n N 2 ) (0 0)

9 Quivers of n edes 331 Teorem 3.2. If P is a projective left R-module, ten 0 P is a projective representation of Q =. Proof. Let, M 2, N 1, N 2 be left R-modules, and k : P N 2 be an R-linear map. Consider te followin diaram (0 P ) 1 n M 2 ) (N 1 1 n N 2 ) (0 0) Ten since P is a projective left R-module, we can complete te followin commutative diaram by. k P M 2 N 2 0 Ten 0 : 0 completes te followin diaram (0 P ) 0 1 n M 2 ) (N 1 1 n N 2 ) (0 0) is a projective representa- as a commutative diaram. Hence, 0 P tion. Teorem 3.3. If P is a projective left R-module, ten P i 1 i n n P is a projective representation of Q = were i k is te kt injection.

10 332 Sanwon Park Proof. Let, M 2, N 1, N 2 be left R-modules and : P N 1 be a R-linear map. Consider te followin diaram (P i 1 i n n P ) M 2 ) (N 1 N 2 ) (0 0) Since P is a projective left R-module we can complete te followin diaram by. P N 1 0 Define H((a 1, a 2,, a n )) = f 1 ((a 1 ))+f 2 ((a 2 ))+ +f n ((a n )). Ten f 1 ((a)) = H(i 1 (a)), f 2 ((a)) = H(i 2 (a)),, f n ((a)) = H(i n (a)). Tus we can complete te followin diaram (P i 1 i n n P ) H M 2 ) (N 1 N 2 ) (0 0) is a projective repre- as a commutative diaram. Hence, P i 1 i n n P sentation. Teorem 3.4. If M 2 is an projective representation of a quiver Q = ten and M 2 are projective left R-modules. Proof. First we sow tat is a projective left R-module. Let S and N be left R-modules and : S be an R-linear map and we consider te followin diaram N S 0

11 Quivers of n edes 333 Ten since M1 M 2 is a projective representation, tere exist : N wic completes te followin M 2 ) (N 0) (S 0) (0 0) Terefore, is a projective left R-module. Let : M 2 S be an R-linear map and we consider te followin diaram M 2 N S 0 Define G : n i=1s i were p k (G(m)) = (f k (m)), for k = 1,..., n and G(m) = ((f 1 (m)), (f 2 (m)),, (f n (m))), and p k ((a 1, a 2,, a n )) = a k, and consider te followin diaram M 2 ) G ( n i=1n i q 1 q n N) ( n S p 1 p n S) (0 0) Ten since M1 M 2 is a projective representation, tere exist : n N and α : M 2 N suc tat te followin diaram M 2 α ( n N i 1 i n N)

12 334 Sanwon Park as a commutative diaram.tus, α : M 2 N completes te followin diaram M 2 N S 0 Terefore, M 2 is a projective left R-module. α References [1] R. Diestel, Grap Teory, G.T.M. No.88, Spriner-Verla, New York (1997). [2] E. Enocs, I. Herzo, S. Park, Cyclic quiver rins and polycyclic-by-finite roup rins, Houston J. Mat. (1), 25 (1999) [3] E. Enocs, I. Herzo, A omotopy of quiver morpism wit applications to representations, Canad J. Mat. (2), 51 (1999), [4] S. Park, Projective representations of quivers, IJMMS(2), 31 (2002), [5] S. Park, D. Sin, Injective representation of quiver, Commun. Korean Mat. Soc. (1), 21 (2006), [6] R. S. Pierce, Associative Alebras, G.T.M. No.173, Spriner-Verla, New York (1982). [7] J. Rotman, An Introduction to Homoloical Alebra, Academic Press Inc., New York (1979). Department of Matematics, Don-A University, Pusan, Korea swpark@dona.ac.kr