Definition 1.2 An algebra A is called a division algebra if every nonzero element a has a multiplicative inverse b ; that is, ab = ba = 1.

Transcription

1 1 Semisimple igs ad modules The mateial i these otes is based upo the teatmets i S Lag, Algeba, Thid Editio, chaptes 17 ad 18 ; J-P See, Liea epesetatios of fiite goups ad N Jacobso, Basic Algeba, II Sectio 1 Algebas Defiitio 11 A algeba A ove a field k is a k -vecto space A togethe with a k -biliea map µ : A x A A The map µ is the multiplicatio opeatio i A, ad we shall assume that multiplicatio is associative We shall assume i additio that A has a uit elemet 1 (ie µ(1,a) = µ(a,1) = a fo all a A) ad that k lies i the (multiplicative) cete of A Evey algeba A is a ig with a vecto space stuctue attached to the opeatio + Defiitio 12 A algeba A is called a divisio algeba if evey ozeo elemet a has a multiplicative ivese b ; that is, ab = ba = 1 Defiitio 13 A map f betwee two k -algebas A ad B is a algeba homomophism if f is a k -liea map of vecto spaces such that f(aa*) = f(a) f(a*) fo all elemets a, a* i A Examples 14 1) Let V be a vecto space ove a field k, ad let Ed k (V) deote the set of k -liea tasfomatios of V Clealy Ed k (V) is a vecto space ove k, ad the multiplicatio opeatio i Ed k (V) is give by compositio 2) Let M (k) deote the collectio of x matices whose elemets lie i a field k M (k) is a vecto space of dimesio 2 ove k i which the additio opeatio + is the usual oe ad multiplicatio is matix multiplicatio If V is a -dimesioal vecto space ove k, the by fixig a basis of V we obtai a algeba isomophism betwee Ed k (V) ad M (k) 3) Goup algebas k [G] Let G be ay goup, ot ecessaily fiite, ad let k [G] deote the set of fomal fiite sums Σ a g G g g, whee the coefficiets a g lie i k Note that k k [G] ad G k [G]

2 2 Additio is defied i the obvious way by ( Σ a g G g g) + ( Σ b g G g g) = ( Σ (a g G g + b g )g Give elemets x of k, h of G ad ξ = Σ a g G g g of k [G] we defie xξ = ξx = Σ xa g G g g ad hξ = Σ a g G g hg The defiitio of hξ exteds i the obvious way to a multiplicatio o k [G] that makes k [G] a algeba Note : 1) k always lies i the cete of k [G] 2) The elemets of the goup G ae liealy idepedet i k [G] with espect to k Hece, if G is a fiite goup, the k [G] has fiite dimesio G ove k Moeove, the goup algeba k [G] is fiite dimesioal ove k G is a fiite goup Example 15 Cetal elemets i Ç[G] Let G be a fiite goup, ad let {ρ i : G GL(V i ), 1 i } be a complete set, up to equivalece, of ieducible complex fiite dimesioal epesetatios of G a) Fo a class fuctio f : G Ç defie e f = Σ f(g) g Ç[G] d g G b) Fo 1 i defie e i Ç[G] by e i = i G Σ χ g G i (g) g, whee χi is the chaacte of the epesetatio ρ i ad d i is the dimesio of V i Assetio The elemet e f lies i the cete of Ç[G] fo each class fuctio f : G Ç The elemets e i lie i the cete of Ç[G] fo 1 i, ad ae a Ç-basis fo {e f, f a class fuctio o G} We shall see late i Popositio 67 that evey elemet of the cete of Ç[G] is oe of the elemets e f, f : G Ç a class fuctio o G of the assetio (cf See, Theoem 8, p 34) If f is ay class fuctio o G, the it is outie to show that e f commutes with d all elemets i G ad hece with all elemets i Ç[G] The fuctio f i (g) = i G χ i (g) is a class fuctio o G fo each i, ad hece e i lies i the cete of Ç[G] fo all i The chaactes {χ i : G Ç, 1 i } ae a othoomal basis fo the class fuctios o G 1 elative to the ie poduct (ƒ ψ) = Σ G ƒ(g)ψ(g) If f : G Ç is a class g G fuctio o G, the so is f ad we may wite f = Σ a i=1 i χ i, whee a i Ç Hece e f = _ Σ b i=1 i e i, whee b i = (a i G ) / di

3 3 It emais oly to show that the elemets {e 1, e 2, e } ae liealy idepedet i Ç[G] Fom the defiitio oe sees that each elemet e i is ozeo sice the chaacte χ i that defies it is a ozeo fuctio o G ad the elemets of G ae liealy idepedet i 2 Ç[G] (Recall that χ i (1) = dim V i ) By the coollay below, e = ei fo 1 i ad i e i e j = 0 if i j Suppose that 0 = Σ a i=1 i e i fo some elemets a i Ç The 0 = e j (Σ a i=1 i e i ) = a j e j Hece a j = 0 fo each j sice e j is ozeo æ The elemets e i i Ç[G] also have special meaig fo the decompositio of a abitay complex epesetatio ρ : G GL(V) ito isotypic compoets Popositio Let G be a fiite goup, ad {ρ i : G GL(V i ), 1 i } be a complete set, up to equivalece, of ieducible complex fiite dimesioal epesetatios of G Let e i = d i G Σ χ g G i (g) g, whee χi is the chaacte of the epesetatio ρ i ad d i is the dimesio of V i Let ρ : G GL(V) be a complex epesetatio of G Let W i be the diect sum of all ieducible submodules of V that ae equivalet to V i The a) V = W 1 W 2 W b) ρ(e i ) Ed(V) is the pojectio map V W i if W i {0} c) ρ(e i ) = 0 if W i = {0} d) ρ j (e i ) = δ ij Id Fo ay ξ Ç[G] χ j (e i ξ) = χ j (ξe i ) = χ j (ξ )δ ij of the Popositio The assetio i a) is obvious To pove b) ad c) we ote that the map ρ(e i ) commutes with ρ(g) fo all g G sice e i lies i the cete of Ç[G] If W is a ieducible G-submodule of V, the ρ(e i ) = λ i Id o W by Schu's Lemma fo some λ i Ç Takig the tace of the expessio above we fid that λ i = (1 / d) χ(e i ), whee d is the dimesio of W ad χ is the chaacte of ρ Sice W is equivalet to V j as a G-module fo some j, 1 j d, it follows that d = d j ad χ = χ j We compute χ j (e i ) = i G Σ χ g G i (g) χj (g) = d i (χ j χ i ) = d i δ ij Hece λ i = (1 / d j ) χ j (e i ) = (d i / d j ) δ ij We coclude that ρ(e i ) = Id o W if W is a ieducible G-submodule of V that is equivalet to V i, ad ρ(e i ) = 0 o W if W is a ieducible G-submodule of V that is equivalet to V j fo some j i This completes the poofs of b) ad c) If oe applies c) of the popositio i the case that ρ = ρ j, the it follows immediately that ρ j (e i ) = δ ij Id, which poves the fist assetio of d) If ξ Ç[G] is abitay, the fom the fist pat of d) ad the fact that e j is a cetal elemet

4 4 of Ç[G] we obtai ρ j (e i ξ) = ρ j (ξe i ) = ρ j (ξ)ρ j (e i ) = δ ij ρ j (ξ) Takig taces i the equatio above completes the poof of d) æ Coollay Let G be a fiite goup, ad let {e 1, e 2, e } be the cetal elemets i Ç[G] defied above The 1) 1 = e 1 + e e 2 2) e = ei fo 1 i i 3) e i e j = 0 if i j We begi by ecallig some facts about the egula epesetatio eg : G GL(Ç ), whee = G, ad its chaacte χ eg : G Ç a) χ eg = Σ d i=1 i χ i, whee d i = dim V i Equivaletly, the ieducible module V i has multiplicity d i i the egula epesetatio b) χ eg (g) = 0 if g 1, ad χ eg (1) = G c) eg : G GL(Ç ) is ijective Fom the defiitio of the elemets {e i } ad the facts above we compute e 1 + e e = (1 / G ) Σ {Σ d g G i=1 i χ i (g) } g = (1 / G ) Σ χ g G eg (g) g = 1 This poves 1) Now let Ç = W 1 W 2 W be the decompositio of the egula epesetatio ito isotypic compoets as i the popositio above Sice each ieducible submodule V i appeas with positive multiplicity fo ρ = eg it follows fom b) 2 of the popositio that ρ(e i ) is the pojectio of V oto W i fo 1 i Hece ρ(e ) = i ρ(e i ) 2 = ρ(e i ) fo all i ad ρ(e i e j ) = ρ(e i )ρ(e j ) = 0 if i j by the popeties of pojectio maps The assetios 2) ad 3) of the coollay ow follow sice ρ is ijective æ 16 Fiite dimesioal divisio algebas Fo a abitay field k thee may, i geeal, be a ifiite umbe of k-divisio algebas that ae fiite dimesioal ove k Fo example, let k = Œ, the field of atioal umbes, ad let a Ç be a oot of a ieducible polyomial f(x) of degee i Œ[x] We shall see that Œ ad a ae cotaied i subfield Œ(a) of Ç such that Œ(a) is a - dimesioal vecto space ove Œ If k = Â, the oe has the followig stikig esult of Fobeius (poof omitted)

5 5 Theoem (Fobeius) Let A be a fiite dimesioal divisio algeba ove  The A is isomophic to oe of the followig algebas : 1) A =  (1-dimesioal ove Â) 2) A = Ç (2-dimesioal ove Â) 3) A = Ó (4-dimesioal ove Â, the quateios) The fist two examples ae fields, but the quateios ae ot commutative with espect to multiplicatio 17 A-modules ae vecto spaces Let (M,+) be a abelia goup, ad let Hom (M) deote the additive abelia goup of homomophisms of (M,+) Hom (M) becomes a ig with the idetity map Id as uit if compositio is take as the multiplicative opeatio If R is a ig, the a R-module is a abelia goup (M,+) togethe with a ig homomophism ρ : R Hom (M) Now let A be a algeba ove a field k, ad let M be a otivial A-module We asset that the homomophism ρ : A Hom(M) is ijective o k with ρ(1)(m) = m fo all m M Idetifyig k with ρ(k ) we the obtai a k -vecto space stuctue o M If ρ(z*) = 0 fo some ozeo elemet z* of k, the ρ(1) = ρ((1/z*) (z*)) = ρ(1/z*) ο ρ(z*) = 0 Fo ay elemet z of k we the have ρ(z) = ρ(z 1) = ρ(z) ο ρ(1) = 0 This cotadicts the hypothesis that ρ is a ozeo homomophism Theefoe ρ : k Hom(M) is ijective A simila agumet shows that ρ(1) = Id o M The ext obsevatio is useful i the discussio of goup algebas k [G], whee G is a fiite goup Popositio Let A be a fiite dimesioal algeba ove a field k, ad let M be a otivial A- module The M is fiitely geeated as a A-module A is fiite dimesioal as a k - vecto space If M is fiite dimesioal as a vecto space ove k, the ay fiite k-basis ı fo M is also a fiite geeatig set fo M as a A-module sice k A Covesely, suppose that M is a fiitely geeated A-module, ad let {x 1, x 2,, x } be a fiite geeatig set Let {a 1, a 2, a N } be a k-basis of A If ı = {a i (x j ) : 1 i, 1 j N}, the it is easy to

6 6 check that ı is a (fiite) k-spaig set fo M Hece ı cotais a fiite k-basis fo M æ Sectio 2 Stuctue of Ed R (M) I this sectio let R be a ig with 1 Let M be a R-module, ad let f : R Hom(M) be the homomophism that defies the actio of R o M A ig R is a divisio ig if evey ozeo elemet of R has a ivese Commutats ad bicommutats Defiitio 21 Ed R (M) = { ƒ Hom(M) : ƒ commutes with f() fo all R} Equivaletly, we may let deote f() ad defie Ed R (M) = {ƒ Hom(M) : ƒ(m) = ƒ(m) fo all R ad all m M} Note that R' = Ed R (M) is a subig of Hom(M), ad M becomes a module ove R' The ig R' is called the commutat of R If R'' = Ed R' (M) = { ƒ Hom(M) : ƒ commutes with all all ' R'}, the R'' is called the bicommutat of R If is a elemet of R, the f() commutes with the elemets of R' ad hece is a elemet of R'' The map f: R Hom(M) theefoe has image i R'', ad we obtai by estictio a ig homomophism f : R R'' The Jacobso Desity Theoem i sectio 4 says that the image f(r) is a "lage" subset of R'' if M is semisimple ove R I the sequel we will typically suppess the otatio f() ad just use as is customay The ig R the becomes a subig of R'' Simple R-modules Defiitio 22 A R-module M is simple if the oly R-submodules ae {0} ad M Popositio 23 Let M be a simple R-module, ad let D = Ed R (M) The D is a divisio ig If R is a algeba ove a field k, the D is a divisio algeba ove k Let T be ay ozeo elemet of Ed R (M) It follows immediately fom the defiitio of Ed R (M) that N = Ke(T) ad N* = Im(T) ae both R-submodules of M Sice M is simple ad T is ozeo it follows that Ke(T) = {0} ad Im(T) = M Hece T is ivetible If R is a algeba ove a field k, the it is easy to see that D = Ed R (M) is a algeba ove k æ The case that R is a algeba ove a field k

7 7 We obseved i sectio 1 that if R is a algeba ove a field k, the a R-module is also a vecto space ove k Regadig k as a algeba, the the algeba of k-liea tasfomatios of M is pecisely Ed k (M) = {T Hom(M) : T commutes with x fo all x k} I this cotext it is useful to obseve the followig Popositio 24 Let R be a algeba ove a field k, ad let M be a R-module The R, R' = Ed R (M) ad R'' = Ed R' (M) ae all subalgebas of Ed k (M) Sice k lies i the cete of R it follows immediately that R Ed k (M) Next, obseve that R' = Ed R (M) Ed k (M) sice k R Fially, k R' sice k Z(R), the cete of R, ad hece R'' = Ed R' (M) Ed k (M) æ Examples If R is a fiite dimesioal algeba ove a field k, the ay simple R- module M is a fiite dimesioal vecto space ove k by the discussio i sectio 17 ; if m is ay ozeo elemet of M, the Rm is a ozeo R-submodule of M ad hece equal to M I this case we ca ofte add to the statemet of Popositio 23 that D = Ed R (M) is a divisio algeba Popositio 25 Let R be a fiite dimesioal algeba ove a field k ad let M be a simple R- module 1) Ed R (M) is a fiite dimesioal algeba ove k 2) If k is algebaically closed, the Ed R (M) = {λid : λ k} ~ = k 3) If k = Â, the Ed R (M) ~ = Â, Ç o Ó 1) By Popositio 24 ad the discussio above M is a fiite dimesioal vecto space ove k, ad Ed R (M) is a subalgeba of Ed k (M), whose k-dimesio is fiite 2) If T is ay elemet of Ed R (M), the T has a eigevalue i k sice k is algebaically closed By defiitio R leaves ivaiat the eigespace V λ of T, ad hece V λ = V sice V is R-simple We coclude that T = λ Id,which poves 1) 3) Sice D = Ed R (M) is a divisio algeba by Popositio 23 ad is fiite dimesioal ove  by 1) the esult follows fom the Fobeius theoem i sectio 16 Examples of fiite dimesioal algebas ove  1) Let R = Ç ad M = Ç The R may be egaded as a 2-dimesioal algeba ove Â, ad M is a simple R-module sice R acts tasitively o M = R I this case Ed R (M) = Ed Ç (Ç) ~ = Ç

8 8 2) Let R = Ó ad M = Ó The R is a 4-dimesioal algeba ove Â, ad M is a simple R-module elative to left multiplicatio sice R acts tasitively o M Let Ó deote the 4-dimesioal subalgeba of Ed  (M) obtaied fom the left multiplicatio of Ó o M Let Ó op deote the 4-dimesioal subalgeba of Ed  (M) obtaied fom the ight multiplicatio of Ó o M Sice Ó ad Ó op commute it follows that Ó op Ed R (M) ad equality holds sice 4 = dim Ó op  dim  Ed R (M) 4 by the Fobeius theoem (Oe may also pove equality by a tedious diect calculatio) It follows that Ed R (M) ~ = Ó Opposite igs ad Ed R (R) If M = R ad R acts o itself o the left, the oe has a simple ad explicit desciptio of Ed R (R) i tems of the opposite ig R op We defie R op = R as a set, the additio i R ad R op is the same but the multiplicatio # of R op is defied by x#y = y x, whee is the give multiplicatio opeatio i R Popositio 26 1) Fo evey elemet x of R the map f(x) : R R give by f(x)() = x is a elemet of Ed R (R) 2) The map f : R Ed R (R) is a bijectio that satisfies f(xy) = f(y) ο f(x) ad f(x+y) = f(x) + f(y) fo all x,y i R 3) If I : R op R is the idetity map, the g = f ο I : R op Ed R (R) is a ig isomophism The poof of 1) is outie sice ight ad left multiplicatio maps always commute Assetio 3) follows immediately fom 2) We pove 2) The assetios f(x+y) = f(x) + f(y) ad f(xy) = f(y) ο f(x) fo all x,y i R ae obvious If f(x) = f(y) fo x,y i R, the x = f(x)(1) = f(y)(1) = y If ƒ is ay elemet of Ed R (R) ad x = ƒ(1), the ƒ = f(x) sice ƒ() = ƒ( 1) = ƒ(1) = x = f(x)() fo all Ræ The ext elemetay esult is basic fo the stuctue theoy of semisimple igs that we discuss ext Popositio 27 Let M be a R-module ad a positive itege Let M deote the diect sum of copies of M The Ed R (M ) is ig isomophic to M (R'), the collectio of x matices with elemets i R' = Ed R (M) Fo 1 i let π i : M M deote pojectio oto the i th facto ad defie

9 9 I i : M M by I i (m) = (0,, 0, m, 0,, 0), whee m sits i the i th positio It is easy to check that π i ad I j commute with the R-actio o M ad M fo all i ad j Hece if ƒ is ay elemet of Ed R (M ), the A ij (ƒ) = π i ο ƒ ο I j is a elemet of R' = Ed R (M) fo all 1 i,j Let A(ƒ) deote the x matix i M (R') detemied by a elemet ƒ of Ed R (M ) If we egad the elemets (m 1, m 2,, m ) of M as " colum vectos " the it is easy to check that ƒ(m 1, m 2,, m ) = A(ƒ) (m1, m 2,, m ) (" matix multiplicatio ") fo all elemets (m 1, m 2,, m ) of M ad elemets ƒ of Ed R (M ); that is, ƒ(m 1, m 2,, m ) = (y 1, y 2,, y ), whee y i = Σ A j=1 ij (ƒ) m j Hece the map ƒ A(ƒ) is a ijective ig homomophism of Ed R (M ) ito M (R') Covesely, give a elemet B of M (R') we defie a map ƒ i Hom (M ) by ƒ(m 1, m 2,, m ) = B(m 1, m 2,, m ) (matix multiplicatio) It follows that ƒ commutes with the R-actio o M sice the eties of B lie i R' = Ed R (M) This poves that ƒ Ed R (M ) ad B = A(ƒ) Hece ƒ A(ƒ) is a ig isomophism of Ed R (M ) oto M (R') æ Sectio 3 Basic stuctue of semisimple igs ad modules The basic stuctue theoem fo semisimple igs, due to Weddebu, says that evey semisimple ig R is a diect poduct R 1 x R 2 x x R of fiitely may simple igs Moeove, a simple ig R is isomophic to a matix algeba M (D), whee D is a divisio algeba ad is a positive itege that deped o R The atue of this depedece will be made pecise late If R = Ç[G], whee G is a fiite goup, the the simple igs R i that ae factos of R ae pecisely the simple algebas Ed Ç (V i ), whee {V 1, V 2, V } is a complete list (up to equivalece) of the complex ieducible G-modules I this sectio we discuss defiitios ad basic popeties of semisimple igs ad modules Popositio 31 The followig ae equivalet fo a ig R with 1 ad a R-module M 1) Fo evey R-submodule N thee exists a R-module N' such that N N' = M 2) M is a diect sum of fiitely may simple R-submodules See Lag, Algeba, pp Defiitio 32

10 10 a) A R-module M is said to be semisimple if eithe the coditios of Popositio 31 hold b) A ig R with 1 is said to be semisimple if it is semisimple as a R-module, whee R acts o itself by left multiplicatio c) A left ideal J of a ig R is said to be simple if J cotais o pope left ideals I of R d) A ig R with 1 is simple if R = L 1 L 2 L, whee {L i }ae simple left ideals of R that ae all isomophic (I paticula ay simple ig is semisimple) Popositio 33 If R is a semisimple ig, the evey R-module M is semisimple See Lag, Algeba, p 651 Lemma 34 Let L be a simple ideal of a abitay ig R with 1, ad let M be a simple R- module The eithe L is isomophic to M as a R-module o LM = {0} Suppose that LM {0}, ad let m be a ozeo elemet of M such that N = Lm {0} Note that RN = RLm = Lm = N sice L is a left ideal of R Hece N is a ozeo submodule of M ad must equal M The map f : L M give by f() = m is a sujective homomophism of R-modules It follows that Ke(f) = {0} ad f is a isomophism sice Ke(f) is a R-submodule of the simple R-module L Coollay 35 Let R be a semisimple ig, ad let M be a simple R-module The 1) M is isomophic as a R-module to some simple left ideal of R 2) If R is a simple ig, the ay two simple R-modules ae isomophic 3) If R is a simple ig, the ay two simple left ideals ae isomophic By the defiitio of a semisimple ig we ca fid simple left ideals L 1, L 2,, L of R such that R = L 1 L 2 L If M is ot isomophic to L i fo ay i, the by the lemma above L i M = {0} fo evey i We coclude that RM = {0}, which cotadicts the fact that R cotais 1 This poves 1) If R is a simple ig, the the simple left ideals {L i } above ca be chose to be all isomophic Assetio 2) ow follows fom the poof of 1) Assetio 3) follows fom 2) sice simple left ideals ae simple R-modules æ

11 11 Examples 36 1) We begi with a example of a simple ig R, which will tu out to be the oly example Let D be a divisio ig, ad let M (D) deote the algeba of x matices with eties i D Let D deote the space of -tuples (d 1, d 2,, d ), d i D Let M (D) act o D o the left by matix multiplicatio, egadig the elemets of D as colum vectos Fo 1 i let L i deote the matices i M (D) whose j th colum is zeo fo all j i Equivaletly, L i = {X M (D) : X(e j ) = {0} fo all j i}, whee {e 1, e 2,, e } deotes the usual atual basis of D It is clea fom the secod desciptio that each L i is a left ideal of M (D), ad it is ot difficult to show that each L i is simple Clealy, M (D) = L 1 L 2 L If L is a left ideal of M (D), the LX is a left ideal isomophic to L fo ay ivetible elemet X of M (D) Moeove, L is simple LX is simple Let P ij be the pemutatio matix i M (D) such that P ij e k = e k if k / {i,j}, P ij e i = e j ad P ij e j = e i 2 It is easy to see that L i P ij = L j ad P ij = Id It follows that the simple left ideals {Li } ae all isomophic, ad hece M (D) is a simple ig 2) Let G be a fiite goup ad let R = k [G], the goup algeba ove a field k whose chaacteistic does ot divide G, the ode of G The R is a semisimple algeba (cf Lag) We have aleady see this i the case that k = Ç Fo a poof i the geeal case see Lag 3) Cliffod algebas These ae semisimple algebas that ae fiite dimesioal ove  o Ç, ad they play a impotat ole impotat i geomety ad physics We descibe biefly oe classical example ove  Fo futhe discussio see Lag pp , Fulto ad Hais, Repesetatio Theoy a Fist Couse, Spige, GTM # 129, 1991, pp o Lawso ad Michelsoh, Spi Geomety, Piceto Uivesity Pess,1989, Chapte 1 Stat with  equipped with some ie poduct <, >, say the stadad dot poduct Let {e 1, e 2, e } be a othoomal basis of  elative to <, >, say the stadad basis elative to the dot poduct The choice of ie poduct ad othoomal basis does't matte sice the esultig algebas will all be equivalet ad of dimesio 2 ove  Let C () deote the set of fiite fomal sums Σ i1 i 2 ij e i1 i 2 ij, whee the i1 i 2 ij ae eal umbes Multiplicatio of the expessios e i1 i 2 ij ad e m1 m 2 mk is defied by ( e i1 i 2 ij ) (e m1 m 2 mk ) = e i1 i 2 ij e m1 m 2 mk, subject to the ules e i e j =

12 12 2 e j e i if i j, ad e i = 1 By expadig vectos v ad w of  i tems of the othoomal basis {e 1, e 2, e } ad usig the multiplicatio ules above we obtai the impotat elatios (*) vw + wv = 2 <v,w> fo all vectos v ad w i  v 2 = v 2  fo all vectos v i  The secod elatio i (*) follows fom the fist by settig v = w The elatios (*) also show that multiplicatio does ot deped o the choice of othoomal basis i  By defiitio C () cotais the eal umbes  ad also Â, coespodig to the expessios Σ i e i By ispectio ay expessio e i1 i 2 ij, whee j >, cotais at least two equal idices ad ca be educed by the multiplicatio ules to a expessio e m1 m 2 mk, whee k ad m m +1 fo all, 1 k Fo each itege 1 k let V k deote the vecto space of all eal liea combiatios of elemets of the fom e m1 m 2 mk, whee {m 1,m 2, m k } is ay subset of k elemets i {1, 2, } such that m m +1 fo all, 1 k Hece C () = V o V 1 V, whee V o = Â, V 1 =  ad V k has dimesio ( k ) = (!) / (k!) (-k)! fo evey k Note that V has dimesio 1 ad cosists of all eal multiples of the elemet e 1 e 2 e Addig up the dimesios of the {V i } we see (by iductio o ) that dim C () = 2 fo evey The multiplicatio defied above is associative, ad C () becomes a algeba ove  We sketch a poof that C () is a semisimple algeba fo evey The elatios (*) above show that left multiplicatio o C () by a ozeo elemet v of  is a ivetible liea tasfomatio of C () Let Pi() deote the subgoup of GL(C ()) geeated by all left multiplicatios by uit vectos v i  Oe ca show that Pi() is a compact subgoup of GL(C ()) Now let ρ : C () GL(U) be ay algeba homomophism (ie a epesetatio of C ()), whee U is a fiite dimesioal eal vecto space The ρ is cotiuous ad ρ(pi()) is a compact subgoup of GL(U) The compactess of ρ(pi()) implies, by a stadad aveagig pocess simila i spiit to that used fo fiite subgoups of GL( k ), that thee exists a ie poduct <, > o U such that ρ(pi()) is a subgoup of the othogoal goup of {U, <, >} This meas that < ρ(g)u, ρ(g)u* > = < u, u* > fo all vectos u,u* i U ad all elemets g i Pi() Let ρ : C () GL(U) be ay algeba homomophism ad let <, > be a ivaiat ie poduct fo ρ(pi()) o U as above If v is a uit vecto i Â, the ρ(v) 2 = Id sice v 2 = 1 by the elatios (*) above The fact that ρ(v) 2 = Id meas that ρ(v) = ρ(v) 1 = ρ(v) t We coclude that ρ(v) is skew symmetic as well as othogoal elative to <, > if v is a uit vecto i  It follows immediately that ρ(v) is skew symmetic elative to <, > fo all v i  sice ρ(λv) = λρ(v) fo all v i  ad all eal umbes λ

13 13 Summay : Ay algeba homomophism ρ : C () GL(U) admits a ie poduct <, > o U such that the tasfomatios i ρ(â ) ae all skew symmetic elative to <, > If W is a subspace of U ivaiat ude ρ(c ()), the W is ivaiat ude ρ(â ), whee W deotes the othogoal complemet of W i U elative to <, > The algeba ρ(c ()) is geeated by ρ(â ) sice the algeba C () is geeated by  Hece W is ivaiat ude ρ(c ()) if W is ivaiat ude ρ(c ()) This meas that evey fiite dimesioal eal C ()-module U is semisimple Apply this fact ow i the case that U = C () ad ρ : C () Ed(C ()) is the homomophism give by ρ(g) = left multiplicatio by g We coclude that C () is a semisimple algeba Example C (2) ~ = Ó Let {e 1, e 2 } be the stadad oth0omal basis fo  2 Fom the desciptio above C (2) =  Â(e 1 ) Â(e 2 ) Â(e 1 e 2 ), ad the multiplicatio ules ae e 1 e = 2 e 2 e 1 ad 2 e 1 = 2 e2 = 1 If we defie I = e1, J = e 2 ad K = e 1 e 2, the the multiplicatio ules fo C (2) show that I 2 = J 2 = K 2 = 1; IJ = JI = K; JK = KJ = I ad KI = IK = J This shows that C (2) is isomophic to the quateios Ó Sectio 4 The Jacobso Desity Theoem ad applicatios Fix a ig R with 1 ad a R-module M Let f : R Hom(M) be the homomophism that defies the actio of R o M We ecall fom sectio 2 that the ig R' = Ed R (M) is called the commutat of R, ad R'' = Ed R' (M) is called the bicommutat of R Moeove, we obseved that f(r) R'' o simply R R'' Hom (M) if we suppess the otatio f as is customay If R is semisimple, the we ca say much moe Theoem 41 (Jacobso Desity Theoem) Let M be a semisimple R-module ove a ig R with 1 Let R' = Ed R (M) ad let R'' = Ed R' (M) Let X be ay fiite subset of M ad let '' be ay elemet of R'' The thee exists a elemet of R such that x = ''x fo evey x i X Lemma 42 If N is ay R-submodule of M, the N is also a R''-submodule of M of Lemma 42 Let N be ay R-submodule of M Sice M is semisimple ove R thee is a R- submodule N' such that M = N N' If π : M N is the pojectio map, the π is R- liea sice N ad N' ae R-submodules, o equivaletly, π is a elemet of R' Ay

14 14 elemet '' i R'' commutes with π, ad hece ''N = ('' ο π ) (M) = (π ο '')(M) π(m) = N æ of Theoem 41 (followig Boubaki) We fist coside the case that X is a sigle elemet {x} of M The set N = Rx is a R-submodule of M ad hece also a R''-submodule by the lemma above The elemet x lies i N sice R cotais 1 If '' R'' is ay elemet, the ''x N = Rx, ad hece thee exists R such that x = ''x Now let X = {x 1, x 2,, x } be ay subset of M with elemets, ad let M = M M M ( times) Note that M is a R-module, whee the R-actio o M is the diagoal actio give by (m 1, m 2,, m ) = (m 1, m 2,, m ) fo R ad (m 1, m 2,, m ) M Moeove, M is semisimple ove R sice M is semisimple ove R Let R'() deote Ed R (M ) ad let R''() deote Ed R'() (M ) Let R' ad R'' act o M by the diagoal actio ξ(m 1, m 2,, m ) = (ξm 1, ξm 2,, ξm ) If ƒ R'(), the by Popositio 26 thee exists a matix A = A(ƒ) M (R') such that fo all (m 1, m 2,, m ) M we have (*) ƒ(m 1, m 2,, m ) = A(m 1, m 2,, m ) = (y 1, y 2,, y ), whee y i = Σ A j=1 ij m j If '' R'' is ay elemet, the '' commutes o M with the elemets {A ij } R', ad it follows immediately fom (*) that '' commutes o M with all elemets ƒ of R'() This poves that R'' R''() By the case = 1 above applied to the elemet x = (x 1, x 2,, x ) M we kow that fo evey elemet '' R'' R''() thee exists a elemet R such that (x 1, x 2,, x ) = x = ''x = (''x 1, ''x 2,, ''x ) Hece x i = ''x i fo 1 i æ Applicatios of the Desity Theoem Popositio 43 Let A be a semisimple algeba that is fiite dimesioal ove a field k Let M be a fiitely geeated A-module, ad let R = ρ(a), whee ρ : A Hom (M) is the ig homomophism that defies the actio of A Let R' = Ed R (M) ad R'' = Ed R' (M) The 1) R = R'' Ed k (M) 2) If M is a simple A-module ad k is algebaically closed, the R = Ed k (M) 1) We showed i Popositio 24 that R, R' ad R'' ae all subalgebas of Ed k (M) Note that R is a fiite dimesioal algeba ove k sice A has this popety Hece M is a fiite dimesioal vecto space ove k by the discussio i sectio 17 Let ı be a k-basis of M, ad let '' R'' be give By the desity theoem thee exists a elemet R such that x = ''x fo all x ı Sice ad '' ae elemets of Ed k (M)

15 15 with the same values o a basis it follows that = '' Hece R'' R ad we obseved ealie that the evese iequality holds fo ay ig R 2) Let M be a simple A-module, o equivaletly a simple R-module, ad let T be a elemet of R' The elemets of R commute with T ad hece leave ivaiat evey eigespace of T Sice M is R-simple it follows that T = c Id fo some c k Hece R' = k Id, ad fom 1) we coclude that R = R'' = Ed R' (M) = Ed k (M) Coollay 44 Let G be a fiite goup, ad let ρ : G GL(V) be a ieducible epesetatio of G o a fiite dimesioal complex vecto space V The ρ(ç[g]) = Ed Ç (M) The goup algeba Ç[G] is a semisimple algeba, ad V is a simple Ç[G]-module Now apply 2) of Popositio 43 æ Coollay 45 Let G be a fiite goup, ad let {ρ i : G GL(V i ), 1 i } be a complete set, up to equivalece, of ieducible complex fiite dimesioal epesetatios of G Let V = V 1 V 2 V, ad let ρ = ρ 1 + ρ ρ : G GL(V) deote the coespodig epesetatio Let W = Ed Ç (V 1 ) x Ed Ç (V 2 ) x x Ed Ç (V ) Ed Ç (V) The 1) If R = ρ(ç[g]), the R' = {T W : T = λ i Id o each V i fo some λ i Ç} 2) ρ : Ç[G] W is a algeba isomophism 1) We ote that R' Ed Ç (V) sice Ç R The subspaces {V i : 1 i } ae the isotypic compoets of V sice the epesetatios {ρ i : G GL(V i ), 1 i } ae iequivalet By the Popositio i sectio 15 the pojectios π i : V V i ae elemets d of R = ρ(ç[g]); specifically, π i = i G Σ χ g G i (g) ρ(g), whee χi is the chaacte of the epesetatio ρ i ad d i is the dimesio of V i Hece the elemets of R' leave ivaiat each subspace V i sice they commute with the pojectios {π i : V V i, 1 i } This meas that R' W by the defiitio of W O V i the elemets of R' must be multiples of the idetity sice they commute with ρ i (G) ad each V i is a ieducible G-module This completes the poof of 1) 2) It is evidet that ρ : Ç[G] W is a algeba homomophism If d i is the 2 dimesio of V i, the dim Ç Ed Ç (V i ) = d i fo all i As vecto spaces ove Ç both Ç[G] 2 ad W have dimesio G = Σ d i=1 i It suffices to pove that ρ : Ç[G] W is

16 16 sujective Let R = ρ(ç[g]) The R = R'' Ed Ç (V) by Popositio 43, ad R W by the defiitios of R ad ρ By 1) it follows that R'' = W æ Remak I Popositio 610 below we shall obtai a explicit fomula fo the ivese of the isomophism ρ : Ç[G] W Sectio 5 Stuctue of simple igs We begi this sectio with a useful basic esult Popositio 51 Let R be a simple ig with 1 1) If L ad L' ae simple left ideals, the L' = La fo some elemet a of R 2) LR = R fo ay left ideal L of R 3) R has o pope 2-sided ideals We egad R as a R-module by lettig R act o itself o the left The R-submodules ae the pecisely the left ideals of R 1) By Coollay 35 thee exists a R-module isomophism ƒ : L L' Sice R is semisimple thee exists a left ideal M of R such that R = L M The pojectio π : R L lies i Ed R (R) as does the compositio ƒ ο π : R L' R By Popositio 26 thee exists a elemet a R such that (ƒ ο π)(x) = xa fo all x R I paticula, L' = (ƒ ο π)(l) = La 2) If L is a left ideal of R, the R = L 1 L 2 L k, whee each L i is a left ideal isomophic to L By 1) we ca fid elemets x 1, x 2,, x k i R such that L i = Lx i fo all i Hece L i LR fo all i, ad it follows that LR R = L 1 L 2 L k LR 3) Let I be a ozeo 2-sided ideal of R By Popositio 33, I is a semisimple R- module, ad hece I cotais a simple R-submodule L, which is a simple left ideal of R Hece R = LR IR I by 2), ad we coclude that I = R æ We ow pove the mai esult of this sectio Theoem 52 Let R be a simple ig Let L be a simple left ideal of R, ad let D = Ed R (L), whee L is egaded as a simple R-module with R actig o the left Let deote the umbe of simple left ideals i a diect sum decompositio of R The D is a divisio ig ad R is ig isomophic to M (D op ), whee D op deotes the divisio ig opposite to D Remak

17 17 Ay two left ideals of a simple ig R ae isomophic to each othe by Coollay 36 o Popositio 51 Hece D is uiquely detemied up to isomophism, ad D is a divisio ig by Popositio 23 D-modules have o ozeo tosio elemets sice the ozeo elemets of D ae ivetible Hece ay fiitely geeated D-module M is a fee D- module whose ak depeds oly o M Assumig the theoem above, the simple ig R is a fee D-module of ak 2 This will pove that the umbe of simple left ideals of R i a diect sum decompositio of R is idepedet of the decompositio Fom the theoem above, Popositiio 25 ad the fact that D op is isomophic to D fo D = Â, Ç o Ó we obtai Coollay 53 Let R be a simple, fiite dimesioal algeba ove a field k 1) If k is algebaically closed, the R is isomophic as a algeba to M (k) fo some positive itege 2) If k = Â, the R is isomophic as a algeba to M (D) fo some positive itege, whee D = Â, Ç o Ó of the theoem Let L be a simple left ideal of R Sice R is a diect sum of simple left ideals, all of which ae R-isomophic to L by Coollay 35, we coclude that R is isomophic to L, the diect sum of copies of L fo some positive itege By Popositios 23, 26 ad 27 we obtai R op ~ = EdR (R) ~ = Ed R (L ) ~ = M (D), whee D = Ed R (L) is a divisio algeba Hee ~ = meas ig isomophism If ƒ : (R, ) (S, ) is a isomophism of igs, the it is easy to check that ƒ : (R,#) (S,#) is a isomophism of igs, whee x#y = y x i both R ad S Hece R = (R op ) op ~ = M (D) op by the pevious paagaph The theoem will follow immediately fom the ext esult Lemma Let R be a ig with 1 The M (R) op is ig isomophic to M (R op ) The poof is elemetay, but oe must be caeful sice fou diffeet multiplicatio opeatios ae ivolved, two fo R ad R op ad two matix multiplicatios fo M (R) ad M (R) op Recall that S ad S op ae equal as sets ad have the same additive opeatio + fo ay ig S Hee if deotes the multiplicatio opeatio i R, the the multiplicatio

18 18 opeatio # i R op is defied by x#y = y x If ο deotes the usual matix multiplicatio, the (A ο B) ij = Σ Aik Bkj fo A,B M k=1 (R), ad (C ο D) ij = Σ Cik # D k=1 kj fo C,D M (R op ) Let * deote the opposite matix multiplicatio i M (R) op give by A * B = B ο A fo A,B M (R) Give a matix A i M (R) op we let ψ(a) be that matix i M (R op ) such that [ψ(a)] ij = A ji fo all i,j Assetio The map ψ : (M (R) op, *) (M (R op ), ο) is a isomophism of igs Let A ad B M (R) op be give It is outie to show that ψ(a+b) = ψ(a)+ψ(b), ad we omit the details Fo ay iteges 1 i,j we have [ψ(a * B)] ij = [ψ(b ο A )] ij = (B ο A ) ji = Σ Bjk Aki sice the eties of A ad B lie i R Sice the k=1 eties of ψ(a) ad ψ(b) lie i R op we have [ψ(a) ο ψ(b)] ij = Σ [ψ(a)]ik # [ψ(b)] k=1 kj = Σ Aki # B k=1 jk = Σ B k=1 jk Aki = [ψ(a * B)] ij Hece ψ(a * B) = ψ(a) ο ψ(b) fo all A,B i M (R) op, which poves the lemma sice ψ is a bijectio æ As aothe coollay to Theoem 52 we obtai a desciptio of the multiplicative cete of a simple ig Coollay 54 Let R be a simple ig, ad let Z(R) deote the multiplicative cete of R The Z(R) is isomophic to Z(D), the cete of the divisio ig D = Ed R (L), whee L is a simple left ideal of R I paticula, Z(R) is a field Examples If D =  o Ç, the Z(D) = D, while if D = Ó, the Z(D) ~ =  Fom Coollaies 53 ad 54 we obtai immediately Coollay 55 Let R be a simple, fiite dimesioal algeba ove a field k 1) If k is algebaically closed, the Z(R) is isomophic as a algeba to k 2) If k = Â, the R is isomophic as a algeba to eithe  o Ç of Coollay 54 By Theoem 52 we kow that R is ig isomophic to M (D op ) fo some positive itege Sice Z(D op ) = Z(D) it suffices to pove the followig Lemma

19 19 Let D be a divisio ig, ad R = M (D) deote the ig of x matices with coefficiets i D The Z(R) = {λ Id : λ Z(D)}, whee Id deotes the idetity matix ad Z(D) deotes the multiplicative cete of D of the Lemma Clealy, if λ Z(D), the A = λ Id lies i the cete of R Covesely, let A = (A ij ) be a elemet i the cete of R We show fist that A ij Z(D) fo all i,j Let x D be give, ad let B = x Id The A ij x = (AB) ij = (BA) ij = xa ij fo all i,j Hece A ij Z(D) fo all i,j sice x D was abitay Next we asset that A ij = 0 if i j Let i ad j be distict with 1 i,j Let B = diag (x 1,x 2,, x ) be a diagoal matix such that x i ad x j ae distict ozeo elemets of D The A ij x j = (AB) ij = (BA) ij = x i A ij = A ij x i sice A ij Z(D) Hece A ij = 0 sice D is a divisio ig ad x i, x j ae distict ozeo elemets i D We have poved that if A Z(D), the A is a diagoal matix diag λ 1,λ 2,, λ ), whee λ i Z(D) fo all i We show that eties {λ i }ae all equal, which will coclude the poof Give iteges i j let B M (D) be a matix such that B ij 0 The λ i B ij = (AB) ij = (BA) ij = B ij λ j = λ j B ij sice { λ k } Z(D) Hece λ i = λ j sice B ij 0 æ Sectio 6 Stuctue of semisimple igs By defiitio a semisimple ig R is the diect sum of fiitely may simple left ideals {L α } By the poof of Coollay 35 ay simple left ideal of R is isomophic to oe of the ideals {L α } Hece thee exists a fiite collectio {L 1, L 2, L } of oisomophic simple left ideals such that ay simple left ideal L is isomophic to exactly oe of the {L i } Fo 1 i let R i deote the sum of all simple left ideals isomophic to L i By defiitio a elemet of R i is a fiite sum Σ x β A β, whee each x β lies i a simple left ideal L isomophic to L i Hece R i is also a left ideal of R fo 1 i Theoem 61 1) Each R i is a simple ig ad a two sided ideal i R fo 1 2) R i R j = 0 fo i j 3) Thee exist elemets e i R i, 1 i, such that a) 1 = Σ ei i=1 b) e i y i = y i e i fo evey elemet y i i R i, 1 i 4) R = R 1 R 2 R Moeove, the map ( 1, 2,, ) is a ig isomophism of R oto the diect poduct

20 20 R 1 x R 2 x x R 5) The decompositio of R i 4) is uique up to the ode of the factos ; that is, if R = R 1 * R 2 * R m * is a diect sum of commutig simple subigs, the m = ad afte eodeig, R i * = e i R = R e i fo all i, whee {e 1, e 2,, e } ae the elemets fom 3) Remak 62 The elemets {e 1, e 2,, e } belog to the multiplicative cete of R by 2) ad 3) of the theoem Example 63 Let G be a fiite goup ad let R = Ç[G] Let {ρ i : G GL(V i ), 1 i } be a complete list, up to equivalece, of ieducible complex epesetatios of G I Coollay 45 we saw that R = Ç[G] is isomophic to Ed Ç (V 1 ) x Ed Ç (V 2 ) x x Ed Ç (V ) Sice Ed Ç (V i ) is a simple ig fo 1 i the uiqueess assetio i 5) shows the followig : 1) Each simple ig R i i the decompositio of R = Ç[G] is isomophic to Ed Ç (V i ), whee V i is a ieducible complex G-module 2) The umbe of simple igs i the decompositio of R = Ç[G] is the umbe of cojugacy classes of G Fom the discussio i sectio 15 we also coclude d 3) e i = i G Σ χ g G i (g) g, whee χi is the chaacte of the epesetatio ρ i ad d i is the dimesio of V i Example 64 Let R be a semisimple algeba that is fiite dimesioal ove a field k 1) If k is algebaically closed, the R is isomophic to a diect poduct of simple igs R 1 x R 2 x x R, whee each simple ig R i is isomophic to a matix algeba M i (k) fo some positive iteges { i } 2) If k = Â, the R is isomophic to a diect poduct of simple igs R 1 x R 2 x x R, whee each simple ig R i is isomophic to a matix algeba M i (D) fo some positive iteges { i }, whee D = Â, Ç o Ó of example 64 This is a immediate cosequece of Theoem 61 ad Coollay 53 æ of Theoem 61 Let i ad j be distict iteges, 1 i,j If x α ad x β ae elemets of simple left ideals L α ad L β cotaied i R i ad R j espectively, the x α x β L α L β = {0} by Lemma 34 This poves

21 21 (a) R i R j = {0} if i j We ote that (b) R = R 1 + R R sice R is a diect sum of simple left ideals {L α } ad each L α is cotaied i some R i Fo 1 j we have R j R j R = R j (R 1 + R R ) = R j R j R j The equality assetios follow fom (a) ad (b) while the iclusio assetios follow sice R cotais 1 ad R j is a left ideal Hece all iclusios ae equalities, which poves (c) Each R i is a two sided ideal i R Next, we asset (d) R = R 1 R 2 R, diect sum By (b) we may wite (*) 1 = e 1 + e e, whee e i R i fo all i We allow the possibility that e i = 0 fo some i ad that the decompositio (*) is ot uique (Neithe of these possibilities actually happes) It suffices to show that if 0 = Σ xj, whee x j=1 j R j fo all j, the x j = 0 fo all j If x R i, the by (a) x = 1 x = Σ ej x = e j=1 i x Hece fo ay itege i we have 0 = e 0 = i Σ ei x j=1 j = e i x i = x i This poves (d) (Similaly, if x R i, the x = x 1 = Σ xej = xe j=1 i ) (e) Wite (*) 1 = e 1 + e e, whee e i R i fo all i The (i) The elemets e i ae uique 2 (ii) e i = ei fo evey i, ad e i x i = x i = x i e i fo all x i R i (iii) e i e j = 0 if i j The assetio (i) follows fom (d), ad (iii) follows fom (a) Assetio (ii) was poved i the poof of (d) The fact that the map ( 1, 2,, ) is a ig isomophism of R oto the diect poduct R 1 x R 2 x x R is a immediate cosequece of (a) ad (d) It emais oly to pove the uiqueess assetio (5) of the theoem Wite R = R 1 R 2 R, diect sum, as i (d) above Suppose we ca wite R = R 1 * R 2 * R m *, diect sum, whee each R j * is a simple ig By the defiitio of simple ig each R j * is a diect sum of simple left ideals that ae isomophic to each othe By the discussio above the set {L 1, L 2,, L } of left ideals of R cotais exactly oe of each isomophism type of left ideals Moeove, fo 1 i, the ig R i is the sum of all left ideals isomophic to L i Hece each R j * is cotaied i R i fo some uique i It follows that m, ad sice R = R 1 R 2 R = R 1 * R 2 * R m * we coclude that m = ad R j * = R σj fo all j, whee σ is some pemutatio of

22 22 {1, 2, } Fially, fom (a), (d) ad (e) above we have e i R = R e i = R i fo evey i This completes the poof of the theoem æ Cetes of semisimple igs Popositio 65 Let R be a semisimple ig with 1, ad let R = R 1 R 2 R be its decompositio ito simple igs R i, whee R i R j = {0}if i j Let Z(R) deote the multiplicative cete of R The Z(R) = Z(R 1 ) Z(R 2 ) Z(R ), whee Z(R i ) is a field F i fo each i Let z Z(R) ad wite z = Σ zi, whee z i=1 i R i fo evey i If x j R j, the x j z j = Σ xj z i=1 i = x j (Σ zi ) = x i=1 j z = zx j = Σ zi x i=1 j = z j x j Hece z j Z(R j ) fo all j This poves that Z(R) = Z(R 1 ) Z(R 2 ) Z(R ), which is isomophic to the diect poduct Z(R 1 ) x Z(R 2 ) x x Z(R ) By Coollay 54 each ig Z(R i ) is a field æ If R is a semisimple algeba that is fiite dimesioal ove a field k, the we ca say moe Note that each simple facto R i of R must also be a fiite dimesioal algeba ove k Coollay 66 Let R be a semisimple algeba that is fiite dimesioal ove a field k Let R = R 1 R 2 R be its decompositio ito simple fiite dimesioal algebas R i ove k, whee R i R j = {0}if i j The 1) If k is algebaically closed, the Z(R) ~ = k x k x x k, -times 2) If k = Â, the Z(R) ~ = F 1 x F 2 x x F, whee F i = Â o Ç fo each i Both assetios ae immediate cosequeces of Popositio 65 ad Coollay 55 æ Cetes of goup algebas Let G be a fiite goup ad let k be a field whose chaacteistic does ot divide G The goup algeba k [G] is a fiite dimesioal algeba ove k, ad k[g] is also semisimple (cf Lag) We poved this i the case k = Ç ad the poof also woks fo the case k = Â We may shape the esults above still futhe i the case k = Ç

23 23 Popositio 67 Let G be a fiite goup, ad let R = Ç[G] The 1) Z(R) ~ = Ç x Ç x x Ç, times, whee is the umbe of cojugacy classes of G 2) The elemets {e 1, e 2, e } defied i example 15 fom a basis fo Z(R), ad 2 these elemets satisfy the elatios e = ei fo 1 i ad e i i e j = 0 if i j 1) By assetio 2) i the discussio of example 63 the umbe of simple igs R i i the decompositio of R is pecisely the umbe of cojugacy classes i G Assetio 1) ow follows fom assetio 1) of Coollay 66 2) By the discussio i example 15 we kow that the elemets {e 1, e 2, e } ae liealy idepedet i Ç[G] ad Ç-spa {e 1, e 2, e } Z(R) Equality holds sice Z(R) has dimesio ove Ç by 1) Futhe emaks o the cete of Ç[G] Let G be a fiite goup, ad let k be a field whose chaacteistic does ot divide the ode of G Thee is a completely diffeet desciptio of the multiplicative cete of k [G], which we elate to the discussio above i the case that k = Ç I k [G] a cojugacy class is defied to be Σ σ, the sum of all elemets i a σ Ç cojugacy class Ç of G Popositio 68 If R = k [G], the the cojugacy classes fom a basis ove k fo Z(R) Let Ç 1, Ç 2, Ç deote the cojugacy classes i G Let x i k [G] deote the sum of all elemets i the cojuagcy class Ç i fo 1 i We show fist that the elemets {x 1, x 2, x } ae liealy idepedet i k [G] Suppose 0 = Σ a i=1 i x i fo some elemets {a 1, a 2, a } i k The 0 = Σ a i=1 i x i = Σ a σ Ç σ σ, whee a σ = a i if σ Ç i Sice the elemets of G ae liealy idepedet i k [G] it follows that a σ = 0 fo all σ G ad hece a i = 0 fo all i, 1 i Next we show that the elemets {x 1, x 2, x } lie i the cete of R = k [G] It suffices to show that x i τ = τ x i fo all i ad all τ G sice the elemets of G geeate the algeba k [G] We compute x i τ = Σ στ = τ{ Σ τ 1 στ } = τ{ Σ ξ } (substitutig ξ σ Ç i σ Ç i ξ Ç i

24 24 = τ 1 στ) = τ x i Let z k [G] be a elemet fom the cete of k [G] We complete the poof by showig that z is a k - liea combiatio of {x 1, x 2, x } Wite z = Σ a σ G σ σ, whee {a σ } k Fix a elemet µ G The µz = Σ a σ G σ (µσ) = Σ a σ G σ (µσµ 1 )µ = { Σ a ξ G (µ 1 ξµ) ξ}µ, substitutig ξ = µσµ 1 Sice zµ = ( Σ a ξ)µ ad zµ = µz it follows ξ G ξ that Σ a ξ G (µ 1 ξµ) ξ = Σ a ξ Sice G is a basis fo k [G] this implies ξ G ξ (*) a (µ 1 ξµ) = a fo all ξ G ξ Sice µ G was abitay we coclude that thee exist costats {a 1, a 2, a } i k such that a σ = a i fo all σ Ç, 1 i Hece z = Σ a σ G σ σ = Σ ai x i=1 i æ I the case that k = Ç we obtai a explicit desciptio of the tasitio matix betwee the two bases of Z(Ç[G]) that we have discussed : the elemets {e 1, e 2, e }, which wee defied i sectio 15, ad the cojugacy classes {x 1, x 2, x } defied above Popositio 69 Let G be a fiite goup, ad let Ç 1, Ç 2, Ç be the cojugacy classes of G Fo 1 i, fix a elemet τ i i each cojugacy class Ç i, ad let x i Ç[G] deote the sum of the elemets i the cojugacy class Ç i Let {ρ i : G GL(V i ), 1 i } be a complete list, up to equivalece, of the complex ieducible epesetatios of G, ad let χ i : G Ç deote the chaacte of ρ i 1 Fo 1 i,j defie A ij = (d i / G ) χ i (τ j ), whee di = dim V i Fo 1 i let d e i = i G Σ χ g G i (g) g The (*) e i = Σ A j=1 ij x j Remak If g ad h ae cojugate elemets i a goup G, the g 1 ad h 1 ae also cojugate 1 elemets i G Hece χ i (τ j ) does ot deped o the choice of τj i Ç j sice the chaactes {χ i } ae class fuctios of the popositio Let 1 i be give Sice the elemets of G fom a basis of Ç[G] we ca wite e i = Σ σ G a σ σ fo some costats {a σ } Ç Let χ eg : G Ç deote the chaacte of the

25 25 egula epesetatio of G As i sectio 15 we ecall that i) χ eg (g) = 0 if g 1, ii) χ eg (1) = G ad iii) χ eg = Σ d i=1 i χ i, whee d i = dim V i Fom i) ad ii) we compute χ eg (e i τ 1 ) = Σ σ G a σ χ eg (στ 1 ) = G a τ fo all τ G Fom iii) ad fom d) of the popositio i sectio 15 we obtai χ eg (e i τ 1 ) = Σ d j=1 j χ j (e i τ 1 ) = d i χ i (e i τ 1 ) = d i χ i (τ 1 ) Hece a τ = (d i / G )χ i (τ 1 ) If τ belogs to the cojugacy class Ç j, which cotais τ j, the χ i (τ 1 1 ) = χ i (τ j ) ad aτ = (d i / G )χ i (τ 1 1 ) = (d i / G )χ i (τ j ) = A ij Hece a τ = a σ = A ij fo ay two elemets τ, σ i Ç j, which completes the poof of (*) æ Although it is somewhat out of place logically, the ext esult, which descibes the ivese of the algeba homomophism i Coollay 45, is simila i appeaace ad poof to the esult above Popositio 610 Let G be a fiite goup, ad let {ρ i : G GL(V i ), 1 i } be a complete set, up to equivalece, of ieducible complex fiite dimesioal epesetatios of G Let V = V 1 V 2 V, ad let ρ = ρ 1 + ρ ρ : G GL(V) deote the coespodig epesetatio Let W = Ed Ç (V 1 ) x Ed Ç (V 2 ) x x Ed Ç (V ) Ed Ç (V) The σ = ρ 1 : W Ç[G] is give by σ(t) = Σ g G b g (T) g whee fo each elemet g of G, b g Hom (W,Ç) is give by b g (T) = 1 Σ d G i=1 i Tace Vi {ρ i (g 1 ) ο T i }, with T = (T 1, T 2, T ), T i Ed Ç (V i ), a abitay elemet of W, d i = dim V i Recall that ρ : Ç[G] W is a algeba isomophism by Coollay 45 It is easy to see that b g is a liea fuctio o W fo each elemet g of G Hece σ : W Ç[G] is liea It suffices to show that (σ ο ρ)(h) = h fo all elemets h of G sice G is a basis of

26 26 Ç[G] ove Ç Fo h,g G we compute b g (ρ(h)) = 1 Σ d G i=1 i Tace Vi {ρ i (g 1 ) ο ρ i (h)} = 1 Σ d G i=1 i Tace Vi {ρ i (g 1 1 h)} = Σ d G i=1 i χ i (g 1 h) Hece σ(ρ(h)) = Σ g G b g (ρ(h)) g = 1 G Σ g G { Σ i=1 d i χ i (g 1 h)}g = 1 G Σ g G {χ eg (g 1 h)} g = h by the popeties of χ eg that wee used i the poof of Popositio 69 ad also i sectio 15 We have poved that (σ ο ρ)(h) = h fo all elemets h of G æ