Ring Algebras

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(1)Emre and. Lily. Representation Theory of Algebras Talk 8 Algebras and Modules I this talk. In. will. we. where. modules Since all. k. introduce. he is. and algebras field closed. algebraically are also rings we'll start off algebras some basics from by recalling theory and ring to move on k algebras Then we'll defininghow we can connect give them you plenty of examples on to Quivers through path algebras And lastly we'll relate them to path algebras modules and presentQuivers as well an. and. 4. Basics from. 1. Since. Ring Theory. this section. is. meant we'll use. later. that things not into brief delve and give all of the proofs some. Def. 121. Def. too. refresher we'll. on. keep. details. many. it. 1 10 12 be a with ring I C R is a subgroup A ideal cleft ideal right of the 12 additive I sat group r a e I Va c I V reR resp r a c TI If a ideal is also a left ideal then we'll simply right an refer to it as ideal of 12 fo short. let. examples I. a. as. 4.71. 03 E R. Given Im is an. an. itself. R. and ideal. I. c. are. both ideals of 12. and. R. all finite sums of elements. ideal. m c. a. of 12. 4.71 I C 12 An ideal it Im _0 some for. is. called. MEN. a. IN. so. am. I ai C I. nilpotent so.

(2) Def. CA 21. cat. is. at. k. If. 03. is. k. 4.31 The radical. of all. Lemma. then. let. a c. 2. Vb cR. 131 41 El. Gl. b c R. 4. has. a a. o. Ideal. ing. only. J R IER. ideals. intersection. the. in. and ring statements. I. 12. a C. are. R. equivalent. inverse right two sided inverse. ab. 14.2 is also. rad 12. maximal 21 race R. 31. ab. r n. ideals. its. I. has is an element of the intersection of all left maximal ideals r ba inverse v b c 12 has a left r ba Vb c R i two sided has inverse t a. Corollary 1 1. a. following. red R. n. be. 12. the. right. is. an. quotient. then. field. nad R. maximal. CA 71. a. either. 12. ICR. ideals. e R. that. ring the. a. Rft. 21. holds. commutative maximal iff is a field. In. ideal. right u g. if. maximal is called sit it I c TER. Remark. Def. left. A proper. red. If. I. left is. c. I e raid. to equal ideals. an. 121rad R. 12. an. ideal. I. ideal. O is. the. intersection. of. of 12. nilpotent. then. all.

(3) 4.2. Algebras. In this section we'll define algebras a and detailed couple properties give including the path algebra of a. and. quiver. Def. A. 9.41 k. unity of a 41. let k be vector. k. addition. in in. addition. is. also. A. examples. closed algebraically CA t 1 with a. an. A. algebra 7 set. their. ringhas. structure. the. where. space. the vector space A. coincides. the. ring A cat scalar multiplication in the vector space is compatible with the multiplication ring F a be A. the. Remark. dimension of. dimension. examples 11. Ha bl. V be k. of. the. Cla. the. k. a. field. with A ie. b afb.to 1a.b b A. the. space. A. is. k CX. in. X. is. algebra vector. 4.2. The. a. k. ring of polynomials Its algebra unity 7. polynomial Scalar. is the. constant. by teh of is done by b multiplication polynomial by multiplying each coefficient We can see that multiplication and easily are scalar multiplication in k x compatible textgCxlGk x bek then ls.ec l gCx1 fCx1 CSgcx1 HCx1gCxHb t.ctcxl.gcxll 21. The set. is. a. k. identity. of. all. algebra. matrix. nxn. Matrices over h Matuncle its unit being the.

(4) how Matu Chl. It should be clear. has. a. Again it. is. and. Scalar. k. just the. multiplication to. We. look at the. triangular a sub also. ing have. ring as usual. same. check multiplication. 1Len A i3 cMatanCiel be k lb Al D CA B A. 31. a. structure. vector Space is. easy. is. set. A DI b. SB. of. comptatibility. Lowe. or. upper. These matrices form clearly and all of thereto c matrices a k structure. Algebra the unity multiplication and scalar multiplication in a e the k same Matan. as. d. The set of all 3 3 matrices of the forms also form. The. a. O. O x. k algebra matrix is in the. Let usidentity just check. the. t.CA D. a o. multiplication albicide v and Sek. K. any. o. HE. eH EI. L. B tt outta. cans. 0. set. D. A. O. 0. f. cuter. Sau. w. Hy. no. duty. O. II 4 7. 0. z c. 0 e. k.

(5) Y'at. eoEEI I L. b.int. 9891 u u. III. we. but. A is. if. if. as. the. a. set does not k algebra same the of elements. on. set. a. in. multiplication. be A or. b. b. a. her Aor is also D. Take. underlying then of. So a. every the. if. any. we. bi. FE Sibi. means. two. in the. h. IE. fulfill. the. vector. a c A i 7 n. take a. defined. as. algebra bn. bz. a. space of the is. a. linear. a. a c A. two. Ee. b bi. for. is. satisfy. FI. basis. of. the. k a Jeb a A combination. arbitrary bi b. C. h. elements i4. n. bib bib. specify how to multiply then multiplication completely determined. we that if basis elements. algebra. vector Space. multiplication in A. an. b. underlying Aor A. A OP is. then their product must a a b bi Sibil. this. is not. a. multiplication in Aor. Remarks. set. algebra then the opposite algebra AOP. an. defined. is i.e But. this. of. E. IE. EEE. tooth elements. are. u. therefore this properties of a G. I. o.

(6) Recall. Q is a CQ quadruple consisting of the following data Q is a set of vertices Q is a set of arrows A. S Q. its. t. its. do. it. end. scat I. with j. il. this. with. x. start to. useful. Def. k. E. end. by. c. c. c. 1. j. il. c. definition. new. point. an. x. to. arrow. by drawing like so. paths j l da di. a. lk. concatenation of twopaths a. we. ar can. t.lk. d. start. with paths Together can use this to construct namely the path algebra. algebra. 9.5 et We define. C. the. c. Qi. to. arrow. Given two. quiver xz arl. multiplication defining ou remark before we a. its. a. tccl.sc. by. given. arrow. an. tix. denote. we. an. map which sends. point. Q be. c e. map which sends. Qo. its. from. et. a. starting is point a. Q. represent. we. Def. is. sit 1. Qa. quiver. Q be. for. of Q kQ path quiver algebra with basis comprised of as the algebra in the Q and paths multiplication quiver c c two basis elements given by the. C c. a. c c. 0. the. f. scc. I. of. tlc. otherwise. the. two elements product is of any determined by path algebra. therefore. in. if. all. b c 1. be. c. K bi. c c.

(7) Lemma CA 3 kQ. is. paths proof let. the. a. be. ei. examples Crl. c. c. Hcl. if. i. a. s. for. J E k. ei i then remain. CCc. only. since. C. every vertex in Q in the sum then every appears ill c. path. foei. Ea. Ek. Lee. a. similarly. algebra path all constant. c. a. c e. to. q Haki. c. L da. a. Ea. Efi path. c. of. Lei. since. 0,15. sum. a. iea. then. h Q. of. element. unity given by r. a c. them. the. once. appear. Ea.ci Ebc c c E b c. EocEScei C a DM. CA 3. Q be the quiver the then of Q paths thus the path algebra k. et. multiplication. is. simply. a. are a. d. Q e. a. has basis. at. Itt. 13,04 521 I. I. e. kid. site N. Then K Q is to the algebra of isomorphic over 4 This can be shown through polynomials the basis elements fairly simply 6. k. k Q. Eos.si ecas.at. i. x. Eos. cecastt. x. t. t. E as e at. 1.

(8) r n z da z x Q be the p s quire then K Q is isomorphic to the set of all. let. 21. upper triangular matrices Since each path in GQ is to j. i. uniquely Ci the. with. unique path. note ie. e. let. 3. b on. Ci. Sazpaz Gz Ca. the other land S. elate's Also. t. bzzpzz.czz.cz. I. 1s. ios. we'll. demonstrate. be concatenated a For example take can't Len. therefore. a. b. era. cz. t.mn. I. E. s. ecb. _0. we x. and. go.gg. ty. an. example. b p. bij ai. reisjes. 13353. cars. 423. 122.123 Css. eca.sl. go.fi j. example that when 2 points. the get c etat. gI8. and. O matrix. b. c. e. Tbl 8oo8E. 23. a. Jazpas 43. fog an. In. j. We have. z. j. j. i. cecal ecb a. by. ticijl. at. Ela b. to. i. ski. zc.ztlyzqztbzzczztbz.cz. ez. j. i. e.EE. bGkQ where. a. of K Q s.t. Ck. f. to. demonstrate. since a. sends. b. to. Mat. eiEciil 1 Tuo. that. bn.cn t. i. nxn.EE. we'll. b. a. j En. and. K Q. y. from path straight then we can. a. from. it. 7y. hth. each basis element. denote. fi then. n e i. is. du. ecol. eca.b.

(9) Q be the. let. 131. then. b Q is. path. isomorphic. c. kQ has the basis e. cz e. is. ea. the structure and similar. Def. sa. y. ap so. 121. to. a. to. ya. for this isomorphism is won't. we. k let A D be two k linear D map 4 A. homomorphism f. e. t a. a. t.fr. c. 4.61 a. is. i I ieIstttIE. I. twheese.ae. quiver. of. I c A. algebras. f Ca. a. I. f. go. into. algebras is a. very detail here then. if a. fca. used such homomorphisms in examples 4.3 already a we'll Now where you can problem you give try to work with these maps as well. We. Problem A 5 KG. let. be. 6 E. bg.gl. a. group. and. let. tag C le. finitely many by aer nonzero be the h algebra with basis G and multiplication the group operation given by KG is called the group algebra of G Show that. 7 12. geo. KZ is isomorphic to the algebra of over k Laurent polynomials in one variable k. 2. 2. is. isomorphic to. KEXY x 7.

(10) Solutions. UI. define. 111. k Z. s. I. Eez Sa n. l. I. kC. X. E. Laurent. Polynomials i X. X. s. nez. need to show that I is a bijective let a be KE p E k homomorphism. k. we. a. b. la n. z. Icp at b Elp Ez tin 2,41 th't x p z k Chea. Io is. e. k Z. in. unity. the. atm. n m. 101. OIC. z. z. oI Ez Six. t. X. is. a. k. hint. 2,5 nl. Iola I. Iocbl s bn bi Knez. o. so. then Io is. 2. Eez a. X. e. q h. c. surjective. k. x. KZ. bl. algebra x. oI Iz1Ezbciti.u.n 1. 4 1. six 1. Ez. n. b. Six. zs. then. X. and. k. I. phis. 2,6. Io is injective. et. o. Jux. Ez. p Iola It. unity in k. y. algebra Lo. pbntb4ln. in this. since. EzlEzbu.ti1xh Ezlnxhl I. n. t.in. n. o. I. Iola b. t. 0. is. bi. z. algebra. ILEza.in. a. in. an C k. b. knee. Ez an X. that. let us recall that for any plxlek Cx First off can find one 91 1 next c k Cx using long derision. sat. next. pCx1 six. Cx. it. qcx. hence he. where. thx. n. six I lick.

(11) next. hence in we. can. in. multiplying replace. Ei. six. I. that. note. b. by. elements. so. x ti. Eoi. t. than. for a b e k 21h21 similarly b i a b bi E EE j then a b i.IE jsibj itj1modn E fEnnSibj. Lobi b lie bath so. let. now. bio tha k 12. 4. od. in. in. xitjcmo.nl. bubo th. t. in. n. EE bit. x'I. b lie. Hobie. kC two. k. cxK. mu. une N u. ibi Xk. Xk. hi. where k t. t. b. k. lui. k. 2. ii i. six. to slow that 4 is a bijective k algebra let a b a k 2 2 e homomorphism p Le need. we. b. Iii. a. 4G. b1. a. p. 4 is le linear. x. lol bob. EI. b. r. s. xi. y is. a. b bi the. b bi k. Iii 1. 4. l. Xi. xi. k 21h21 I KC the unity in. 44E.jsi.it. 4ca.bt 41. t. sits Ii. I. p 4cal 4161. the unity of. 0 is. 4101. Iii. 411. psitt Ix. Again. Ii i. EI. Ii i 11. lift didotbit. t. t. ti to bit bi. bi. t. tbh b. t. In Ii. 4cal 4lb I algebra homomorphism x. xn ay. x.

(12) FEI t. 4lb. 4cal. EIolii et. Yi. i o ai. EI ai. so. Ii. i. Xi. then ai. x. 61242. c. 41. and. 4 is surjective. Def. tiso. a n. b. a. k. e. b. x. c. h. ti. il. ai. O. n n. six DB. let D be a k vector subspace of A CA 7 a sub 13 is then algebra if 13 contains 1. Prop s t 9.91 the I. proof. b b. fb b e D. and. since hence 0xkxh. already he is a. maximal. race. Xk Act. consider. IT. Kb. 11 I. nilpotent I. a. ideals of lexoxkx xk. the. since. of. 1 A. A a. EradA. 0,4 lex. are. Lex. 1. at I. cA. Tint means. then again. from. then. Corollary 9.2. sole. its. Xk. of A. are. xkoxh. Lex. ideal. ideals. by. ideals 4. 0. is the. definition. 1 2. we. Kho. then. Tlr Sa 1 1714 G Tcb Thal ICH Mail has an inverse in AII. Tic Ce ball. r. lemma lyslow. but we ve therefore raid A. xk. radical. the maximal. all. ideal. kxkx. AE. radial them from Len a d c EA l ba has an iverse. a c. Lich. 13. know that I field we know o. intersection. let. is. Algebra. ad A we. IEA. If. c. c. 4.7. radial I. 1. Tlc. Tical C 0. radial 51 Tical o. a c I DBU.

(13) If Q is a without oriented quiver then rad kQ is the ideal generated in Q arrows. Corollary 9.51 cycles by all A path of. the form. give by. il. proof. x. dz. x. i. H. di. a. l. ti. a. s is an. oriented. cycle. denote by Ro the ideal generated by all arrows in s.at Q l be the We te't Q will integer largest i L of c a contain length path any product of t.tn a ro s will be 0 This means 12 0 hence Ro is a nilpotent ideal kQ no Also ei Ro l ie Q S is a basis for so k C number of copies of h is IQ I kQhzo I kx then by Prop 4.9 Ro rad kQ we. Bms. Remark 14.61. It's important the condition Q fulfills For this quiver Q i the path algebra kQ is And polynomials belt. polynomial ideal. we. A. 3. Modules. In. this. section. 12 ring and finish. modules. with. with. over. see. for that. we'll. define. x. we'll. 1. an a. a. also. example of. this corollary that of having no oriented cycles for example a. in. the every linear. isomorphic to since. k generates ad kCx3 O. a c. modules. present. path algebra. a. o. over. a. maximal. examples between two. some. phis. a.

(14) 9.81 let 12 be a ring with a 1 0 a righta 12 module M is an abelian group together with binary operation called the right R action. Def. MXR m. St. um. at 21. r. M. s m r c M and. I l. m. cnn.i mz.lv Cr tri. Mnr. m. 31 91. ma C r. ref. I. me. 12. left. have that. by multiplying the and following the. is 12 module defined of M from the left elements n axioms Cal accordingly a. we. re. cnn.ru rz. m. c. r. marz. t m. r. m. Tr. examples 14.51 41. I. If. R is. C. ideal then I right where the right R in 12 In. a. is a action. right R module is given by multiplication particular a c 12 ideal the namely by generated is a right 12 module rc 123 a.ir a 12. 21. If. then an. the. Q A. by. is. a. for any. and quiver ie vertex. module. dimensional and whose. one. ai. Vc. sci 1. cA. EQ. whose. h vector A action. c. m e. let's check that. A Q. m. we. is. its can. abelian. algebra path define. group. generated space is given. ei. O. is. by. if c e otherwise. fullfils the module axioms it's enouph to only use two paths c c c A sci. In this case since other cases follow by le linearity of c c c A et m uz c he then m.eitm.ci mne.ie. l. mzeic. cnn.eitnzeilc. c. c. mutmyei mrei. myeictw.es. c. if c c otherwise. untrue. 4. t mzei. 1 and Sci. o. male. other. it ise. c. e.

(15) Cata l. mac. une. etc. 2ns ei. c. C. t w e c. c 1C. une. m. 3. If. Q is. then for an. equal. by. m. Eci. ti bye k. t.ie. e. e. and. t. c. hate. o. c ei. path tie. c c. c. C. m. A. i 2. j. if. qeie I. kQ. is. in Qa. c. e. otherwise. o. its. more. algebra path can define. we. abelian group is k vector space generated. right. in A. A. action. t.ie. tht. if. Iida. is. if. c c. e. ej. ifotheII.se. 0. that this right action coincides with the we defined on the path algebra multiplication makes sense For example it lie x since we see. the constant which from i to j. Ci. i. is. j. is. a. ain. path. means. o. o. c c _e. e. me. e. c. whose. a. c. otherwise. otherwise. two dimensional. and. o clee c ee. c. whose. Mcat. a. o o e. cotheruse c. quiver ar o any. given by tr. c. if. mneicl. a. module to the. A. c. he. if. 0. 1. more. c. e. nei 2 nei 0. Cure Ic. c.ci. e. c. nei 0. Clc. une. in eic. c'I. c. more. I. c. otherwise. o. more. c ei c o. e. at i. and. their. a is. an. arrow. concatenation. meal fulfills the module axioms Showing that is very similar to how we proceeded in 21 we here further detail won't go into. so.

(16) Sli. Mls. and have may and. as. appear. Def. seen. j. later. 91. d. night them. This on. is. the elements. You pie quiver rep. They. 4. likely. said to be generated. is. M. to you. si. coincidence. again. in. Stil the. as. no. A module. familiar. seem. to. if. m. men. by. every as m arm taint exist s sat aier M is called finitelygenerated if it is generated a finite set of elements there. Remark. If. it is. mz. by generated tusk mz Rt. ne m Rt for example. by only Def. 9. one. et. ro. the ideal element. A map l f m if h Cutie I. N. m. em. he Llm 1. h cha. The kernel of. a R ac R. N be two is. If A. is. A modules. underlying. Prop. A 81 then. proof. et. If. bae 12 it Ll il. hcm.tn. h Sr al. n. finitely generated. R modules a. a. h. Ke Chl men I 41 1 03 In CLI LC I 1 men coke Chi NII ch. k algebra then a morphism of two is also a homomorphism of the a. k vector spaces. L. Ke Chi. m. is. then. ms. z. morphism of 12 modules we la e. called. is the set the image of L is the set is and the co kernel of 4. Remark. m. by. c. M. i. Inchi. ke.CL. him 7th. b him. N is. b me. 4.0. a. and. this. linear. a. map. morphism of. coke. he A Oto. 0. A modules are A modules. 41. 1Leni O. 1. mat Zeke CLI in C Ice CLI.

(17) mutual. nd. mu Ct that m. m. Ct. 1. but. an. A nodule. m. b I Iz. the directly from E. follows. I. t. in. beerChl is. met. t. m m. fact that Ke CLI A. module. M. au. Deters. the proofs for In 111 and Coke L are and easy So we won't bore you. very similar with this here. A kQ be a path algebra And let example 9.61 andlet Mcat be A modules as defined in. Scj. Lee je Q and EQ with THI j. examples 19.51121 and 131. Then there. is. morphism. a. L. SC I me. I. 7 Mca 1. s. L is indeed. Let's check that. te k et m CEA za k Ll tune tune l Ll Infante b here e I thcinze I. h ma. LCune l hlm. h me. c c. c c. m. 4101 a c. Lchnejlc. therefore h is. a. na. mad O. no. plisa. ofmodules. hematuria Sm.at my. L is. if. mix 0. a. if. h Linea. c. e. otherwise c. c. otherwise. morphism of modules.

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