PROJECTIVE REPRESENTATIONS OF QUIVERS

Transcription

1 IJMMS 31: II. S ttp://ijmms.indawi.com Hindawi ublisin Corp. ROJECTIVE RERESENTATIONS OF QUIVERS SANGWON ARK Received 3 Auust 21 We prove tat 2 is a projective representation o a quiver Q = i and only i and 2 are projective let R-modules, is an injection, and 2 is a summand. Ten, 1 2 we eneralize te result so tat a representation M 1 n 2 n 1 M n 1 M n o a quiver Q = is projective representation i and only i eac M i is a projective let R-module and te representation is a direct sum o projective representations. 2 Matematics Subject Classiication: 16E3, 13C11, 16D8. 1. Introduction. A quiver is just a directed rap. We allow multiply edes and edes oin rom a vertex to te same vector. Oriinally a representation o a quiver assined a vertex space to eac vertex and a linear map to eac ede or arrow wit te linear map oin rom te vector space assined to te initial vertex to te one assined to te terminal vertex. For example, a representation o a quiver is V 1 V2. Ten we can deine a morpism o two representations o te same vector. Now, instead o vector spaces we can use let R-modules and also instead o linear maps we can use R-linear maps. In tis paper, we study te properties o projective representations o quivers. Representations o quivers is a new topic in module teory and is recently developed in [1, 2]. Deinition 1.1. A representation 2 o a quiver Q = is sa to be projective i every diaram o representations can be completed to a commutative diaram as ollows: Lemma 1.2. I 2 is a projective representation o a quiver Q =, ten and 2 are projective let R-modules.

2 98 SANGWON ARK roo. Let M and N be let R-modules, α : N an R-linear map, and β : M N an onto R-linear map. Ten, since 2 is a projective representation, we can complete te diaram α M N as a commutative diaram. Tus is a projective let R-module. Let : 2 N be an R-linear map and let : M N be an onto R-linear map. Ten, since 2 is a projective representation, we can complete te diaram M M N N as a commutative diaram. Tus 2 is a projective let R-module. Lemma 1.3. I is a projective let R-module, ten a representation o a quiver Q = is a projective representation. roo. Te lemma ollows by completin te diaram 1.5 k as a commutative diaram. Remark 1.4. A representation oaquiverq = is not a projective representation i, because we cannot complete te diaram 1.6 as a commutative diaram. Lemma 1.5. I is a projective let R-module, ten a representation o a quiver Q = is a projective representation.

3 ROJECTIVE RERESENTATIONS OF QUIVERS 99 roo. Let M 1,, N 1, and be let R-modules and let : M 1 and : N 1 be R-linear maps. Let k : N 1 be an R-linear map and coose k : as an R-linear map. And conser te ollowin diaram: 1.7 k k Ten, since is a projective let R-module, tere exists a map α : M 1. Now coose α : as an R-linear map. Ten α and α complete te above diaram as a commutative diaram. Tereore is a projective representation. 2. Direct sum o projective representations Teorem 2.1. A representation 2 o a quiver Q = is projective i and only i and 2 are projective let R-modules, is an injection, and 2 is a summand. roo. Conser te ollowin diaram: Since 2 is a projective representation, we can complete te above diaram as a commutative diaram as ollow: Tus = 1. Tereore, 2 2 ker and ker ker. 2.3 Tis completes te proo. Now let Q = be a quiver wit n vertices and n 1arrows. Ten, we can easily eneralize te results o Lemmas 1.3 and 1.5 as ollows: te

4 1 SANGWON ARK representations n, n 1 n 1, , are all projective representations o a quiver Q =,ieac i 1 is a projective let R-module. We can also eneralize Lemma 1.2 so tat i M 1 2 n 1 n M n 1 M n o a quiver Q = is projective representation, ten eac M i is a projective let R-module. 1 2 Teorem 2.2. A representation M 1 M 3 o a quiver Q = is projective i and only i M 1,, and M 3 are projective let R-modules, 1 M 1 is a summand o, and 2 is a summand o M 3. Tat is, M roo. Te diaram M M 3 M 1 M can be completed to a commutative diaram by : M 1 M 1, 21 : M 1, and :M 3. Ten we can et 21 1 = so tat M 1 ker 21.Nowte diaram M M 3 1 M2 M2 2.7 can be completed to a commutative diaram by 1 : M 1,:, and 32 : M 3. Ten, we can et 32 2 = M2 so tat M 3 Ker 32. Tereore, M 3 Ker 32 M 1 Ker 21 Ker 32.Hence, M 3 1 Tis completes te proo

5 ROJECTIVE RERESENTATIONS OF QUIVERS Now, we can easily eneralize Teorem 2.2 so tat a representation M 1 n 2 n 1 M n 1 M n o a quiver Q = is projective representation i and only i eac M i is a projective let R-module and te representation is te direct sum o te ollowin projective representations: n, n 1 n 1, ,. Remark 2.3. Te representations o a quiver Q = :,,. 2.1 are not projective representations i. Acknowledment. Tis work was supported by Korea Science and Enineerin Foundation KOSEF Reerences [1] E. E. Enocs and I. Herzo, A omotopy o quiver morpisms wit applications to representations, Canad. J. Mat , no. 2, [2] E. E. Enocs, I. Herzo, and S. ark, Cyclic quiver rins and polycyclic-by-inite roup rins, Houston J. Mat , no. 1, Sanwon ark: Department o Matematics, Don-A University, usan, , Korea address: swpark@mail.dona.ac.kr

6 Advances in Operations Researc Advances in Decision Sciences Journal o Applied Matematics Alebra Journal o robability and Statistics Te Scientiic World Journal International Journal o Dierential Equations Submit your manuscripts at International Journal o Advances in Combinatorics Matematical ysics Journal o Complex Analysis International Journal o Matematics and Matematical Sciences Matematical roblems in Enineerin Journal o Matematics Discrete Matematics Journal o Discrete Dynamics in Nature and Society Journal o Function Spaces Abstract and Applied Analysis International Journal o Journal o Stocastic Analysis Optimization