Lectures and problems in representation theory

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1 Lectures and problems n representaton theory by Pavel Etngof and students of the 2004 Clay Mathematcs Insttute Research Academy: Oleg Goldberg, Tanka Lu, Sebastan Hensel, Alex Schwendner, Elena Udovna, and Mtka Vantrob Aprl 23, Introducton What s representaton theory? To say t n one sentence, t s an exctng area of mathematcs whch studes representatons of assocatve algebras. Representaton theory has a wde varety of applcatons, rangng from physcs (elementary partcles) and chemstry (atoms, molecules) to probablty (card shuffles) and number theory (Fermat s last theorem). Representaton theory was born n 1896 n the work of the German mathematcan F. G. Frobenus. Ths work was trggered by a letter to Frobenus by R. Dedeknd. In ths letter Dedeknd made the followng observaton: take the multplcaton table of a fnte group G and turn t nto a matrx X G by replacng every entry g of ths table by a varable x g. Then the determnant of X G factors nto a product of rreducble polynomals n x g, each of whch occurs wth multplcty equal to ts degree. Dedeknd checked ths surprsng fact n a few specal cases, but could not prove t n general. So he gave ths problem to Frobenus. In order to fnd a soluton of ths problem (whch we wll explan below), Frobenus created representaton theory of fnte groups. The general content of representaton theory can be very brefly summarzed as follows. An assocatve algebra over a feld k s a vector space A over k equpped wth an assocatve blnear multplcaton a, b ab, a, b A. We wll always consder assocatve algebras wth unt,.e., wth an element 1 such that 1 a = a 1 = a for all a A. A basc example of an assocatve algebra s the algebra EndV of lnear operators from a vector space V to tself. Other mportant examples nclude algebras defned by generators and relatons, such as group algebras and unveral envelopng algebras of Le algebras. A representaton of an assocatve algebra A (also called a left A-module) s a vector space V equpped wth a homomorphsm ρ : A EndV,.e., a lnear map preservng the multplcaton and unt. A subrepresentaton of a representaton V s a subspace U V whch s nvarant under all operators ρ(a), a A. Also, f V 1, V 2 are two representatons of A then the drect sum V 1 V 2 has an obvous structure of a representaton of A. A representaton V of A s sad to be rreducble f ts only subrepresentatons are 0 and V tself, and ndecomposable f t cannot be wrtten as a drect sum of two nonzero subrepresentatons. Obvously, rreducble mples ndecomposable, but not vce versa. Typcal problems of representaton theory are as follows: 1

2 1. Classfy rreducble representatons of a gven algebra A. 2. Classfy ndecomposable representatons of A. 3. Do 1 and 2 restrctng to fnte dmensonal representatons. As mentoned above, the algebra A s often gven to us by generators and relatons. For example, the unversal envelopng algebra U of the Le algebra sl(2) s generated by h, e, f wth defnng relatons he eh = 2e, hf fh = 2f, ef fe = h. (1) Ths means that the problem of fndng, say, N-dmensonal representatons of A reduces to solvng a bunch of nonlnear algebrac equatons wth respect to a bunch of unknown N by N matrces, for example system (1) wth respect to unknown matrces h, e, f. It s really strkng that such, at frst glance hopelessly complcated, systems of equatons can n fact be solved completely by methods of representaton theory! For example, we wll prove the followng theorem. Theorem 1.1. Let k = C be the feld of complex numbers. Then: () The algebra U has exactly one rreducble representaton V d of each dmenson, up to equvalence; ths representaton s realzed n the space of homogeneous polynomals of two varables x, y of degree d 1, and defned by the formulas ρ(h) = x x y y, ρ(e) = x y, ρ(f) = y x. () Any ndecomposable fnte dmensonal representaton of U s rreducble. That s, any fnte dmensonal representaton of U s a drect sum of rreducble representatons. As another example consder the representaton theory of quvers. A quver s a fnte orented graph Q. A representaton of Q over a feld k s an assgnment of a k-vector space V to every vertex of Q, and of a lnear operator A h : V V j to every drected edge h gong from to j. We wll show that a representaton of a quver Q s the same thng as a representaton of a certan algebra P Q called the path algebra of Q. Thus one may ask: what are ndecomposable fnte dmensonal representatons of Q? More specfcally, let us say that Q s fnte f t has fntely many ndecomposable representatons. We wll prove the followng strkng theorem, proved by P. Gabrel about 30 years ago: Theorem 1.2. The fnteness property of Q does not depend on the orentaton of edges. The graphs that yeld fnte quvers are gven by the followng lst: A n : D n : 2

3 E 6 : E 7 : E 8 : The graphs lsted n the theorem are called (smply laced) Dynkn dagrams. These graphs arse n a multtude of classfcaton problems n mathematcs, such as classfcaton of smple Le algebras, sngulartes, platonc solds, reflecton groups, etc. In fact, f we needed to make contact wth an alen cvlzaton and show them how sophstcated our cvlzaton s, perhaps showng them Dynkn dagrams would be the best choce! As a fnal example consder the representaton theory of fnte groups, whch s one of the most fascnatng chapters of representaton theory. In ths theory, one consders representatons of the group algebra A = C[G] of a fnte group G the algebra wth bass a g, g G and multplcaton law a g a h = a gh. We wll show that any fnte dmensonal representaton of A s a drect sum of rreducble representatons,.e. the notons of an rreducble and ndecomposable representaton are the same for A (Maschke s theorem). Another strkng result dscussed below s the Frobenus dvsblty theorem: the dmenson of any rreducble representaton of A dvdes the order of G. Fnally, we wll show how to use representaton theory of fnte groups to prove Burnsde s theorem: any fnte group of order p a q b, where p, q are prmes, s solvable. Note that ths theorem does not menton representatons, whch are used only n ts proof; a purely group-theoretcal proof of ths theorem (not usng representatons) exsts but s much more dffcult! Ths text s based on a mn-course gven by Pavel Etngof at the 2004 Clay Mathematcs Insttute Research Academy. The remanng authors, who were partcpants of the Academy, mproved and extended the ntal lecture notes, and added solutons of homework problems. The goal of the text s not to provde a systematc ntroducton to representaton theory, but rather to convey to the reader the sprt of ths fascnatng subject, and to hghlght ts beauty by dscussng a few strkng results. In other words, the authors would lke to share wth the reader the fun they had durng the days of the Academy! 1. The text contans many problems, whose solutons are gven n the last secton. These problems were gven as homework durng the CMI Research Academy. Sometmes they are desgned to llustrate a noton or result from the man text, but often contan new materal. The problems are a very mportant part of the text, and we recommend the reader to try to solve them after readng the approprate portons. The only serous prerequste for readng ths text s a good famlarty wth lnear algebra, and some profcency n basc abstract algebra (groups, felds, polynomals etc.) The necessary materal s dscussed n the frst seven chapters of Artn s Algebra textbook. Perhaps the only basc noton from lnear algebra we ll need whch s not contaned n standard texts s that of tensor product of 1 For more about the subject, we recommend the reader the excellent textbook of Fulton and Harrs Representaton theory 3

4 vector spaces; we recommend the reader to solve the preparatory problems below to attan a better famlarty wth ths noton. Acknowledgments. The authors are very grateful to the Clay Mathematcs Insttute (and personally to Davd Ellwood and Vda Salah) for hosptalty and wonderful workng condtons. They also are very ndebted to the Academy fellows Josh Nchols-Barrer and Vctor Ostrk, whose mathematcal nsghts and devoton were crucal n makng the Academy a success and n creatng ths text. 1.1 Preparatory problems on tensor products Recall that the tensor product V W of vector spaces V and W over a feld k s the quotent of the space V W whose bass s gven by formal symbols v w, v V, w W, by the subspace spanned by the elements (v 1 +v 2 ) w v 1 w v 2 w, v (w 1 +w 2 ) v w 1 v w 2, (av) w a(v w), v (aw) a(v w), where v V, w W, a k. Problem 1.3. (a) Let U be any k-vector space. maps V W U and lnear maps V W U. Construct a natural bjecton between blnear (b) Show that f {v } s a bass of V and {w j } s a bass of W then {v w j } s a bass of V W. (c) Construct a natural somorphsm V W Hom(V, W ) n the case when V s fnte dmensonal. (d) Let V be a vector space over C. Let S n V be the subspace of V n (n-fold tensor product of V ) whch conssts of the tensors that are symmetrc under permutaton of components. Let Λ n V be the subspace of V n whch conssts of the tensors whch are antsymmetrc,.e., s j T = T, where s j s the permutaton of and j. (These spaces are called the n-th symmetrc, respectvely exteror, power of V ). If {v } s a bass of V, can you construct a bass of S n V, Λ n V? If dmv = m, what are ther dmensons? (e) Let A : V W be a lnear operator. Then we have A n : V n W n, and ts restrctons S n A : S n V S n W, Λ n A : Λ n V Λ n W. Suppose V = W and has dmenson N, and assume that the egenvalues of A are λ 1,..., λ N. Fnd T r(s n A), T r(λ n A). In partcular, show that T r(λ N A) = det(a). Problem 1.4. Let J N be the lnear operator on C N gven n the standard bass by the formula J N e = e 1 for > 1, J N e 1 = 0. Thus J N s a Jordan block of sze N. Fnd the Jordan normal form of the operator B = J N 1 M + 1 N J M on C N C M, where 1 L denotes the dentty operator on C L. Hnt. Compute dmensons of kernels of B j for all j. Problem 1.5. Hlbert s problem. It s known that f A and B are two polygons of the same area then A can be cut by fntely many straght cuts nto peces from whch one can make B. Davd Hlbert asked n 1900 whether t s true for polyhedra n 3 dmensons. In partcular, s t true for a cube and a regular tetrahedron of the same volume? The answer s no, as was found by Dehn n The proof s very beautful. Namely, to any polyhedron A let us attach ts Dehn nvarant D(A) n 4

5 V = R (R/Q) (the tensor product of Q-vector spaces). Namely, D(A) = a l(a) β(a) π, where a runs over edges of A and l(a), β(a) are the length of a and the angle at a. (a) Show that f you cut A nto B and C by a straght cut, then D(A) = D(B) + D(C). (b) Show that α = arccos(1/3)/π s not a ratonal number. Hnt. Let p n, n 1 be the sequence defned by the recursve equaton p n+1 = p 2 n 2, p 0 = 2/3. Show that p n = 2cos2 n πα. On the other hand, show that p n must take nfntely many dfferent values. From ths, derve that α cannot be ratonal. (c) Usng (a) and (b), show that the answer to Hlbert s queston s negatve. (Compute the Dehn nvarant of the regular tetrahedron and the cube). 2 Basc notons of representaton theory 2.1 Algebras Let k be a feld. Unless stated otherwse, we wll assume that k s algebracally closed,.e. any nonconstant polynomal wth coeffcents n k has a root n k. The man example s the feld of complex numbers C, but we wll also consder felds of characterstc p, such as the algebrac closure F p of the fnte feld F p of p elements. Defnton 2.1. An assocatve algebra over k s a vector space A over k together wth a blnear map A A A, (a, b) ab, such that (ab)c = a(bc). Defnton 2.2. A unt n an assocatve algebra A s an element 1 A such that 1a = a1 = a. Proposton 2.3. If a unt exsts, t s unque. Proof. Let 1, 1 be two unts. Then 1 = 11 = 1. From now on, by an algebra A we wll mean an assocatve algebra wth a unt. We wll also assume that A 0. Example 2.4. Here are some examples of algebras over k: 1. A = k. 2. A = k[x 1,..., x n ] the algebra of polynomals n varables x 1,..., x n. 3. A = EndV the algebra of endomorphsms of a vector space V over k (.e., lnear maps from V to tself). The multplcaton s gven by composton of operators. 4. The free algebra A = k x 1,..., x n. The bass of ths algebra conssts of words n letters x 1,..., x n, and multplcaton s smply concatenaton of words. 5. The group algebra A = k[g] of a group G. Its bass s {a g, g G}, wth multplcaton law a g a h = a gh. Defnton 2.5. An algebra A s commutatve f ab = ba for all a, b A. 5

6 For nstance, n the above examples, A s commutatve n cases 1 and 2, but not commutatve n cases 3 (f dm V > 1), and 4 (f n > 1). In case 5, A s commutatve f and only f G s commutatve. Defnton 2.6. A homomorphsm of algebra f : A B s a lnear map such that f(xy) = f(x)f(y) for all x, y A, and f(1) = Representatons Defnton 2.7. A representaton of an algebra A (also called a left A-module) s a vector space V together wth a homomorphsm of algebras ρ : A EndV. Smlarly, a rght A-module s a space V equpped wth an anthomomorphsm ρ : A EndV ;.e., ρ satsfes ρ(ab) = ρ(b)ρ(a) and ρ(1) = 1. The usual abbrevated notaton for ρ(a)v s av for a left module and va for the rght module. Then the property that ρ s an (ant)homomorphsm can be wrtten as a knd of assocatvty law: (ab)v = a(bv) for left modules, and (va)b = v(ab) for rght modules. Example V = V = A, and ρ : A EndA s defned as follows: ρ(a) s the operator of left multplcaton by a, so that ρ(a)b = ab (the usual product). Ths representaton s called the regular representaton of A. Smlarly, one can equp A wth a structure of a rght A-module by settng ρ(a)b := ba. 3. A = k. Then a representaton of A s smply a vector space over k. 4. A = k x 1,..., x n. Then a representaton of A s just a vector space V over k wth a collecton of arbtrary lnear operators ρ(x 1 ),..., ρ(x n ) : V V (explan why!). Defnton 2.9. A subrepresentaton of a representaton V of an algebra A s a subspace W V whch s nvarant under all operators ρ(a) : V V, a A. For nstance, 0 and V are always subrepresentatons. Defnton A representaton V 0 of A s rreducble (or smple) f the only subrepresentatons of V are 0 and V. Defnton Let V 1, V 2 be two representatons over an algebra A. A homomorphsm (or ntertwnng operator) φ : V 1 V 2 s a lnear operator whch commutes wth the acton of A,.e. φ(av) = aφ(v) for any v V 1. A homomorphsm φ s sad to be an somorphsm of representatons f t s an somorphsm of vector spaces. Note that f a lnear operator φ : V 1 V 2 s an somorphsm of representatons then so s the lenar operator φ 1 : V 2 V 1 (check t!). Two representatons between whch there exsts an somorphsm are sad to be somorphc. For practcal purposes, two somorphc representatons may be regarded as the same, although there could be subtletes related to the fact that an somorphsm between two representatons, when t exsts, s not unque. Defnton Let V = V 1, V 2 be representatons of an algebra A. Then the space V 1 V 2 has an obvous structure of a representaton of A, gven by a(v 1 v 2 ) = av 1 av 2. Defnton A representaton V of an algebra A s sad to be ndecomposable f t s not somorphc to a drect sum of two nonzero representatons. 6

7 It s obvous that an rreducble representaton s ndecomposable. On the other hand, we wll see below that the converse statement s false n general. One of the man problems of representaton theory s to classfy rredcble and ndecomposable representatons of a gven algebra up to somorphsm. Ths problem s usually hard and often can be solved only partally (say, for fnte dmensonal representatons). Below we wll see a number of examples n whch ths problem s partally or fully solved for specfc algebras. We wll now prove our frst result Schur s lemma. Although t s very easy to prove, t s fundamental n the whole subject of representaton theory. Proposton (Schur s lemma) Let V 1, V 2 be rreducble representatons of an algebra A over any feld F. Let φ : V 1 V 2 be a nonzero homomorphsm of representatons. Then φ s an somorphsm. Proof. The kernel K of φ s a subrepresentaton of V 1. Snce φ 0, ths subrepresentaton cannot be V 1. So by rreducblty of V 1 we have K = 0. The mage I of φ s a subrepresentaton of V 2. Snce φ 0, ths subrepresentaton cannot be 0. So by rreducblty of V 2 we have I = V 2. Thus φ s an somorphsm. Corollary (Schur s lemma for algebracally closed felds) Let V be a fnte dmensonal rreducble representaton of an algebra A over an algebracally closed feld k, and φ : V V s an ntertwnng operator. Then φ = λ Id (the scalar operator). Proof. Let λ be an egenvalue of φ (a root of the characterstc polynomal of φ). It exsts snce k s an algebracally closed feld. Then the operator φ λid s an ntertwnng operator V V, whch s not an somorphsm (snce ts determnant s zero). Thus by Schur s lemma ths operator s zero, hence the result. Corollary Let A be a commutatve algebra. Then every rreducble fnte dmensonal representaton V of A s 1-dmensonal. Remark. Note that a 1-dmensonal representaton of any algebra s automatcally rreducble. Proof. For any element a A, the operator ρ(a) : V V s an ntertwnng operator. Indeed, ρ(a)ρ(b)v = ρ(ab)v = ρ(ba)v = ρ(b)ρ(a)v (the second equalty s true snce the algebra s commutatve). Thus, by Schur s lemma, ρ(a) s a scalar operator for any a A. Hence every subspace of V s a subrepresentaton. So 0 and V are the only subspaces of V. Ths means that dm V = 1 (snce V 0). Example A = k. Snce representatons of A are smply vector spaces, V = A s the only rreducble and the only ndecomposable representaton. 2. A = k[x]. Snce ths algebra s commutatve, the rreducble representatons of A are ts 1-dmensonal representatons. As we dscussed above, they are defned by a sngle operator ρ(x). In the 1-dmensonal case, ths s just a number from k. So all the rreducble representatons of A are V λ = k, λ k, whch the acton of A defned by ρ(x) = λ. Clearly, these representatons are parwse non-somorphc. The classfcaton of ndecomposable representatons s more nterestng. To obtan t, recall that any lnear operator on a fnte dmensonal vector space V can be brought to Jordan normal 7

8 form. More specfcally, recall that the Jordan block J λ,n s the operator on k n whch n the standard bass s gven by the formulas J λ,n e = λe + e 1 for > 1, and J λ,n e 1 = λe 1. Then for any lnear operator B : V V there exsts a bass of V such that the matrx of B n ths bass s a drect sum of Jordan blocks. Ths mples that all the ndecomosable representatons of A are V λ,n = k n, λ k, wth ρ(x) = J λ,n. The fact that these representatons are ndecomposable and parwse non-somorphc follows from the Jordan normal form theorem (whch n partcular says that the Jordan normal form of an operator s unque up to permutaton of blocks). Ths example shows that an ndecomposable representaton of an algebra need not be rreducble. Problem Let V be a nonzero fnte dmensonal representaton of an algebra A. Show that t has an rreducble subrepresentaton. Then show by example that ths does not always hold for nfnte dmensonal representatons. Problem Let A be an algebra over an algebracally closed feld k. The center Z(A) of A s the set of all elements z A whch commute wth all elements of A. For example, f A s commutatve then Z(A) = A. (a) Show that f V s an rreducble fnte dmensonal representaton of A then any element z Z(A) acts n V by multplcaton by some scalar χ V (z). Show that χ V : Z(A) k s a homomorphsm. It s called the central character of V. (b) Show that f V s an ndecomposable fnte dmensonal representaton of A then for any z Z(A), the operator ρ(z) by whch z acts n V has only one egenvalue χ V (z), equal to the scalar by whch z acts on some rreducble subrepresentaton of V. Thus χ V : Z(A) k s a homomorphsm, whch s agan called the central character of V. (c) Does ρ(z) n (b) have to be a scalar operator? Problem Let A be an assocave algebra, and V a representaton of A. By End A (V ) one denotes the algebra of all morphsms of representatons V V. Show that End A (A) = A op, the algebra A wth opposte multplcaton. Problem Prove the followng Infnte dmensonal Schur s lemma (due to Dxmer): Let A be an algebra over C and V be an rreducble representaton of A wth at most countable bass. Then any homomorphsm of representatons φ : V V s a scalar operator. Hnt. By the usual Schur s lemma, the alegbra D := End A (V ) s an algebra wth dvson. Show that D s at most countably dmensonal. Suppose φ s not a scalar, and consder the subfeld C(φ) D. Show that C(φ) s a smple transcendental extenson of C. Derve from ths that C(φ) s uncountably dmensonal and obtan a contradcton. 2.3 Ideals A left deal of an algebra A s a subspace I A such that ai I for all a A. Smlarly, a rght deal of an algebra A s a subspace I A such that Ia I for all a A. A two-sded deal s a subspace that s both a left and a rght deal. Left deals are the same as subrepresentatons of the regular representaton A. Rght deals are the same as subrepresentatons of the regular representaton of the opposte algebra A op, n whch the acton of A s rght multplcaton. Below are some examples of deals: If A s any algebra, 0 and A are two-sded deals. An algebra A s called smple f 0 and A are ts only two-sded deals. 8

9 If φ : A B s a homomorphsm of algebras, then ker φ s a two-sded deal of A. If S s any subset of an algebra A, then the two-sded deal generated by S s denoted S and s the span of elements of the form asb, where a, b A and s S. Smlarly we can defne S l = span{as} and S r = span{sb}, the left, respectvely rght, deal generated by S. 2.4 Quotents Let A be an algebra and I a two-sded deal n A. Then A/I s the set of (addtve) cosets of I. Let π : A A/I be the quotent map. We can defne multplcaton n A/I by π(a) π(b) := π(ab). Ths s well-defned because f π(a) = π(a ) then π(a b) = π(ab + (a a)b) = π(ab) + π((a a)b) = π(ab) because (a a)b Ib I = ker π, as I s a rght deal; smlarly, f π(b) = π(b ) then π(ab ) = π(ab + a(b b)) = π(ab) + π(a(b b)) = π(ab) because a(b b) ai I = ker π, as I s also a left deal. well-defned, and A/I s an algebra. Thus multplcaton n A/I s Smlarly, f V s a representaton of A, and W V s a subrepresentaton, then V/W s also a representaton. Indeed, let π : V V/W be the quotent map, and set ρ V/W (a)π(x) := π(ρ V (a)x). Above we noted the equvalence of left deals of A and subrepresentatons of the regular representaton of A. Thus, f I s a left deal n A, then A/I s a representaton of A. Problem Let A = k[x 1,..., x n ] and I A be any deal n A contanng all homogeneous polynomals of degree N. Show that A/I s an ndecomposable representaton of A. Problem Let V 0 be a representaton of A. We say that a vector v V s cyclc f t generates V,.e., Av = V. A representaton admttng a cyclc vector s sad to be cyclc. Show that (a) V s rreducble f and only f all nonzero vectors of V are cyclc. (b) V s cyclc f and only f t s somorphc to A/I, where I s a left deal n A. (c) Gve an example of an ndecomposable representaton whch s not cyclc. Hnt. Let A = C[x, y]/i 2, where I 2 s the deal spanned by homogeneous polynomals of degree 2 (so A has a bass 1, x, y). Let V = A be the space of lnear functonals on A, wth the acton of A gven by (ρ(a)f)(b) = f(ba). Show that V provdes a requred example. 2.5 Algebras defned by generators and relatons A representaton V of A s sad to be generated by a subset S of V f V s the span of {as a A, s S}. If f 1,..., f m are elements of the free algebra k x 1,..., x n, we say that the algebra A := k x 1,..., x n / {f 1,..., f m } s generated by x 1,..., x n wth defnng relatons f 1 = 0,..., f m = 0. 9

10 2.6 Examples of algebras Throughout the followng examples G wll denote a group, and k a feld. 1. The group algebra k[g], whose bass s {e g g G}, and where multplcaton s defned by e g e h = e gh. A representaton of a group G over a feld k s a homomorphsm of groups ρ : G GL(V ), where V s some vector space over k. In fact, a representaton of G over k s the same thng as a representaton of k[g]. 2. The Weyl algebra, k x, y / yx xy 1. A bass for the Weyl algebra s {x y j } (show ths). The space C[t] s a representaton of the Weyl algebra over C wth acton gven by xf = tf and yf = df/dt for all f C[t]. Thus, the Weyl algebra over C s the algebra of polynomal dfferental operators. Defnton. A representaton ρ : A End V s fathful f ρ s njectve. C[t] s a fathful representaton of the Weyl algebra. 3. The q-weyl algebra over k, generated by x +, x, y +, y wth defnng relatons y + x + = qx + y + and x + x = x x + = y + y = y y + = 1. One customarly wrtes x + as x, x as x 1, y + as y, and y as y 1. Problem Let A be the Weyl algebra, generated over an algebracally closed feld k by two generators x, y wth the relaton yx xy = 1. (a) If chark = 0, what are the fnte dmensonal representatons of A? What are the two-sded deals n A?. (b) Suppose for the rest of the problem that chark = p. What s the center of A? (c) Fnd all rreducble fnte dmensonal representatons of A. Problem Let q be a nonzero complex number, and A be the algebra over C generated by X ±1 and Y ±1 wth defnng relatons XX 1 = X 1 X = 1, Y Y 1 = Y 1 Y = 1, and XY = qy X. (a) What s the center of A for dfferent q? If q s not a root of unty, what are the two-sded deals n A? (b) For whch q does ths algebra have fnte dmensonal representatons? (c) Fnd all fnte dmensonal rreducble representatons of A for such q. 2.7 Quvers Defnton A quver Q s a drected graph, possbly wth self-loops and/or multple arrows between two vertces. Example

11 We denote the set of vertces of the quver Q as I, and the set of edges as E. For an edge h E, let h, h denote the source and target, respectvely, of h. h h h Defnton A representaton of a quver Q s an assgnment to each vertex I of a vector space V and to each edge h E of a lnear map x h : V h V h. It turns out that the theory of representatons of quvers s part of the theory of representatons of algebras n the sense that for each quver Q, there exsts a certan algebra P Q, called the path algebra of Q, such that a representaton of the quver Q s the same as a representaton of the algebra P Q. We shall frst defne the path algebra of a quver and then justfy our clam that representatons of these two objects are the same. Defnton The path algebra P Q of a quver Q s the algebra generated by p for I and a h for h E wth the relatons: 1. p 2 = p, p p j = 0 for j 2. a h p h = a h, a h p j = 0 for j h 3. p h a h = a h, p a h = 0 for h Remark It s easy to see that I p = 1, and that a h2 a h1 = 0 f h 1 h 2. Ths algebra s called the path algebra because a bass of P Q s formed by elements a π, where π s a path n Q (possbly of length 0). If π = h n h 2 h 1 (read rght-to-left), then a π = a hn a h2 a h1. If π s a path of length 0, startng and endng at pont, then a π s defned to be p. Clearly a π2 a π1 = a π2 π 1 (where π 2 π 1 s the concatenaton of paths π 2 and π 1 ) f the fnal pont of π 1 equals the ntal pont of π 2 and a π2 a π1 = 0 otherwse. We now justfy our statement that a representaton of a quver s the same as a representaton of the path algebra of a quver. Let V be a representaton of the path algebra P Q. From ths representaton of the algebra P Q, we can construct a representaton of Q as follows: let V = p V and let x h = a h ph V : p h V p h V. Smlarly, let (V, x h ) be a representaton of a quver Q. From ths representaton, we can construct a representaton of the path algebra P Q : let V = V, let p : V V V be the projecton onto V, and let a h = h x h p h : V V h V h V where h : V h V s the ncluson map. It s clear that the above assgnments V (p V) and (V ) V are nverses of each other. Thus, we have a bjecton between somorphsm classes of representatons of the algebra P Q and of the quver Q. Remark In practce, t s generally easer to consder a representaton of a quver as n Defnton The above serves to show, as stated before, that the theory of representatons of quvers s a part of the larger theory of representatons of algebras. We lastly defne several prevous concepts n the context of quvers representatons. Defnton A subrepresentaton of a representaton (V, x h ) of a quver Q s a representaton (W, x h ) where W V for all I and where x h (W h ) W h and x h = x h Wh : W h W h for all h E. 11

12 Defnton The drect sum of two representatons (V, x h ) and (W, y h ) s the representaton (V W, x h y h ). As wth representatons of algebras, a representaton (V ) of a quver Q s sad to be rreducble f ts only subrepresentatons are (0) and (V ) tself, and ndecomposable f t s not somorphc to a drect sum of two nonzero representatons. Defnton Let (V, x h ) and (W, y h ) be representatons of the quver Q. A homomorphsm ϕ : (V ) (W ) of quver representatons s a collecton of maps ϕ : V W such that y h ϕ h = ϕ h x h for all h E. Problem Let A be a Z + -graded algebra,.e., A = n 0 A[n], and A[n] A[m] A[n + m]. If A[n] s fnte dmensonal, t s useful to consder the Hlbert seres h A (t) = dm A[n]t n (the generatng functon of dmensons of A[n]). Often ths seres converges to a ratonal functon, and the answer s wrtten n the form of such functon. For example, f A = k[x] and deg(x n ) = n then Fnd the Hlbert seres of: h A (t) = 1 + t + t t n +... = 1 1 t (a) A = k[x 1,..., x m ] (where the gradng s by degree of polynomals); (b) A = k < x 1,..., x m > (the gradng s by length of words); (c) A s the exteror algebra k [x 1,..., x m ], generated by x 1,..., x m wth the defnng relatons x x j + x j x = 0 for all, j (gradng s by degree). (d) A s the path algebra P Q of a quver Q as defned n the lectures. Hnt. The closed answer s wrtten n terms of the adjacency matrx M Q of Q. 2.8 Le algebras Let g be a vector space over a feld k, and let [, ] : g g g be a skew-symmetrc blnear map. (So [a, b] = [b, a].) If k s of characterstc 2, we also requre that [x, x] = 0 for all x (a requrement equvalent to [a, b] = [b, a] n felds of other characterstcs). Defnton (g, [, ]) s a Le algebra f [, ] satsfes the Jacob dentty [ [a, b], c ] + [ [b, c], a ] + [ [c, a], b ] = 0. (2) Example Some examples of Le algebras are: 1. R 3 wth [u, v] = u v, the cross-product of u and v 2. Any space g wth [, ] = 0 (abelan Le algebra) 3. Any assocatve algebra A wth [a, b] = ab ba 4. Any subspace U of an assocatve algebra A such that [a, b] U for all a, b U 5. sl(n), the set of n n matrces wth trace 0 For example, sl(2) has the bass ( ) ( ) e = f = wth relatons [e, f] = h, [h, f] = 2f, [h, e] = 2e. h = ( )

13 ( 0 ) 6. The Hesenberg Le algebra H of matrces It has the bass x = y = c = wth relatons [y, x] = c and [y, c] = [x, c] = The algebra aff(1) of matrces ( 0 0 ) Its bass conssts of X = ( ) and Y = ( ), wth [X, Y ] = Y. 8. so(n), the space of skew-symmetrc n n matrces, wth [a, b] = ab ba Defnton Let g 1, g 2 be Le algebras. A homomorphsm ϕ : g 1 g 2 of Le algebras s a lnear map such that ϕ([a, b]) = [ϕ(a), ϕ(b)]. Defnton A representaton of a Le algebra g s a vector space V wth a homomorphsm of Le algebras ρ : g End V. Example Some examples of representatons of Le algebras are: 1. V=0 2. Any vector space V wth ρ = 0 3. Adjont representaton V = g wth ρ(a)(b) = [a, b] def = ab ba That ths s a representaton follows from Equaton (2). It turns out that a representaton of a Le algebra g s the same as a representaton of a certan assocatve algebra U(g). Thus, as wth quvers, we can vew the theory of representatons of Le algebras as part of the theory of representatons of assocatve algebras. Defnton Let g be a Le algebra wth bass x and [, ] defned by [x, x j ] = k ck j x k. The unversal envelopng algebra U(g) s the assocatve algebra generated by the x s wth the relatons x x j x j x = k ck j x k. Example The assocatve algebra U(sl(2)) s the algebra generated by e, f, h wth relatons he eh = 2e hf fh = 2f ef fe = h. Example The algebra U(H), where H s the Hesenberg Le algebra of Example , s the algebra generated by x, y, c wth the relatons yx xy = c yc cy = 0 xc cx = 0. The Weyl algebra s the quotent of U(H) by the relaton c = 1. Fnally, let us defne the mportant noton of tensor product of representatons. Defnton The tensor product of two representatons V, W of a Le algebra g s the space V W wth ρ V W (x) = ρ V (x) Id + Id ρ W (x). It s easy to check that ths s ndeed a representaton. 13

14 Problem Representatons of sl(2). Accordng to the above, a representaton of sl(2) s just a vector space V wth a trple of operators E, F, H such that HE EH = 2E, HF F H = 2F, EF F E = H (the correspondng map ρ s gven by ρ(e) = E, ρ(f) = F, ρ(h) = H. Let V be a fnte dmensonal representaton of sl(2) (the ground feld n ths problem s C). (a) Take egenvalues of H and pck one wth the bggest real part. Call t λ. Let V (λ) be the generalzed egenspace correspondng to λ. Show that E V (λ) = 0. (b) Let W be any representaton of sl(2) and w W be a nonzero vector such that Ew = 0. For any k > 0 fnd a polynomal P k (x) of degree k such that E k F k w = P k (H)w. (Frst compute EF k w, then use nducton n k). (c) Let v V (λ) be a generalzed egenvector of H wth egenvalue λ. Show that there exsts N > 0 such that F N v = 0. (d) Show that H s dagonalzable on V (λ). (Take N to be such that F N = 0 on V (λ), and compute E N F N v, v V (λ), by (b). Use the fact that P k (x) does not have multple roots). (e) Let N v be the smallest N satsfyng (c). Show that λ = N v 1. (f) Show that for each N > 0, there exsts a unque up to somorphsm rreducble representaton of sl(2) of dmenson N. Compute the matrces E, F, H n ths representaton usng a convenent bass. (For V fnte dmensonal rreducble take λ as n (a) and v V (λ) an egenvector of H. Show that v, F v,..., F λ v s a bass of V, and compute matrces of all operators n ths bass.) Denote the λ + 1-dmensonal rreducble representaton from (f) by V λ. Below you wll show that any fnte dmensonal representaton s a drect sum of V λ. (g) Show that the operator C = EF + F E + H 2 /2 (the so-called Casmr operator) commutes wth E, F, H and equals λ(λ+2) 2 Id on V λ. Now t wll be easy to prove the drect sum decomposton. Assume the contrary, and let V be a representaton of the smallest dmenson, whch s not a drect sum of smaller representatons. (h) Show that C has only one egenvalue on V, namely λ(λ+2) 2 for some nonnegatve nteger λ. (use that the generalzed egenspace decomposton of C must be a decomposton of representatons). () Show that V has a subrepresentaton W = V λ such that V/W = nv λ for some n (use (h) and the fact that V s the smallest whch cannot be decomposed). (j) Deduce from () that the egenspace V (λ) of H s n + 1-dmensonal. If v 1,..., v n+1 s ts bass, show that F j v, 1 n + 1, 0 j λ are lnearly ndependent and therefore form a bass of V (establsh that f F x = 0 and Hx = µx then Cx = µ(µ 2) 2 x and hence µ = λ). (k) Defne W = span(v, F v,..., F λ v ). Show that V are subrepresentatons of V and derve a contradcton wth the fact that V cannot be decomposed. (l) (Jacobson-Morozov Lemma) Let V be a fnte dmensonal complex vector space and A : V V a nlpotent operator. Show that there exsts a unque, up to an somorphsm, representaton of sl(2) on V such that E = A. (Use the classfcaton of the representatons and Jordan normal form theorem) (m) (Clebsch-Gordan decomposton) Fnd the decomposton nto rreducbles of the representaton V λ V µ of sl(2). Hnt. For a fnte dmensonal representaton V of sl(2) t s useful to ntroduce the character χ V (x) = T r(e xh ), x C. Show that χ V W (x) = χ V (x) + χ W (x) and χ V W (x) = χ V (x)χ W (x). 14

15 Then compute the character of V λ and of V λ V µ and derve the decomposton. Ths decomposton s of fundamental mportance n quantum mechancs. (n) Let V = C M C N, and A = J M (0) Id N + Id M J N (0). Fnd the Jordan normal form of A usng (l),(m), and compare the answer wth Problem 1.4. Problem (Le s Theorem) Recall that the commutant K(g) of a Le algebra g s the lnear span of elements [x, y], x, y g. Ths s an deal n g (.e. t s a subrepresentaton of the adjont representaton). A fnte dmensonal Le algebra g over a feld k s sad to be solvable f there exsts n such that K n (g) = 0. Prove the Le theorem: f k = C and V s a fnte dmensonal rreducble representaton of a solvable Le algebra g then V s 1-dmensonal. Hnt. Prove the result by nducton n dmenson. By the nducton assumpton, K(g) has a common egenvector v n V, that s there s a lnear functon χ : K(g) C such that av = χ(a)v for any a K(g). Show that g preserves common egenspaces of K(g) (for ths you wll need to show that χ([x, a]) = 0 for x g and a K(g). To prove ths, consder the smallest vector subspace U contanng v and nvarant under x. Ths subspace s nvarant under K(g) and any a K(g) acts wth trace dm(u)χ(a) n ths subspace. In partcular 0 = Tr([x, a]) = dm(u)χ([x, a]).). Problem Classfy rreducble representatons of the two dmensonal Le algebra wth bass X, Y and commutaton relaton [X, Y ] = Y. Consder the cases of zero and postve characterstc. Is the Le theorem true n postve characterstc? Problem (hard!) For any element x of a Le algebra g let ad(x) denote the operator g g, y [x, y]. Consder the Le algebra g n generated by two elements x, y wth the defnng relatons ad(x) 2 (y) = ad(y) n+1 (x) = 0. (a) Show that the Le algebras g 1, g 2, g 3 are fnte dmensonal and fnd ther dmensons. (b) (harder!) Show that the Le algebra g 4 has nfnte dmenson. Construct explctly a bass of ths algebra. 3 General results of representaton theory 3.1 The densty theorem Theorem 3.1. (The densty theorem) Let A be an algebra, and let V 1, V 2,..., V r be parwse nonsomorphc rreducble fnte dmensonal representatons of A, wth homomorphsms ρ : A End V. Then the homomorphsm ρ 1 ρ 2 ρ r : A r End V =1 s surjectve. Proof. Let V N1,...,N r = (V 1 V 1 ) (V }{{} 2 V 2 ) (V }{{} r V r ). }{{} N 1 copes N 2 copes N r copes Let p j : V N1,...,N r V be the projecton onto V j, the j th copy of V. We shall need the followng two lemmas: 15

16 Lemma 3.2. Let W V = V N1,...,N r be a subrepresentaton. Then, there exsts an automorphsm α of V such that p N (α(w )) = 0 for some. Proof of Lemma 3.2. We nduct on N = r =1 N. The base case, N = 0, s clear. For the nductve step, frst pck an rreducble nonzero subrepresentaton Y of W (whch clearly exsts by Problem 2.18). As Y 0, there exst some, j such that p j Y : Y V s nonzero. Assume, wthout loss of generalty, that j = 1. As p 1 Y s nonzero, by Schur s lemma, p 1 Y s an somorphsm, and Y = V. As Y can be somorphc to only one V l, p lm Y = 0 for all l, m. As p m Y (p 1 Y ) 1 : V Y V s a homomorphsm, by Schur s lemma for algebracally closed felds, p m Y (p 1 Y ) 1 = λ m Id and p m Y = λ m p 1 Y for some scalar λ m. We now defne an automorphsm γ : V V. For v V, we wrte v = (v lm ) where v lm = p lm (v) V l. We now let γ(v) = (v lm ), where v lm = v lm for l, v 1 = v 1, v m = v m λ m v 1 for 2 m N. Clearly γ s an automorphsm. Suppose that v Y. As p lm Y = 0 for all l, m, v lm = v lm = p lm (v) = 0 for l, m. As p m Y = λ m p 1 Y, v m = λ m v 1 and v m = v m λ m v 1 = 0 for m 1. Thus, p lm (γ(y )) = v lm = 0 unless l = and m = 1. Also, p 1(γ(Y )) = Y. Consder the map ϕ : V = V N1,...,N,,...,N r V N1,...,N 1,...,N r (v lm ) (v lm for (l, m) (, 1)) whch has V 1 (the frst copy of V ) as ts kernel. But ths s also γ(y ), so ker ϕ = γ(y ). As γ(y ) γ(w ), ker ϕ γ(w ) = γ(y ) and ϕ(γ(w )) = γ(w )/γ(y ) = W/Y. By the nductve hypothess, there exsts some β : V N1,...,N 1,...,N r V N1,...,N 1,...,N r such that for some l, p ln l (β(ϕ(γ(w )))) = 0, where N l = N l for l and N = N 1. Defne α = (Id V1 β) γ (where Id V1 s the dentty on the frst copy of V ). As Id V1 ϕ = Id V, Id V1 β = (Id V1 0) + (β ϕ). Thus, p lnl (α(w )) = p ln l (β(ϕ(γ(w )))) = 0 (as (l, N l ) (, 1)). Ths completes the nductve step. Lemma 3.3. Let W V N1,...,N r be a subrepresentaton. Then, W = V M1,...,M r for some M N. Proof of Lemma 3.3. We nduct on N = r =1 N. The base case, N = 0 s clear, as then W = 0. If W = V N1,...,N r, then we smply have M = N. Otherwse, by Lemma 3.2, there exsts an automorphsm α of V N1,...,N r such that p N (α(w )) = 0 for some. Thus, α(w ) V N1,...,N 1,...,N r. By the nductve assumpton, W = α(w ) = V M1,...,M r. Proof of Theorem 3.1. Frst, by replacng A wth A/ ker ρ 1 ρ 2 ρ r, we can assume, wthout loss of generalty, that the map ρ 1 ρ 2 ρ r s njectve. As A s now somorphc to ts mage, we can also assume that A End V 1 End V r. Thus, A s a subrepresentaton of End V 1 End V r. Let d = dm V. As End V = V V }{{}, we have A V d1,...,d r. Thus, by Lemma 3.3, d copes A = V M1,...,M r for some M d. Thus, dm A = r =1 M d. 16

17 Next, consder End A (A). As the V s are parwse non-somorphc, by Schur s lemma, no copy of V n A can be mapped to a dstnct V j. Also, by Schur, End A (V ) = k. Thus, End A (A) = Mat M (k), so dm End A (A) = r =1 M 2. By Problem 2.20, End A(A) = A, so r =1 M 2 = dm A = r =1 M d. Thus, r =1 M (d M ) = 0. As M d, d M 0. Next, as 1 A, the map ρ : A End V s nontrval. As End V s a drect sum of copes of V, A must contan a copy of V. Thus M > 0, and we must have d M = 0 for all, so M = d, and A = r =1 End V. 3.2 Representatons of matrx algebras In ths secton we consder representatons of algebras A = Mat d (k). Theorem 3.4. Let A = r =1 Mat d (k). Then the rreducble representatons of A are V 1 = k d 1,..., V r = k dr, and any fnte dmensonal representaton of A s a drect sum of copes of V 1,..., V r. In order to prove Theorem 3.4, we shall need the noton of a dual representaton. Defnton 3.5. (Dual representaton) Let V be a representaton of A. Then the dual space V s a rght A-module (or equvalently, a representaton of A op ) wth the acton f a = (v f(av)), as (f (ab)) (v) = f((ab)v) = f(a(bv)) = (f a)(bv) = ((f a) b)(v). Proof of Theorem 3.4. Frst, the gven representatons are clearly rreducble, as for any v, w V 0, there exsts a A such that av = w. Next, let X be an n dmensonal representaton of A. Then, X s an n dmensonal representaton of A op. But (Mat d (k)) op = Matd (k) wth somorphsm ϕ(x) = X T, as (AB) T = B T A T. Thus, A = A op and X s an n dmensonal representaton of A. Defne φ : A } {{ A } X n copes by φ(a 1,..., a n ) = a 1 x a nx n where {x } s a bass of X. φ s clearly surjectve, as k A. Thus, the dual map φ : X A n s njectve. But A n = A n. Hence, Im φ = X s a subrepresentaton of A n. Next, as Mat d (k) = V d, A = V d1,...,d r and A n = V nd1,...,nd r. By Lemma 3.3, X = V M1,...,M r = M 1 V 1 M r V r for some M as desred. 3.3 Fnte dmensonal algebras Defnton 3.6. The radcal I of a fnte dmensonal algebra A s the set of all elements of A whch act by 0 n all rreducble representatons of A. It s denoted Rad(A). Proposton 3.7. Rad(A) s a two-sded deal. Proof. If I, a A, v V, where V s any rreducble representaton of A, then av = a 0 = 0 and av = v = 0 where v = av. 17

18 Theorem 3.8. A fnte dmensonal algebra A has only fntely many rreducble representatons V up to somorphsm, these representatons are fnte dmensonal, and A/I = End V, where I = Rad(A). Proof. Frst, for any rreducble representaton V of A, and for any nonzero v V, Av V s a fnte dmensonal subrepresentaton of V. (It s fnte dmensonal as A s fnte dmensonal.) As V s rreducble and Av 0, V = Av and V s fnte dmensonal. Next, suppose that we had nfntely many non-somorphc rreducble representatons. Let V 1, V 2,..., V r be any r nontrval non-somorphc rreducble representatons, wth r > dm A. By Theorem 3.1, the homomorphsm ρ : A End V s surjectve. But ths s mpossble as dm End V r > dm A. Thus, A has only fntely many non-somorphc rreducble representatons. Next, let V 1, V 2,..., V r be all non-somorphc rreducble fnte dmensonal representatons of A. By Theorem 3.1, the homomorphsm ρ : A End V s surjectve. The kernel of ths map s exactly I. Corollary 3.9. (dm V ) 2 dm A, where the V s are the rreducble representatons of A. Proof. As dm End V = (dm V ) 2, Theorem 3.8 mples that dm A dm I = dm End V = (dm V ) 2. As dm I 0, (dm V ) 2 dm A. Defnton A fnte dmensonal algebra A s sad to be semsmple f Rad(A) = 0. Proposton For a fnte dmensonal algebra A over an algebracally closed feld k, the followng are equvalent: 1. A s semsmple 2. (dm V ) 2 = dm A, where the V s are the rreducble representatons of A 3. A = Mat d (k) for some d 4. Any fnte dmensonal representaton of A s completely reducble (that s, somorphc to a drect sum of rreducble representatons) 5. A (as a vector space) s a completely reducble representaton of A Proof. As dm A dm I = (dm V ) 2, clearly dm A = (dm V ) 2 f and only f I = 0. Thus, (1) (2). Next, by Theorem 3.8, f I = 0, then clearly A = Mat d (k) for d = dm V. Thus, (1) (3). Conversely, f A = Mat d (k), then A = End U for some U s wth dm U = d. Clearly 18

19 each U s rreducble (as for any u, u U 0, there exsts a A such that au = u ), and the U s are parwse non-somorphc representatons. Thus, the U s form a subset of the rreducble representatons V of A. Thus, dm A = (dm U ) 2 (dm V ) 2 dm A. Thus, (3) (2)( (1)). Next (3) (4) by Theorem 3.4. Clearly (4) (5). To see that (5) (3), let A = n V. Consder End A (A) (endomorphsms of A as a representaton of A). As the V s are parwse nonsomorphc, by Schur s lemma, no copy of V n A can be mapped to a dstnct V j. Also, by Schur, End A (V ) = k. Thus, End A (A) = Mat n (k). But End A (A) = A op by Problem 2.20, so A op = Mat n (k). Thus, A = ( Mat n (k)) op = (Mat n (k)) op. But (Mat n (k)) op = Matn (k) wth somorphsm ϕ(x) = X T, as (AB) T = B T A T. Thus, A = Mat n (k). Let A be an algebra and V a fnte-dmensonal representaton of A wth acton ρ. Then the character of V s the lnear functon χ V : A k gven by χ V (a) = tr V (ρ(a)). If [A, A] s the span of commutators [x, y] := xy yx over all x, y A, then [A, A] ker χ V. Thus, we may vew the character as a mappng χ V : A/[A, A] k. Theorem (1) Characters of rreducble fnte-dmensonal representatons of A are lnearly ndependent. (2) If A s a fnte-dmensonal semsmple algebra, then the characters form a bass of (A/[A, A]). Proof. (1) If V 1,..., V r are nonsomorphc rreducble fnte-dmensonal representatons of A, then ρ V1 ρ Vr : A End V 1 End V r s surjectve by the densty theorem, so χ V1,..., χ Vr are lnearly ndependent. (Indeed, f λ χ V (a) = 0 for all a A, then λ Tr(M ) = 0 for all M End k V. But each tr(m ) can range ndependently over k, so t must be that λ 1 = = λ r = 0.) (2) Frst we prove that [Mat d (k), Mat d (k)] = sl d (k), the set of all matrces wth trace 0. It s clear that [Mat d (k), Mat d (k)] sl d (k). If we denote by E j the matrx wth 1 n the th row of the jth column and 0 s everywhere else, we have [E j, E jm ] = E m for m, and [E,+1, E +1, ] = E E +1,+1. Now {E m } {E E +1,+1 } forms a bass n sl d (k), and ndeed [Mat d (k), Mat d (k)] = sl d (k), as clamed. By semsmplcty, we can wrte A = Mat d1 k Mat dr k. Then [A, A] = sl d1 (k) sl dr (k), and A/[A, A] = k r. By the corollary to the densty theorem, there are exactly r rreducble representatons of A (somorphc to k d 1,..., k dr, respectvely), and therefore r lnearly ndependent characters n the r-dmensonal vector space A/[A, A]. Thus, the characters form a bass. 3.4 Jordan-Holder and Krull-Schmdt theorems To conclude the dscusson of assocatve algebras, let us state two mportant theorems about ther fnte dmensonal representatons. Let A be an algebra over an algebracally closed feld k. Let V be a representaton of A. A (fnte) fltraton of A s a sequence of subrepresentatons 0 = V 0 V 1... V n = V. Theorem (Jordan-Holder theorem). Let V be a fnte dmensonal representaton of A, and 0 = V 0 V 1... V n = V, 0 = V 0... V m = V be fltratons of V, such that the representatons W := V /V 1 and W := V /V 1 are rreducble for all. Then n = m, and there exsts a permutaton σ of 1,..., n such that W σ() s somorphc to W. 19

20 Proof. Frst proof (for k of characterstc zero). Let I A be the annhlatng deal of V (.e. the set of elements that act by zero n V ). Replacng A wth A/I, we may assume that A s fnte dmensonal. The character of V obvously equals the sum of characters of W, and also the sum of characters of W. But by Theorem 3.12, the characters of rreducble representatons are lnearly ndependent, so the multplcty of every rreducble representaton W of A among W and among are the same. Ths mples the theorem. W Second proof (general). The proof s by nducton on dm V. The base of nducton s clear, so let us prove the nducton step. If W 1 = W 1 (as subspaces), we are done, snce by the nducton assumpton the theorem holds for V/W 1. So assume W 1 W 1. In ths case W 1 W 1 = 0 (as W 1, W 1 are rreducble), so we have an embeddng f : W 1 W 1 V. Let U = V/(W 1 W 1 ), and 0 = U 0 U 1... U p = U be a fltraton of U wth smple quotents Z = U /U 1. Then we see that: 1) V/W 1 has a fltraton wth successve quotents W 1, Z 1,..., Z p, and another fltraton wth successve quotents W 2,..., W n. 2) V/Y has a fltraton wth successve quotents W 1, Z 1,..., Z p, and another fltraton wth successve quotents W 2,..., W n. By the nducton assumpton, ths means that the collecton of rreducble modules wth multplctes W 1, W 1, Z 1,..., Z p concdes on one hand wth W 1,..., W n, and on the other hand, wth W 1,..., W m. We are done. Theorem (Krull-Schmdt theorem) Any fnte dmensonal representaton of A can be unquely (up to order of summands) decomposed nto a drect sum of ndecomposable representatons. Proof. It s clear that a decomposton of V nto a drect sum of ndecomposable representaton exsts, so we just need to prove unqueness. We wll prove t by nducton on dm V. Let V = V 1... V m = V 1... V n. Let s : V s V, s : V s V, p s : V V s, p s : V V s be the natural maps assocated to these decompostons. Let θ s = p 1 sp s 1 : V 1 V 1. We have n s=1 θ s = 1. Now we need the followng lemma. Lemma Let W be a fnte dmensonal ndecomposable representaton of A. Then () Any homomorphsm θ : W W s ether an somorphsm of nlpotent; () If θ s : W W, s = 1,..., n are nlpotent homomorphsms, then so s θ := θ θ n. Proof. () Generalzed egenspaces of θ are subrepresentatons of V, and V s ther drect sum. Thus, θ can have only one egenvalue λ. If λ s zero, θ s nlpotent, otherwse t s an somorphsm. () The proof s by nducton n n. The base s clear. To make the nducton step (n 1 to n), assume that θ s not nlpotent. Then by () θ s an somorphsm, so n =1 θ 1 θ = 1. The morphsms θ 1 θ are not somorphsms, so they are nlpotent. Thus 1 theta 1 θ n = θ 1 θ θ 1 θ n 1 s an somorphsm, whch s a contradcton wth the nducton assumpton. By the lemma, we fnd that for some s, θ s must be an somorphsm; we may assume that s = 1. In ths case, V 1 = Imp 1 1 Ker(p 1 1 ), so snce V 1 s ndecomposable, we get that f := p 1 1 : V 1 V 1 and g := p 1 1 : V 1 V 1 are somorphsms. Let B = j>1 V j, B = j>1 V j ; then we have V = V 1 B = V 1 B. Consder the map h : B B defned as a composton of the natural maps B V B attached to these decompostons. We clam that h s an somorphsm. To show ths, t suffces to show that Kerh = 0 20