FACULTY OF MATHEMATICS AND PHYSICS UNIVERSITY OF LJUBLJANA Damjana Kokol Bukov?sek HOMOMORPHISMS OF MATRIX SEMIGROUPS Doctoral dissertation Ljubljana, 2005 FAKULTETA ZA MATEMATIKO IN FIZIKO UNIVERZA V LJUBLJANI Damjana Kokol Bukov?sek HOMOMORFIZMI MATRICNIH POLGRUP Doktorska disertacija Ljubljana, 2005 Contents Abstract 9 Povzetek 11 1 Introduction 25 2 Stateofthe art 33 2.1 Question and examples ....................... 33 2.2 Case m = 1 ............................. 38 2.3 Case m = n ............................. 40 3 Homomorphisms from dimension two to three 43 3.1 Preliminaries ............................ 43 3.2 Main result ............................. 45 3.3 Corollaries .............................. 55 4 More on homomorphisms from dimension two 59 4.1 Preserving rank 1 .......................... 60 4.2 A technical lemma ......................... 61 4.3 Case n = 4 .............................. 68 4.4 Preserving cyclic unipotent ..................... 75 5 Homomorphisms from a dimension to one dimension higher 81 5.1 Singular matrices .......................... 81 5.2 Two possibilities .......................... 84 5.3 Case m = n + 1 ........................... 90 5.4 Case n = 3 and m = 4,5 ...................... 95 5.5 Case m = 6 ............................. 102 Bibliography 107 5 Zahvala Najprej se zahvaljujem mentorju profesorju Matjažu Omladicu, ki me je navdušil za algebro. Predlagal mi je zanimivo temo in me vodil pri nastajanju tega dela. Najlepša hvala profesorju Heydarju Radjaviju in profesorju Tomažu Koširju za skrben pregled in jezikovne nasvete. Zahvaljujem se tudi domacim za vzpodbudo in potrpežljivost. 7 Abstract In this work we study non-degenerate homomorphisms from the multiplicative semigroup of all n-by-n matrices over a field to the semigroup of m-by-m matrices over the same field. A general introduction is given in the first chapter. In the second chapter we first state our main question and give some examples. Further we characterize all homomorphisms from the multiplicative semigroup of all n-by-n matrices over an arbitrary field to the field and all non-degenerate homomorphisms from the multiplicative semigroup of all n-by-n matrices over an arbitrary field to a semigroup of m-by-m matrices over the same field, if m = n. In the third chapter we characterize all non-degenerate homomorphisms from the multiplicative semigroup of all 2-by-2 matrices over an arbitrary field to the semigroup of 3-by-3 matrices over the same field. If the characteristic of the field is not equal to 2 then we have two possibilities. Either it is a symmetric square, combined with a field homomorphism used entrywise and a matrix conjugation, or a direct sum of the identity and the determinant, combined with a field homomorphism, a homomorphism of the multiplicative semigroup of the field and a matrix conjugation. In the characteristic 2 a symmetric square gives rise to two different homomorphisms and we get three possibilities. In the case of the field of real numbers every irreducible non-9 10 degenerate homomorphism is a matrix conjugation of the symmetric square. In the fourth chapter we study non-degenerate irreducible homomorphisms from the multiplicative semigroup of all 2-by-2 matrices over an algebraically closed field of characteristic zero to the semigroup of m-by-m matrices over the same field. If such a homomorphism maps a cyclic unipotent to a cyclic unipotent, it is the composition of a symmetric power, a field homomorphism used entrywise, and a matrix conjugation. In the case m = 4 we characterize all non-degenerate irreducible homomorphisms. In the fifth chapter we prove that every non-degenerate homomorphism from the multiplicative semigroup of all n-by-n matrices over an algebraically closed field of characteristic zero to the semigroup of (n+1)-by-(n+1) matrices over the same field when n = 3 is reducible and that every non-degenerate homomorphism from the multiplicative semigroup of all 3-by-3 matrices over an algebraically closed field of characteristic zero to the semigroup of 5-by-5 matrices over the same field is reducible. Keywords: matrix semigroup, semigroup homomorphism, multiplicative map, irreducibility. Math. Subj. Class. (2000): 08A35, 15A30, 15A69, 20G05 Povzetek V delu študiramo nedegenerirane homomorfizme iz multiplikativne polgrupe vseh n x n matrik nad komutativnim obsegom v polgrupo m x m matrik nad istim obsegom. Naj bo F poljuben komutativen obseg in n naravno število. Oznacimo z M„(F) množico vseh nxn matrik z elementi v F. Množica M„(F) je polgrupa za operacijo množenja matrik. Vprašanje, s katerim se ukvarjamo, se glasi: Kakšni so homomorfizmi polgrup p : Mn(¥) -»¦ Mm{¥), torej preslikave, ki zadošcajo enacbi Mm»(¥) definirajmo preslikavo p : Mn(¥) -> Mm'+m»(¥) s predpisom p(A) = p'(A) 0 p"{A) za vsako matriko A e Mn(¥). Preslikava p je spet homomorfizem polgrup, ki je vedno razcepen. 6. Zunanja potenca: Naj bo k < n naravno število. Vektorski prostor F© je izomorfen zunanji potenci AfcFra vseh antisimetricnih tenzorjev stopnje k. 13 Ce je S = {ei, e2,..., en} baza prostora ¥n, potem je £' = {eh Ae,2 A... Aeifc,l < «i < i2 < ... < «fc < ^} baza prostora AfcFra. Ce matrika A G M„(F) predstavlja linearno preslikavo prostora ¥n, potem AfcA G A^«(F) predstavlja linearno preslikavo, ki deluje \k) na tenzorjih stopnje k takole: (AfcA)(en A en A ... A eifc) = Aeh A Aet2 A ... A Aelk. Elementi matrike AkA so k x k minorji matrike A. Preslikava (p : Mn(¥) -»¦ A^n\(F), definirana s predpisom ip(A) = AkA, je homomorfizem polgrup. Ce je k enak n - 1, potem preslikava p slika iz M„(F) v Mra(F). V tem primeru je Mm»(¥) definirajmo preslikavo

Mm'm»{W) s predpisom 2 in m < n. Naj bo p : Mn{¥) ->¦ Mm{¥) nedegeneriran homomorfizem polgrup, za katerega velja tp(0) = O in tp(I) = I. Potem je m = n in if ima naslednjo obliko: ¦ F homomorfizem obsega in S e Mn{¥) obrnljiva matrika. V tretjem poglavju najprej pokažemo, da smemo brez škode za splošnost predpostaviti, da (p : Mn(¥) -»¦ Mm{¥) preslika 0 v 0 in identiteto v identiteto. Lema 3 Naj bo F poljuben komutativen obseg in f : M„(F) ->¦ Mm{¥) homomorfizem polgrup. Potem ima f obliko ¦ M3(¥) nede- generiran homomorfizem polgrup, za katerega velja (p(0) je char F ^ 2, potem ima p eno od naslednjih oblik: O in (p(I) = I. Ce (a) f G a 6l\ c d S /(a) f(b) /(c) f{d) O O O o n-1 g(ad - bc) kjer je f : F ->¦ F homomorfizem obsega, g : F ->¦ F homomorfizem multip-likativne polgrupe (¥,¦), za katerega velja g(0) = O, g{\) = 1, in S e M3(¥) obrnljiva matrika, ali (b) /r hi\ r %2) %&) M&2) n r w ) =5 ^(2ac) ^+6c) ^2M) s~ C d [ /i(c 2 ) fr(cd) /i(d 2 ) kjer je /i : F ->¦ F homomorfizem obsega in S E M3(¥) obrnljiva matrika. Ce je char F = 2, potem ima p obliko (a), (b) ali (c) f ([' 3)=s %2) O /i(62) %c) %d + 6c) /i(6d) h(c2) O /i(d2) S-1 kjer je h : F -»¦ F homomorfizem obsega in S e M3(¥) obrnljiva matrika. Ce je char F = 2, sta primera (b) in (c) bistveno razlicna: Matrike v sliki p imajo v primeru (b) natanko en skupen netrivialen invarianten podprostor, ki je dimenzije 2. Po drugi strani pa imajo v primeru (c) skupen invarianten podprostor dimenzije 1. Preslikava p je popolnoma razcepna, ce ima vsak invarianten podprostor slike (p invarianten komplement. Naslednje trditve so preproste posledice Izreka 4. _ _ 17 Posledica 5 Naj bo F komutativen obseg s char F ^ 2. Vsak nedegeneriran homomorfizem polgrup p : M2(¥) -»¦ M3(F) je popolnoma razcepen. Posledica 6 Naj bo >p : M2(F) ->¦ M3(F) nerazcepen nedegeneriran homomorfizem polgrup. Potem je char F ^ 2 in (/9 ([: ž])=« %2) %&) /i(62) S h(2ac) h(ad + bc) h(2bd) h(c2) h(cd) h(d2) c-l kjer je h : F ->¦ F homomorfizem obsega in S e M3(¥) obrnljiva matrika. V obsegu realnih števil E je edini nenicelni homomorfizem identiteta. Posledica 7 Naj bo ip : M2(R) -»¦ -MsW nerazcepen nedegeneriran homomorfizem polgrup. Potem je f S a 6 c d kjer je S G M3(R) obrnljiva matrika. a2 ab b2 2ac ad + bc 2bd c2 cd d2 s-1, Edina zvezna homomorfizma obsega kompleksnih števil C sta identiteta in kompleksna konjugacija. Posledica 8 Naj bo p : M2(C) ->¦ Ms{C) zvezen nerazcepen nedegeneriran homomorfizem polgrup. Potem je ip ([: ž])=s h(a2) h(ab) h{b2) S h{2ac) h(ad + bc) h{2bd) h(c2) h(cd) h(d2) c-l kjer je h : C ->¦ C identiteta ali kompleksna konjugacija in S G M3(C) obrnljiva matrika. 18 POVZETEK V cetrtem in petem poglavju se omejimo na primer, ko je komutativni obseg F algebraicno zaprt in ima karakteristiko nic. Obravnavamo samo nerazcepne homomorfizme. Pogosto uporabljamo naslednjo trditev, ki je posledica Burn-sidovega izreka. Trditev 9 Denimo, da je F algebraicno zaprt obseg s karakteristiko nic. Naj bon>2 m S polgrupa v M„(F). Ce obstaja nenicelen linearen funkcional f na M, (F), ki je enak nic na S, potem je polgrupa S razcepna. Na zacetku cetrtega poglavja pokažemo, da nerazcepen nedegeneriran ho-momorfizem polgrup

3 m ip : M2(F) ->¦ Mn{¥) nerazcepen nedegeneriran homomorfizem polgrup. Glede na zgornjo dekompozicijo ima ip naslednjo obliko: C d * f a X d . . V T z * h i w l k / ce je a,b,c^ 0, potem je /(a) S G(c)EG(a)y m ip G a 6l\ c d xTG(a)EG(b) G(c)EG(a)CG(^) xTG(c)EG(d) /(&) G(d)EG(b)y c-l _ _ 19 kjer smo s C ozna?cili matriko yx T + VEG(^-l)V; ?ce jeb^O, potem je f O d S /(a) O O xTG{a)EG{b) /(&) G(f)VG(b)EG(a)E G{d)EG{b)y O f(d) • sicer pa f a 0 O d S f(a) O O O EG(a)EG(d) O O O f(d) c-l S-1 kjer sta f : F -»¦ F in G : F -»¦ Aira_2(F) homomorfizma polgrup, x,y G Fra"2 neni?celna vektorja, S G M„(F) obrnljiva matrika, E G M„_2(F) matrika z lastnostjo E2 = I in V G M„_2(F) matrika s spektrom enakim {!}. Zgornja tehnicna lema nam pomaga pokazati naslednji izrek, ki je karakte-rizacija homomorfizmov iz polgrupe 2x2 matrik v polgrupo 4x4 matrik. Izrek 11 Naj bo p : M2(F) ->¦ M4(F) nerazcepen nedegeneriran homomor-fizem polgrup. Potem ima ip eno od naslednjih oblik: (a) f c d Sg a3 a2b ab2 b3 3a2c a2d+2abc 2abd+b2c 3b2d 3ac2 2acd + bc2 ad2 + 2bcd 3bd2 c3 c2d cd2 d3 -1 S Sg(Sym3A)S-1, _ _ _ _ _ kjer je g : F ->¦ F homomorfizem obsega in S e M4(¥) obrnljiva matrika, ali 20 POVZETEK (b) ip Q a 6l\ c d S -1 S g{a)h{a) g{a)h{b) g{b)h{a) g(b)h(b) g(a)h(c) g(a)h(d) g(b)h(c) g(b)h(d) g{c)h{a) g{c)h{b) g{d)h{a) g{d)h{b) g(c)h(c) g(c)h(d) g(d)h(c) g(d)h(d) = S(g(A) (g)h(A))S~l, kjer sta g, h : F ->¦ F razli?cna homomorfizma komutativnega obsega in S G M4(¥) obrnljiva matrika. Ce v primeru (b) velja g = h, potem je homomorfizem p razcepen, ker je A 0 A = (A V A) 0 (A A A); sicer je

¦ M4(C) zvezen nerazcepen nedegeneriran homomorfizem polgrup. Potem je bodisi ¦ C identiteta ali kompleksna konjugacija in S e M4(C) obrnljiva matrika, bodisi ip a b c d S ad ah bd bb ac ad bc bd ca cb da db cc cd de dd c-l kjer je S G M4(C) obrnljiva matrika. Matrika A G Mn{¥) je unipotentna, ce je njen spekter enak {1}. Matrika A G M„(F) je cikli?cna, ce ima ciklicni vektor; to je tak vektor x G ¥n za katerega množica {x, Ax, A2x,..., An~lx} napenja ves prostor ¥n. Vsaka _ _ _ 21 ciklicna unipotentna matrika v M„(F) je podobna matriki "1 1 0 ••• 0 0" 0 1 1 '•• 0 0 0 0 1 '•• 0 0 : : •. •. •. : o o o ¦•. 1 1 0 0 0 ••• 0 1 Izrek 13 Naj bo n > 3 in p : M2(F) ->¦ Mn{¥) nerazcepen nedegeneriran ho-momorfizem polgrup, ki preslika cikli?cni unipotent v cikli?cni unipotent. Potem je p(A) = Sg(Symn-lA)S-\ kjer je g : F ->¦ F homomorfizem obsega in S e Mn{¥) obrnljiva matrika. V petem poglavju obravnavamo primer, ko je dimenzija matrik v polgrupi iz katere slikamo vsaj 3. Trditev 14 Naj bo (p : M„(F) -»¦ Mm(F) homomorfizem polgrup, ki preslika 0 vO in identiteto v identiteto. Naj bo k = minjrangA; 3 in m < 2n. Naj bo ip : Mn{¥) ->¦ Mm{¥) nedegeneriran homomorfizem polgrup, ki preslika 0 v 0 in identiteto v identiteto. Denimo, da p slika matrike ranga 1 v matrike ranga 1. Potem p slika matrike ranga 2 v matrike ranga 2. 22 POVZETEK Naslednja trditev je ocitna za n = 3 in m < 6. Dokažemo jo še za vecje vrednosti n. Trditev 16 Denimo, da je n > 4 in m < 2n ali pa n = 4 in m < 5. Naj bo p : Mn(¥) -»¦ Mm(F) nedegeneriran homomorfizem polgrup, ki preslika 0 v 0 in identiteto v identiteto. Potem imamo dve mo?znosti: (a) ?ce je rang A = 1, potem je rang p(A) = 1, in ?c e je rang A = 2, potem je rang ^(A) = 2, ali (b) ?ce je rang A < n - 1, potem je p(A) = 0, in ?c e je rang A = n - 1, potem je rang p (A) = 1. Naslednji dve trditvi obravnavata možnosti, ki nam jih da Trditev 16. Trditev 17 Denimo, da je n > 2 in m > n. Naj bo ip : Mn{¥) ->¦ Mm(F) nedegeneriran homomorfizem polgrup, ki preslika 0 v 0 in identiteto v identiteto. Naj p slika matrike ranga 1 v matrike ranga 1 in ma trike ranga 2 v matrike ranga 2. Potem je ¦ F homomorfizem obsega in S e Mm{¥) obrnljiva matrika. Trditev 18 Denimo, da je n > 3 in m > n. Naj bo ip : Mn{¥) ->¦ Mm{¥) nedegeneriran homomorfizem polgrup, ki preslika 0 v 0 in identiteto v identiteto. Naj p slika matrike ranga manj?sega kot n - 1 v 0 in matrike ranga n - 1 v matrike ranga 1. Potem je (A)= J/(Cof(A)) *]s-i * * kjer je f : F -»¦ F homomorfizem multiplikativne polgrupe (F, •) in 5 G Mm(F) obrnljiva matrika. 23 Naslednja izreka sta osrednja rezultata petega poglavja. Izrek 19 Naj bo n > 3. Vsak nedegeneriran homomorfizem polgrup ip : Mn(¥) -> Mn+i(¥) je razcepen. Izrek 20 Naj bom = 4 ali m = 5. Vsak nedegeneriran homomorfizem polgrup

Mm(¥) je razcepen. Na koncu dodamo še nekaj primerov: Obstajata dva bistveno razlicna nedegenerirana nerazcepna homomorfizma polgrup

M6(F): simetricni kvadrat ¦ S2 is a homomorphism of semigroups, if it preserves the operation o, ^(ao6) = ^(a)o^(6) for all a,b € Sl Let F be an arbitrary field and ra be an integer. Denote by Mn(¥) the set of all n-by-n matrices with entries in F. Then Mn(¥) is a semigroup under the multiplication of matrices. In this work we study homomorphisms of these semigroups and try to classify them. The question of classification of semigroup homomorphisms is quite old and it may be difficult. Let us look first at a simple example. Let (E, +) be the additive semigroup of real numbers. A semigroup homomorphism / : E -»¦ E satisfies Cauchy’s functional equation f(x + y) = f(x) + f(y) for all x, y G E. This equation has some simple solutions f(x) = ex for all igR, 25 26 1. INTRODUCTION where c is a real constant. All other solutions are quite wild. The graph of each solution of Cauchy’s equation which is not of this form is everywhere dense in the plane E2. The equation was solved by Hamel in [10] a hundred years ago. He proved that there exists a subset H of E such that every real number x can be expressed in a unique way in the form n fc=i where hk G H and rk are rational. The general solution of Cauchy’s equation is given by choosing the values of / arbitrary on H and defining (n \ n ^rkhk\ =^rkf(hk). fc=i / fc=i The set of real numbers is also a multiplicative semigroup. Its homomorphisms / : E -»¦ E satisfy Cauchy’s power equation g(xy) = g(x)g(y) Every solution of this equation is of the form g(x) = 0 for all xeR or g(x) = 1 for all xeR or 0(0) = 0 and g(x) = e/(log(|:c|)) for all x = 0, or ^(0) = 0 and g(x) = signxe/(log(|:c|)) for all x = 0, where / is a solution of (additive) Cauchy’s equation. 27 Let us now move on to matrices. The set of all matrices Mn(F) is an algebra. It is well-known that every automorphism of this algebra is inner. More precisely, every bijective linear map

0, a2 > 0,..., ara_i > 0 and an arbitrary. Let ?a : G ^ Symail/ 0 Sym»2(AV) 0 ... 0 Sym^-(A™"V) be defined by ?a(A) = Sym°M 0 Sym«2(A2A) 0 ... 0 Sym^-(A™"1^ • (detA)°» for every A e G. Representation ?a is not irreducible, so let Fa be an irreducible subrepresentation of ?a generated by vector v = (Vai(ei)) 0 (Va2(ei A e2)) 0 ... 0 (V^-(ei A ... A e^)) where {e1} e2,..., en} is a basis for V. Vector v is a highest weight vector of the representations ?a and Fa. Every (differentiable) irreducible complex representation of G is isomorphic to Fa for a unique index a = (a1} a2,..., a„_i, an) with a1} ...,ara_i > 0. For more details see [8]. The problem of homomorphisms

2 and R is a semiprime ring or a ring in which all idempotents are central, then

3, and let AcB(X) 32 1. INTRODUCTION and B C B(Y) be standard operator algebras (that is, algebras of bounded linear operators that contain all finite-rank operators). ? emrl in [35] described multiplicative bijective maps of A onto B, while Molnar in [22] dealt with a problem of bijective maps multiplicative under Jordan triple product. It is proved that such a map is necessarily linear or conjugate-linear in the case when X is infinite-dimensional. Lu in [19] proved that if A is a standard operator algebra on a Banach space X with dimX > 1, R any ring, and

Mn(¥) Mm(¥) o(/) = I, E e Mm-k(¥) is idempotent and S G Mm(F) is an invertible matrix. Here either orm-fcmay be 0, i. e. either 0 Fm"_ If p, . Mn(¥j _^ Mm-(W) and (A) = ^(A) 0 if"(A) for every matrix A G M„(F). Map p is again a semigroup homomorphism. The tensor product ¦ F of the multiplicative semigroup (F, •) such that ¦ F by f(x) ip 0 whenever A is singular. Next define x 0 0 • • 0 0 10- • 0 0 0 1- • 0 0 0 0 ••• 1 It follows that / is a semigroup homomorphism. Since the relation 2 and m¦ Mm{¥) be a semigroup homomorphism, which is non-degenerate and has the properties tp(0) = 0 and tp(I) = I. Then m = n and

¦ F is a field homomorphism, and S G Mn(¥) is an invertible matrix, or _ 2.3. Case m < n 41 (b) p(A) = Sf{Col{A))S-1, where f : F ->¦ F is a field homomorphism, and S E M„(F) is an invertible matrix. We will later (especially in chapter 5) extend the proof of this theorem to more general setting. We will give the proof in Section 5.2. Chapter 3 Homomorphisms from dimension two to three In this chapter we characterize all non-degenerate homomorphisms from the multiplicative semigroup of all 2-by-2 matrices over an arbitrary field to the semigroup of 3-by-3 matrices over the same field. If the characteristic of the field is not equal to 2 then we have two possibilities. Either it is a symmetric square, combined with a field homomorphism used entrywise and a matrix conjugation, or a direct sum of the identity and the determinant, combined with a field homomorphism, a homomorphism of the multiplicative semigroup of the field and a matrix conjugation. In the characteristic 2 a symmetric square gives rise to two different homomorphisms and we get three possibilities. In the case of the field of real numbers every irreducible non-degenerate homomorphism is a matrix conjugation of the symmetric square. 3.1 Preliminaries We will first show that there is no loss of generality if we assume that a semigroup homomorphism ? : Mn(F) ? Mm(F) maps 0 to 0 and the identity 43 44 3. FROM DIMENSION TWO TO THREE to the identity. Lemma 3.1 Let ¥ be a field and tp : Mn(¥) ->¦ Mm(F) a semigroup homo-morphism. Then tp has the form ¦ Mk{¥) is a semigroup homomorphism with tpo(0) = 0, ipo{I) = I, E e Mm-k(¥) is idempotent and S E Mm{¥) is an invertible matrix. Here either k or m - k may be 0, i. e. either p0(A) or E may be absent. Proof. Since 0 and / are two commuting idempotents with 0/ = 0, p(0) and p has the form Q a b]\ _ \f(a) /(6)1 : c d~ /(c) /(d) ' where / : F ->¦ F is a field homomorphism and S E M2(¥) is an invertible matrix. The following proposition is a special case of Proposition 2.1 for n = 2. Proposition 3.3 Let ¥ be a field and >p : M2(¥) ->¦ F a semigroup homomorphism. Then >p has the form ^(\a bX\ =h(ad-bc), where h:¥^¥ is a homomorphism of the multiplicative semigroup (F, •). 3.2 Main result The main result of this chapter is the following: Theorem 3.4 Let ¥ be a field and

¦ F is a field homomorphism, g : F ->¦ F is a homomorphism of the multiplicative semigroup (F, •) with g(0) = 0, g{\) = 1 and S E M3(¥) is an invertible matrix, 46 3. FROM DIMENSION TWO TO THREE (b) f ([: 5])=* h(a2) h(ab) h(b2) S h(2ac) h(ad + bc) h(2bd) h(c2) h(cd) h(d2) s-\ where h : F ->¦ F is a field homomorphism and S G M3(¥) is an invertible matrix. Ifchar F = 2 then p has one of the forms (a), (b) or (c) f ([" 3) = s h(a2) 0 h(b2) S h(ac) h(ad + bc) h(bd) h(c2) 0 h(d2) c-l where h : F ->¦ F is a field homomorphism and S G M3(¥) is an invertible matrix. Remark. If char F = 2, the cases (b) and (c) are essentially different: The image of

F are semigroup homomorphisms with /(0) = ^(0) = 0 and /(1) = g(l) = 1. Proof: Matrix al commutes with E12 and E21, so ^(a/) commutes with El3 and S31 and we obtain the asserted form. Step 3. Homomorphism

¦ F is a field homomorphism, h : F ->¦ F is a semigroup homo- morphism, so we are in the case (a) of the Theorem. Proof: Matrix al commutes with every A G M2(F), so M2(F) and ^2 : A*2(F) -? F where and Qa 6l\ _ [/(a) /(6)1 c d /(c) f{d) ^Ja bX =s(a,b,c,d). c d Now, / is a field homomorphism by Proposition 3.2 and s(a,b,c,d) has the form h(ad - be) by Proposition 3.3. From now on we will assume that f(a) = g (a) for every a G F. So (0) (/9 i lo l i o 0 o -1 o 1 o 2 - xy 0 0 0 0 0 0 0 0 so xy = 2. Since char F ^ 2, both x and y are nonzero. If we take ¦ M3(F) is completely reducible. Corollary 3.6 Let p : M2(¥) ->¦ M3(F) be an irreducible non-degenerate semigroup homomorphism. Then char F = 2 and h(a2) h(ab) h(b2) f {[: :]= s- h(2ac) h(ad + be) h(2bd) h(c2) h(cd) h(d2) where h : ¥ ^ ¥ is a field homomorphism and S E M3(¥) is an invertible matrix. If F is the field of real numbers R, then the only nonzero field homomorphism of F is the identity (see [1], page 57). This implies Corollary 3.7 Let p : M2(R) -»¦ M3(R) be an irreducible non-degenerate semigroup homomorphism. Then ' a2 oh b2 ' 2ac ad + be 2bd c2 cd d2 where S E M3(R) is an invertible matrix. ip Q a 6l\ c d S s- If F is the field of complex numbers C we may be interested only in continuous semigroup homomorphism p : M2(¥) -»¦ M3(¥). Then semigroup or field homomorphisms f,g, h : F -»¦ F in the Theorem 3.4 must be continuous. The only continuous field homomorphisms of C are the identity and the complex conjugation (see [1], page 53). Corollary 3.8 Let >p : M2(C) ->¦ M3(C) be a continuous irreducible non-degenerate semigroup homomorphism. Then >p has the form h(a2) h(ab) h(b2) ip ([: s])=* h(2ac) h(ad + be) h(2bd) h(c2) h(cd) h(d2) S- 3.3. Corollaries 57 where h : C ->¦ C is the identity or the complex conjugation and S E M3{C) is an invertible matrix. Chapter 4 More on homomorphisms from dimension two In this chapter we study non-degenerate irreducible homomorphisms from the multiplicative semigroup of all 2-by-2 matrices over an algebraically closed field of characteristic zero to the semigroup of n-by-n matrices over the same field. If such a homomorphism maps a cyclic unipotent to a cyclic unipotent, it is the composition of a symmetric power, a field homomorphism used entrywise, and a matrix conjugation. In the case n = 4 we characterize all non-degenerate irreducible homomorphisms. From now on we will assume that the field F has characteristic zero and is algebraically closed. We have seen in chapter 3 that from dimension two to three we get a different result in characteristic 2 from other characteristics. The situation is similar when going from dimension two to higher dimensions: the results will depend on whether the characteristic zero or finite and small. We will also restrict ourselves to irreducible homomorphisms. We will often use the following proposition which is a consequence of a theorem of Burnside. It is proved in [32], page 27. Proposition 4.1 Assume F is an algebraically closed field of characteristic 59 60 4. MORE ON HOMOMORPHISMS FROM DIMENSION TWO zero. Let n > 2 and S be a semigroup in M„(F). If there exists a nonzero linear functional f on Mn{¥) which vanishes on S, then S is reducible. 4.1 Preserving rank 1 We first show that every irreducible non-degenerate

2 and p : M2(¥) -»¦ Mn(¥) be a semigroup ho-momorphism, which is irreducible and non-degenerate. Then rank tp(A) = 1 whenever rank A = 1. Proof. Since

1, tp is reducible (Proposition 4.1). D 4.2 A technical lemma Let us divide every n-by-n matrix into 3-by-3 block structure where the middle block is (n-2)-by-(n-2). So a b ¦¦¦ c d e * ••• * f g * • • • * i j ¦¦¦ k h l a X d' V T z i w l where T is a (n - 2)-by-(n - 2) matrix. Lemma 4.3 Let n > 3 and p : M2(¥) ->¦ Mra(F) be a semigroup homomor-phism, which is irreducible and non-degenerate. Then it has the following form with respect to the above decomposition: • ifa,b,c^0anddis arbitrary then f c d _ _ _ _ _ 62 4. MORE ON HOMOMORPHISMS FROM DIMENSION TWO where f(a) xTG(a)EG(b) /(&) S G(c)EG(a)y G(c)EG(a)CG(±) G(d)EG(b)y /(c) xTG(c)EG(d) f(d) C = yx T + VEG(^-1)V c-l ifb=0 and a, d are arbitrary then f a 6 0 d S f(a) xTG(a)EG(b) f(b) 0 G(f)VG(b)EG(a)E G(d)EG(b)y 0 0 f(d) c-l; • otherwise f 0 d S f(a) 0 0 0 EG(a)EG(d) 0 0 0 f(d) c-l; where f : F ->¦ F and G : F ->¦ Aira_2(F) are semigroup homomorphisms, x,y E ¥n~2 are nonzero vectors, E,V G M„_2(F) are matrices with E2 = I and the spectrum ofV equal to {1}, and S E M„(F) is an invertible matrix. Proof. Let us denote by E^ the matrix which has 1 in the i-th row and the j-th column, and 0 elsewhere. We will divide the proof into several steps. Step 1. Without loss of generality we may assume that [J h{f] for every a G F where g : F -»¦ F is a semigroup homomorphism and h : F -»¦ F satisfies %&) = %) + h(b) with %) ^ 0 for at least one a G F. We may assume without loss of generality that G(a)=g(a)\1 h^)] Let us choose an a with h(a) ^ 0. The matrix" ° commutes with J ° and is similar to [J °], so ^(^(a)^ = g (a) J [1 -%)] 0 1 Thus E is of the form E = ? ? 0 -1 o -l ' If E = \l f] then y T = [yu0] and the first column of

¦ M4(C) be a semigroup homomorphism, which is irreducible, non-degenerate and continuous. Then ¦ C is the identity or complex conjugation and S E M4(C) is an invertible matrix, or ip ad ah ba bb a b c d =s ac cd ad cb be da bd db CC cd dc dd c-l where S E M4(C) is an invertible matrix. Proof. If (p is continuous, then so are the field homomorphisms g and h. Case (a) of the theorem gives us the first possibility. If we are in the case (b) of the theorem, then g is the identity and h is complex conjugation or the other way around. But the matrices ad ah ba bb ac ad be bd cd cb da db cc cd dc dd and da db ba bb dc ad be bd ca cb da db cc cd dc dd are simultaneously similar for all a, b, c, d, so we obtain the second possibility. D 4.4. Preserving cyclic unipotent 75 4.4 Preserving cyclic unipotent A matrix A e Mn{¥) is unipotent if its spectrum is equal to {1}. A matrix A e Mn(¥) is cyclic if it has a cyclic vector, i. e. a vector x G ¥n for which the set {x, Ax, A2x,..., An~lx} spans all ¥n. Every cyclic unipotent in Mn(¥) is similar to the matrix 1 1 0 0 0 0 1 1 . . . 0 0 0 0 1 . . . 0 0 . . . . . . . . . . . . 0 0 0 . . . 1 1 0 0 0 . . . 0 1 Theorem 4.6 Let n > 3 and p : M2(¥) ->¦ Mn(¥) be a semigroup homomor-phism, which is irreducible, non-degenerate and maps a cyclic unipotent to a cyclic unipotent. Then ¦ F is a field homomorphism and S E Mn(¥) is an invertible matrix. Proof. Without loss of generality we may assume that Denoting f i i 0 ••• 0 0 0 1 1 '•• 0 0 1 0 = 0 0 1 '•• 0 0 0 0 0 '•• 1 1 0 0 0 ••• 0 1 /7i 0]\ 2 = [%ij ] ij=l 76 4. MORE ON HOMOMORPHISMS FROM DIMENSION TWO and using the equality [-11- -i i- -ii- -i 9 1 1] [1 0] [1 0] [1 11 0 1 0 2 0 2 0 1 (4.1) we obtain Xi+lJ = Xij-2 + ^3-i for all i,j = 1, 2,..., n, where x« = 0 if k or / is less than 1 or grater than n. It follows that ip(\0 0] J is upper-triangular and xi+M+i = 2^. So f 0 2 a 1 * * ... 0 2 * ••• 0 0 4 . . . . . 0 0 0 2n-l We may now apply a simultaneous similarity with an upper-triangular matrix, so that f is diagonal. This will change y 0 2 1 1 0 1 but it will still remain upper- triangular. We again apply a simultaneous similarity with a diagonal matrix, so that for Q1 1 "I \ [ ] 0 1J = Kb=i the entries will satisfy wi>i+1 = i. This similarity will leave

n-i wVn 0 1 2 . . . W2,n-1 W2n 0 0 1 . . . w3>n-i w3n 0 2 if 1 1 0 1 2 1 . . . . . 0 0 0 . . 0 0 0 1 n 1 _ 0 4.4. Preserving cyclic unipotent 77 and f al 0] \ 0 2 a 1 0 0 0 0 2 0 0 0 0 4 . . . . . . 0 0 0 0 . . . 2n~ Let us prove that Wij i-i i by induction on j - i. This is true if j = i + 1. From the equality (4.1) it follows that WikWkj ¦ k=i Thus we obtain using the inductive hypothesis that (23~i -2)wt3 = J2 w^k3 = Yl ( ) u ) k=i+l k=i+l (fc-l)!(j-l)! ^ (7^1)!(fc-i)!(fc-l)!(j-fc)! (J - 1)! E (J - «)! (z - 1)! (j - i)! fcfi (fc - i)! (j - fc)! 3 ~ l\ (T~l -i-l 2)- Now, En commutes with 1 0 0 2 1 1 0 1 En so if (En) is diagonal. Since En it follows that 3 is reducible and that every non-degenerate homomorphism from the multiplicative semigroup of all 3-by-3 matrices over an algebraically closed field of characteristic zero to the semigroup of 5-by-5 matrices over the same field is reducible. 5.1 Singular matrices We first look where an non-degenerate irreducible homomorphism sends singular matrices. Proposition 5.1 Let p : Mn(¥) -»¦ Mm(¥) a semigroup homomorphism, which sends 0 to 0 and identity to identity. Let k = minjrankA; 1. Since EiEj for i ^ j has rank less than k, we have p(Et)p(E3) = 0, and lp{E1),lp{E2),...,ip{Et) are disjoint idempotents. We conclude that t(rank ^{Ex)) < m, implying g) < m. D Proposition 5.2 Assume that n > 3 and m < In. Let

3. Then 2{n - 1) + 1 > m. 5.1. Singular matrices 83 Now p(E22 + £33) has products of rank 1 with 4 and m < In or that n = A and m < 5. Let tp : Mra(F) ->¦ Mm(F) be a semigroup homomorphism, which is non-degenerate and sends 0 to 0 and identity to identity. Then we have two possibilities: (a) ifrank A = 1 then rank p(A) = 1, and ifrank A = 2 then rank p(A) = 2, or (b) if rank A < n - 1 then 4, then m < 2n < g). If ra = 4, then m < 5 < Q. So by Proposition 5.1 k = 1 or k = n - 1. Case (a): fc = 1. The matrices En,E22, ....Enn e M„(F) are idempotents of rank 1, so (P„) G Mm(F) are disjoint commuting idempotents with the same rank, say /. Since they are disjoint, nl < m, so / = 1. Thus rank A = n-1 implies rank 2 andm>n. Let p : M„(F) -»¦ Mm{¥) be a semigroup homomorphism, which is non-degenerate and sends 0 to 0 and identity to identity. Suppose that rank A = 1 implies rank tp(A) = 1 and that rank A = 2 implies rank p (A) = 2. Then ^A) = s\^A) *]s-\ * * where / : F ->¦ F is a field homomorphism and S E Mm{¥) is an invertible matrix. Proof. Denote by E^ the matrix which has 1 in the i-th row and the j-th column, and 0 elsewhere. Matrices En,E22,..., Enn e M„(F) are disjoint commuting idempotents of rank 1, so p(En), p(E22), ... 3. Then ^)=^(i?11+i?22)A)=[^;^ ;][^) ;] = [/( 0 A) ;]. Let us now prove that / is additive. For a, b G F we have f(a + b)En = 3 andm>n. Let

¦ F is a homomorphism of the multiplicative semigroup (F, •) and 5 G Mm{¥) is an invertible matrix. Proof. Denote by E^ the matrix which has 1 in the i-th row and the j-th column, and 0 elsewhere. Introduce Pa = I - En e Mn(¥), and let U be the identity matrix in Mi(¥). Further, let Nt be the matrix in Mi(¥), defined by Ni = E12 + ... + Ei_hi. Denote P^ = 1^ 0 Nj_i+1 0 /„_.,¦ if i < j, and The matrices Pn, P22,..., Pnn G Mn(¥) are disjoint commuting idempo-tents of rank n - 1, so ^(£„), p(E22), -, ¥>(£nn) G A^m(F) are disjoint commuting idempotents of rank 1. So they are simultaneously similar to En,E22,...,Enn e Mm{¥). Without loss of generality we may thus assume that p(Pu) = En. Observe that P^ = PikPkj and PikPtj has rank less than n - 1 if k ^ I. We now have SkiSMPij) = 3 and let k = minjrankA; 3. Every non-degenerate semigroup homo-morphism tp : Mn(¥) ->¦ Mn+i(¥) is reducible. Proof. Suppose

2i(4) M^) ' If also 5 G M,(F), then ^ >~ % + 1, we have P^Ri = PjjRiPjj, so E^R,) = Ejjy{Ri)Ejj, thus the only non-zero element in the j-th row of o^ Now let A = A' 0 In_2, where A' = \a b] eM2(¥). * * Multiplying A by P33, ...Pnn on the left or on the right side we obtain c d (ii) / [ d2 c2 dc g I b2 a2 ba \[2db 2ca da + cb where f(x) = g(x2) and g is additive. In this case we have d2 c2 0 dc b2 a2 0 ba 0 0 (ad-bc)2In_2 0 2db 2ca 0 da + cb 3, m > n and

¦ Mm(F) is reducible. Proof. If m = 4, this is a special case of Theorem 5.6, so let m = 5. Suppose

(#2) where the lower-right corner S E 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 ei e2 0 0 0 es e4 e3 e<2\ c-l _ _ _ 5.4. Case n = 3 and m = 4, 5 101 is an involution similar to J -° and the product J -° E is of order three or one. So either E 1 0 0 -1 [l o 0 -l or it has the form S |"-2 61 J. I 46 2 where 6^0. In the first case again