A Class of Semigroups of Finite Representation Type Thawat Changphas a ; Vlastimil Dlab b a

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1 This article was dowloaded by: [Hebrew Uiversity] O: 13 July 2010 Access details: Access Details: [subscriptio umber ] Publisher Taylor & Fracis Iforma Ltd Registered i Eglad ad Wales Registered Number: Registered office: Mortimer House, Mortimer Street, Lodo W1T 3JH, UK Commuicatios i Algebra Publicatio details, icludig istructios for authors ad subscriptio iformatio: A Class of Semigroups of Fiite Represetatio Type Thawat Chagphas a ; Vlastimil Dlab b a Departmet of Mathematics, Kho Kae Uiversity, Kho Kae, Thailad b School of Mathematics ad Statistics, Carleto Uiversity, Ottawa, Otario, Caada To cite this Article Chagphas, Thawat ad Dlab, Vlastimil(2008) 'A Class of Semigroups of Fiite Represetatio Type', Commuicatios i Algebra, 36: 4, To lik to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms ad coditios of use: This article may be used for research, teachig ad private study purposes. Ay substatial or systematic reproductio, re-distributio, re-sellig, loa or sub-licesig, systematic supply or distributio i ay form to ayoe is expressly forbidde. The publisher does ot give ay warraty express or implied or make ay represetatio that the cotets will be complete or accurate or up to date. The accuracy of ay istructios, formulae ad drug doses should be idepedetly verified with primary sources. The publisher shall ot be liable for ay loss, actios, claims, proceedigs, demad or costs or damages whatsoever or howsoever caused arisig directly or idirectly i coectio with or arisig out of the use of this material.

2 Commuicatios i Algebra, 36: , 2008 Copyright Taylor & Fracis Group, LLC ISSN: prit/ olie DOI: / A CLASS OF SEMIGROUPS OF FINITE REPRESENTATION TYPE Thawat Chagphas 1 ad Vlastimil Dlab 2 1 Departmet of Mathematics, Kho Kae Uiversity, Kho Kae, Thailad 2 School of Mathematics ad Statistics, Carleto Uiversity, Ottawa, Otario, Caada Dowloaded By: [Hebrew Uiversity] At: 07:20 13 July 2010 I their study of the class of stadardly stratified algebras with irreducible represetatios, Ágosto et al. (to appear) have itroduced two operators ad actig o the class ad satisfyig the followig relatios: 2 = 2 = ad 1 = 1 I this little ote we are presetig a complete descriptio of idecomposable liear represetatios of the respective semigroups S = a b a 2 = a b 2 = b ab 1 a = ab 1 1 by costructig the graph semigroups T (see Dlab ad Pospíchal, 2002) such that the semigroup algebras KT ad KS are isomorphic. The umber of idecomposable represetatios of S is for >1 (of which 4 are irreducible) ad all idecomposable represetatios of S are uiserial. Key Words: Fiite represetatio type; Graph semigroups; Idecomposable represetatios; Liear represetatios of semigroups Mathematics Subject Classificatio: Primary 20M30, 16G20; Secodary 16G60, 16G70. STRUCTURE OF KS Give the semigroup S = a b a 2 = a b 2 = b ab 1 a = ab 1 attach to S the uity 1 ad deote the resultig semigroup by S = S 1 Received December 21, 2006; Revised Jauary 31, Commuicated by I. P. Shestakov. Address correspodece to Vlastimil Dlab, School of Mathematics ad Statistics, Carleto Uiversity, Ottawa, Otario K1S 5B6, Caada; Fax: (613) ; vdlab@math.carleto.ca 1474

3 A CLASS OF SEMIGROUPS OF FINITE REPRESENTATION TYPE 1475 Thus, S has elemets a 2k 1 = ab k 1 a a 2k = ab k for 1 k 1 b 2k 1 = ba k 1 b b 2k = ba k b 2 1 = ba 1 b ad 1 for 1 k 1 Note that a 1 = a ad b 1 = b Oe ca easily check the followig multiplicatio table for S : Dowloaded By: [Hebrew Uiversity] At: 07:20 13 July 2010 a 1 a s = a s for all 1 s 2 2 a 2r a s = a 2r+1 a s = a t where t = mi 2r + s 2 2 for all 1 r 2 ad 1 s 2 2 a 2 2 a s = a 2 2 for all 1 s 2 2 a 2r 1 b s = a 2r b s = a t where t = mi 2r + s for all 1 r 2 ad 1 s 2 1 b 2r 1 a s = b 2r a s = b t where t = mi 2r + s for all 1 r 1 ad 1 s 2 2 b 2 1 a s = b 2 1 for all 1 s 2 2 b 1 b s = b s for all 1 s 2 1 ad b 2r b s = b 2r+1 b s = b t where t = mi 2r + s 2 1 for all 1 r 1 ad 1 s 2 1 Now, let K be a field ad A = K S the semigroup algebra of S For = 1 clearly, A 1 K K = Kb K 1 b is semisimple, ad for = 2 A 2 is hereditary with a K-basis cosistig of e 1 = b 1 b 2 e 2 = a 1 a 2 e 3 = 1 b 1 + b 2 a 1 + a 2 b 3 e 4 = b 3 Thus A 2 is the path algebra of the quiver g = b 2 b 3 ad h = a 2 b g 3 h 4

4 1476 CHANGPHAS AND DLAB For 3 defie the followig elemets of A : 2 2 e 1 = 1 i+1 b i i=1 2 2 e 2 = 1 i+1 a i i=1 e 3 = 1 e 1 e 2 b 2 1 e 4 = b 2 1 g 1 = b 2 b 3 g 2 = a 2 a 3 ad g 3 = a 2 2 b 2 1 Dowloaded By: [Hebrew Uiversity] At: 07:20 13 July 2010 It is a straightforward calculatio to check the followig multiplicatio table (Table 1). Moreover, we ca show that g 1 g 2 2 = b 2 3 b b 2 1 Ideed, we ca show that for ay 1 t 2 g 1 g 2 t = b 2t+1 b 2t+2 + This is true for t = 1, sice Observe that, i geeral, 2 p=t+1 p b 2p+1 b 2p+2 for suitable itegers p ( ) g 1 g 2 = b 3 b b 5 b 6 b 2t+1 b 2t+2 g 1 g 2 = b 2t+3 b 2t+4 2 b 2t+5 b 2t+6 + b 2t+7 b 2t+8 Table 1 e 1 e 2 e 3 e 4 g 1 g 2 g 3 e 1 e g e 2 0 e g 2 0 e e g 3 e e g 1 0 g g 1 g 2 0 g 2 g g 2 g g g

5 A CLASS OF SEMIGROUPS OF FINITE REPRESENTATION TYPE 1477 where ad equal to 0 or 1, ad thus, proceedig by iductio, follows. I particular, for t = 2, we have g 1 g 2 2 = b 2 3 b b 2 1 Furthermore, oe ca verify easily that g 1 g 2 2 g 1 = b 2 2 b 2 1 g 1 g 2 1 = 0 ad g 2 g 1 g 2 2 = g 2 g 1 2 g 2 = 0 Remark here that g 2 g 1 2 = a 2 3 a 2 2 As a result, we ca formulate the followig theorem. Dowloaded By: [Hebrew Uiversity] At: 07:20 13 July 2010 Theorem. where A 1 ad A 2 For every 3 A A 1 A 2 = KQ 1 / g 2 g 1 2 g 2 is a factor algebra of the path algebra over the quiver is the path algebra over the quiver Q 1 = 1 g 1 2 g 2 Q 2 = 3 4 g 3 Thus, A is of fiite represetatio type: There are 4 5 idecomposable represetatios of A 1 ad 3 idecomposable represetatios of A 2 All these represetatios are uiserial. Proof. It is easy to see that B 1 = e 1 g 1 g 1 g 2 g 1 g 2 g 1 g 1 g 2 2 g 1 e 2 g 2 g 2 g 1 g 2 g 1 g 2 g 2 g 1 2 is a K-basis of A 1 ad the set B 2 = e 3 g 3 e 4 is a K-basis of A 2 Thus, dim K A = dim K A 1 + dim K A 2 ad the theorem follows. Remark 1. Deotig by S 1 ad S 2 the simple represetatios of A 1 correspodig to the idempotets e 1 ad e 2 respectively, we ca display the Auslader-Reite

6 1478 CHANGPHAS AND DLAB quiver as follows: Dowloaded By: [Hebrew Uiversity] At: 07:20 13 July 2010 where U t 1 t 2 2 are the uiserial modules with the compositio series S 1 S 2 S 1 S 2 of legth t ad V t 1 t 2 3 are the uiserial modules with the compositio series S 2 S 1 S 2 S 1 of legth t Here, P 1 ad P 2 are the idecomposable projective represetatios. Remark 2. Note that A is isomorphic to the semigroup algebra of the graph semigroup T defied by the set E = e 1 e 2 e 3 e 4 of the orthogoal idempotets ad the set G = g 1 g 2 g 3 of the geerators i the termiology of Dlab ad Pospíchal (2002). Remark 3. Let us describe the (uique) expressios for the geerators a ad b of S i terms of the basis B 1 B 2 : a = e 2 + g g 2 g 1 g g 2 g 1 2 g g 2 g 1 3 g 2 + g 3 + e 4 with suitable itegers t 1 t 3 ad b = e 1 + g g 1 g 2 g g 1 g 2 2 g g 1 g 2 3 g 1 + g 1 g 2 2 g 1 + e 4 with suitable itegers t 1 t 3 ad The expressios follow immediately usig the relatios g 2 g 1 g 2 t = a 2t+2 a 2t+3 + g 1 g 2 t g 1 = b 2t+2 b 2t+3 + that ca be easily derived from ( ). 3 p=t+1 3 p=t+1 p a 2p+2 a 2p+3 p b 2p+2 b 2p+3

7 A CLASS OF SEMIGROUPS OF FINITE REPRESENTATION TYPE 1479 Table 2 a 2r 1 a 2r b 2r 1 b 2r a 2s 1 a h 3 a h 2 a h 2 a h 1 a 2s a h 1 a h a h 2 a h 1 b 2s 1 b h 2 b h 1 b h 3 b h 2 b 2s b h 2 b h 1 b h 1 b h Dowloaded By: [Hebrew Uiversity] At: 07:20 13 July 2010 FINAL REMARKS Defiig formally a p to be the product abab with p factors (resultig i the equalities a p = a 2 2 for all p 2 2) ad b q to be the product baba with q factors (ad thus havig b q = b 2 1 for all q 2 1), we obtai the followig simple multiplicatio table (Table 2) with h = 2 r + s. From Table 2, oe ca get immediately the followig formulae (for all t 1): ad 2t 1 2t 1 g 1 g 2 t = 1 i b i 2 t+i +1 b 2 t+i +2 2t 2t g 1 g 2 t g 1 = 1 i b i 2 t+i +2 b 2 t+i +3 2t 1 2t 1 g 2 g 1 t = 1 i a i 2 t+i +1 a 2 t+i +2 2t 2t g 2 g 1 t g 2 = 1 i a i 2 t+i +2 a 2 t+i +3 It is the a matter of simple computatios to write dow the formulae for a ad b i terms of e 1 e 2 e 3 e 4 g 1 g 2, ad g 3 explicitly. The reader may fid easily that p = p for all p 1 with 1 = 1 = 1 ad that, writig q = p p 1, 3 q 1 2 p q + i p = q+i+1 q i p q+i for all p 2 i=1 Thus, for istace, 5 = 273 ad 10 = Added December 13, 2007: I a recet letter, Prof. József Peliká of Eötvös Lorád Uiversity i Budapest iforms us that REFERENCES p = 1 2p + 1 ( 3p p Ágosto, I., Dlab, V., Lukács, E. Approximatios of algebras by stadardly stratified algebras. To appear i J. Algebra (accepted). Dlab, V., Pospíchal, T. (2002). Graph Semigroups. Advaces i Algebra: Proc. ICM Satelitte Cof. i Algebra, Hog Kog, 2002, World Scietific, pp )