Hopf Algebras in General and in Combinatorial Physics: a practical introduction

Hopf Algebras in General and in Com binatorial Ph ysics: a pract ical in tro duction G H E Duch amp † , P Blasiak ‡ , A Hor z ela ‡ , K A Penson ♦ and A I Solomon ♦ ,♯ a LIPN - UMR 7030 CNRS - Universit ´ e P aris 13 F-9343 0 V illetaneuse, F rance ‡ H. Niewodnicza ´ nski Institute of Nuclear P hysics, Polish Academy o f Scie nc e s ul. Eliasza- Radziko wskiego 152, PL 313 42 Kr ak´ o w, Poland ♦ Lab oratoir e de Physique Th´ eorique des Liquides, Univ er s it ´ e Pierre et Marie Curie, CNRS UMR 7 600 T our 24 - 2e ´ et., 4 pl. Juss ieu, F 75252 Paris Cedex 05 , F rance ♯ The Op en University , Physics and Astro nomy Department Milton Keynes MK7 6AA, United K ing dom E-mail: ghed@l ipn-u niv.paris13.fr , pawel.blas iak@if j.edu.pl , andrze j.hor zela@ifj.edu.pl , penson@lp tl.jus sieu.fr , a.i.so lomon @open.ac.uk Abstract. This tutorial is intended to give an accessible intro ductio n to Hopf algebras . The mathematical context is that of representation theory , and w e also illustrate the structures with examples taken from com binator ics and quan tum physics, showing that in th is latter case the axioms o f Hopf alg ebra ar ise naturally . The text contains ma n y exe r cises, some taken from ph ysics, aimed at expanding and exemplifying the concepts intro duced. 1. Intro duction Quan tum Theory seen in action is an inte rplay o f mathematical ideas and ph ysical concepts. F rom a mo dern p e rsp ectiv e its formalism and s tructure are founded on the theory of Hilb ert spaces [36, 44]. Using a few basic p ostulat es, the ph ysical no tions of a system and apparatus, as w ell as transformations and measuremen ts, are de scrib ed in terms of linear op erator s. In this w ay the algebra of operat o rs constitutes a prop er mathematical fra mew ork within whic h quan tum theories ma y b e constructed. The structure of this algebra is determined by tw o op erations, namely - addition and m ultiplication of o p erators; and these lie at the ro ot of all fundamental asp ects of Quan tum Theory . V ersion 23 Octo ber 20 1 8 1 The forma lism of quantum theory represen ts the phy sical concepts of states, observ ables and their transformations as ob jects in some Hilb ert space H and subseq uen tly provid es a sc heme for measuremen t predictions. Brieﬂy , v ectors in the Hilb ert space describe states of a s ystem, and linear forms in V ∗ represen t basic observ ables. Bot h concepts com bine in the measuremen t pro cess whic h pro vides a probabilistic distribution of results and is given by the Born rule. Ph ysical information ab out the system is gained b y transforming the sy stem and/or apparatus in v arious w ays a nd p erforming measuremen ts. Sets of transformations usually p oss ess some structure – suc h a s that of a group, semi-group, Lie algebra, etc. – and in general can be handled within the concept of an algebra A . The action of the algebra on the vec tor s pace of states V and observ ables V ∗ is simp ly its repres entation. Hence if an alg ebra is to describ e ph ysical transformations it has to hav e represen tations in all ph ysically relev an t systems. This requiremen t directly leads to the Hopf algebra structures in ph ysics. F rom the mathematical viewp oin t the structure o f the theory , mo dulo details, seems to b e clear. P hy sicists, ho w ev er, need to ha v e some additional prop erties and constructions to mo v e fr eely in this arena. Here w e will sho w how the structure of Hopf algebras en ters in to the game in the contex t of represen tations. The ﬁrst issue at p oint is the construction of tensor pro duct of v ector spaces whic h is needed for the description of comp osite systems. Supp ose, we kno w how some transformatio ns act on individual systems , i.e. w e kno w represen tations of t he alg ebra in eac h v ector space V 1 and V 2 , r espective ly . Hence natural need a rises for a canonical construction of an induced represen tation of this algebra in V 1 ⊗ V 2 whic h w ould describ e its action on the comp osite system. Suc h a sc heme exists and is provided by the c o-pr o d uct in the algebra, i.e. a morphism ∆ : A − → A ⊗ A . The phys ical plausibilit y of this construction requires the equiv alence of represen tations built o n ( V 1 ⊗ V 2 ) ⊗ V 3 and V 1 ⊗ ( V 2 ⊗ V 3 ) – sinc e the comp o sition of t hr ee systems can not dep end on the order in whic h it is done. This requiremen t forces the co- pro duct to b e c o-asso ciative . Another p oin t is connected with the fact that from the physic al point of view the v ector space C repres ents a trivial system hav ing o nly one prop erty – “b ein g itself ” – whic h can not c hange. Hence one should hav e a canonical represen tation of the algebra on a trivial system, denoted by ǫ : A − → C . Next, since the comp osition of an y system with a trivial one can no t in tro duce new represen tations, those on V and V ⊗ C should b e eq uiv alen t. This requiremen t imp oses the c ondition on ǫ to be a c o-unit in the algebra. In this w a y w e motiv ate the need fo r a bi-alge b r a structure in ph ysics. The concept of an an tip o de en ters in the con text of measuremen t. Measuremen t in a system is described in in terms of V ∗ × V and m easuremen t predictions ar e giv en thro ug h the canonical pairing c : V ∗ × V − → C . Observ ables, described in the dual s pace V ∗ , can also b e transformed and represen tations of appropriate algebras are giv en with the help o f an an ti-morphism α : A − → A . Ph ysics requires that transformation prefo r med on the system and a ppa r atus sim ultaneously should not c hange the measuremen t results, hence the pairing should tr ivially transform under the action of the bi-algebra. W e thu s obtain the condition on α to b e an an tip o de , whic h is the last ingr edien t of a Hopf Algebra. 2 Man y Hopf algebras are motiv a ted b y v arious theories (ph ysic al or close to ph ysics) suc h a s renormalization [1 2 , 11, 14, 16, 40], non-comm utative geometry [16 , 17], ph ysical c hemistry [43, 13], computer science [29 ], algebraic com binatorics [14, 30, 41, 19], algebra [31, 32, 33]. 2. Op er at or s 2.1. Gener ali ties Throughout this t ext w e will consider (linear) op erators ω : V − → V , where V is a v ector space o v er k ( k is a ﬁeld of scalars whic h can b e thought of as R or C ). The set of all (linear) op erato r s V − → V is an algebra (see app end ix) whic h will b e denoted by E nd K ( V ). 2.2. What is a r epr e s entation It is not rare in Ph ysics that we consider, instead of a single o p erator, a set or a family of o p erators ( ω α ) α ∈ A and of t en t he index set itself has a structure. In the old b ooks one ﬁnds the family-lik e notatio n, where ρ ( α ) is denoted, sa y , ω α . As a family of op erato r s ( ω α ) α ∈ A is no more than a mapping ρ : A 7→ E nd k ( V ) (see [6], Ch. I I 3.4 remark), w e wprefer to exhibit the mapping by considering it deﬁned as suc h. This will b e precisely the concept of r epr esen tation that w e will illustrate b y familiar examples. Moreo v er using arro ws allo ws, as w e will see more clearly b elo w, for extension a nd factorizat io n pro cedures. • First case: A is a group In this case, w e p ostulate that the action o f the op erators b e compatible with the law s of a group; that is, for all α , β ∈ A , ( ρ ( α.β ) = ρ ( α ) ◦ ρ ( β ) ρ ( α − 1 ) = ( ρ ( α )) − 1 (1) whic h is equiv alen t to ( ρ ( α.β ) = ρ ( α ) ◦ ρ ( β ) ρ (1 A ) = 1 E n d ( V ) (2) Note that eac h of these conditio ns implies that the r a nge of ρ is in Aut k ( V ) (= GL n ( k ), the linear group), the set of one-to -one elemen ts o f E nd k ( V ) (called automorphisms). • Second case: A is a Lie algebra 3 In this case, o ne requires that ρ ([ α, β ]) = ρ ( α ) ◦ ρ ( β ) − ρ ( β ) ◦ ρ ( α ) = [ ρ ( α ) , ρ ( β )] . (3) W e will see that these tw o t ypes of action (of a group or Lie a lg ebra) can b e uniﬁed through the concept of the represen tation o f an algebra (or whic h amoun ts to the same thing, of a mo dule ). In the ﬁrst case, one in v ok es the gr oup algebr a k [ A ] (see app endix). In t he case of a Lie algebra, o ne inv ok es the enveloping algebr a U ( A ) (or U k ( A ) see app endix ). In b oth cases, the original represen tation ρ is extended to a represen tation of an as so ciative algebr a with unit (AA U) a s follow s: G ρ / / can   E nd k ( V ) k [ G ] ˆ ρ 9 9 t t t t t t t t t G ρ / / can   E nd k ( V ) U k ( G ) ˆ ρ 9 9 s s s s s s s s s s So far w e ha v e not deﬁned what a represen tation of AAU is. Keeping the philosoph y of (1) (or (2) ) and (3) , w e can stat e the following deﬁnition: Deﬁnition 2.1 L et ( A , + , · ) b e an AA U. A c ol le ction of op er ators { ρ ( α ) } α ∈A in a v e ctor sp ac e V is said to b e a r epr esentation of A iﬀ the mapping ρ : A 7→ E nd ( V ) is c o m p atible with the op er ations a nd units of A . This me ans that, identic al ly (i. e. for al l α , β ∈ A and λ ∈ K ).      ρ ( α + β ) = ρ ( α ) + ρ ( β ) , ρ ( λα ) = λρ ( α ) , ρ ( α · β ) = ρ ( α ) ◦ ρ ( β ) , ρ (1 A ) = I d V , (4) wher e ◦ de notes c omp osition of op er ators. Remarks 2.2 (i) This is e quivalent to saying that the arr ow ρ : A 7→ E nd ( V ) fr om A to E nd ( V ) is a morphism of algeb r as (with units). (ii) In this c ase, it is sometimes c onven i e nt to de n ote by α.v the action of ρ ( α ) on v (i.e. the element ρ ( α )[ v ] ) for α ∈ A and v ∈ V . (iii) It may happ en (and this often o c curs) that a r epr esen tation has r e lations that ar e no t pr esent in the original algebr a. In this c ase the r epr esentation is said to b e not faithful . Mor e ri g our ously a r epr esentation is said to b e fai thful iﬀ ρ is inje ctive or, which is e quivalent, k er ( ρ ) = ρ − 1 ( { 0 } ) = { 0 } (for alge b r a s and Lie algebr as) and k er ( ρ ) = ρ − 1 ( { 1 V } ) = { 1 G } (for gr oups). Example 2.3 : L et G = { 1 , c, c 2 } b e the cyclic gr oup o f o r der 3 ( c is the cycle 1 → 2 → 3 → 1 ), G a d mits the plane r epr esentation by ρ ( c ) = − 1 / 2 − √ 3 / 2 √ 3 / 2 − 1 / 2 ! (it is the matrix c orr esp on ding to a r otation of 2 π / 3 ) . (5) 4 Thus, ρ ( c 2 ) = − 1 / 2 √ 3 / 2 − √ 3 / 2 − 1 / 2 ! and, of c ourse, ρ (1) = 1 0 0 1 ! . (6) The r epr ese n tation ρ is f a ithful while its extens i o n to the gr oup alge b r a is not, as se en fr om: ρ (1 + c + c 2 ) = 0 0 0 0 ! wher e as 1 + c + c 2 6 = 0 in C [ G ] . (7) Note 2.4 Note t hat the situation i s even wors e for a Lie algebr a, a s U k ( G ) is inﬁn ite dimensional iﬀ G is not zer o. 3. Op er at ions on r epresen tations No w, w e w o uld lik e to see the represen tations of AAU as building blo c ks to construct new ones. The elemen tary op e ratio ns on v ector spaces are: - sums - tensor pro ducts - duals Hence, an imp ortant pro blem is: Giv en represen t a tions ρ i : A 7→ V i on the building blo cks V i ; i = 1 , 2 ho w does one naturally construct represen tations on V 1 ⊕ V 2 , V 1 ⊗ V 2 and V ∗ i . Sums will cause no problem as the sum V 1 ⊕ V 2 of t w o v ector spaces V 1 and V 2 amoun ts to taking their cartesian pro duct V 1 ⊕ V 2 ∼ = V 1 × V 2 . Then, if ρ i : A 7→ V i ; i = 1 , 2 are t w o represen tations o f A then the mapping ρ 1 ⊕ ρ 2 : A 7→ V 1 ⊗ V 2 suc h that ρ 1 ⊕ ρ 2 ( a )[( v 1 , v 2 )] = ( ρ 1 ( a )[ v 1 ] , ρ 2 ( a )[ v 2 ]) (8) whic h can b e sym b olically written ρ 1 ⊕ ρ 2 = ρ 1 0 0 ρ 2 ! , (9) is a represen tation of A in V 1 ⊕ V 2 . Dualization will b e discussed la ter a nd solv ed b y the existence of an antip o de . Now , w e start with the problem o f constructing represen tations o n tensor pro ducts . This will b e solv ed by means of the not io n of “ sche me of actions” whic h is to b e formalized, in our case, b y the concept of comultiplic ation (or copro duct). 5 3.1. Arr ows and addition or multiplic ation formulas Let us giv e ﬁrst some examples where com ultiplication naturally arises. W e b egin with functions a dmitting a n “a ddition fo rm ula” or “multiplication f o rm ula”. This means functions suc h that fo r a ll x , y f ( x ∗ y ) = n X i =1 f (1) i ( x ) f (2) i ( y ) , (10) where ∗ is a certain (associative) op eration on the deﬁning set of f and ( f (1) i , f (2) i ) n i =1 b e t w o (ﬁnite) families of functions on the s ame set (see exercises (11.1) and (11.2) on represen tativ e functions). The ﬁrst examples are ta ken in the function space † R R (with ∗ = +, the ordinary addition of real num b ers ). The fo llo wing functions admit “addition f orm ulas” whic h can b e expressed diagramatically as f ollo ws. Diagram Addition formula R × R + / / cos 1 cos 2 − sin 1 sin 2 # # F F F F F F F F F R cos   R cos( x + y ) = cos( x ) cos( y ) − sin( x ) sin( y ) R × R + / / sin 1 cos 2 + sin 2 cos 1 # # F F F F F F F F F R sin   R sin( x + y ) = sin( x ) cos( y ) + sin( y ) cos( x ) R × R + / / exp 1 exp 2 # # F F F F F F F F F R exp   R exp( x + y ) = exp( y ) exp( x ) R × R + / / P n j =0 ( n j ) pr ( j ) 1 pr ( n − j ) 2 # # F F F F F F F F F R ( ) n   R ( x + y ) n = P n j =0  n j  x j y n − j Another example can b e give n where the domain set (source) is C n × n , the algebra of square n × n matrices with complex co eﬃcien ts. Let a ij : C n × n − → C b e the linear form whic h “tak es” the co eﬃcien t of address ( i, j ) (row i a nd column j ), that is to sa y a ij ( M ) := M [ i, j ]. Then, the la w of multiplication of mat rices sa ys that M N [ i, j ] = P n k =1 M [ i, k ] N [ k , j ], whic h can b e represen ted in the st yle o f Eq. (10) by a ij ( M N ) = n X k =1 a ik ( M ) a k j ( N ) . (11) † As usua l, Y X is the set of a ll mappings from X to Y , see appendix . 6 C n × n × C n × n pr oduct / / P n k =1 ( a ik ) 1 ( a kj ) 2 ) ) S S S S S S S S S S S S S S S S S C n × n a ij   C Remark 3.1 Note that form ula (11) holds when the deﬁnition set (sour c e) is a (multiplic ative) semigr oup of matric es (for exam p le, the s e migr oup of unip otent p ositive matric es). W e no w pro ceed to linear mappings that admit suc h “a dditio n” or, rather, “m ultiplication” f o rm ulas. Derivations : Let A b e an arbitrary algebra with law of multiplication: A ⊗ A µ − → A . (12) A deriv ation o f A is an op erator D : A − → A whic h follows the Leibniz rule, that is for all x, y ∈ A , one has D ( xy ) = D ( x ) y + xD ( y ) (Leibniz rule) . (13) In the spirit of what has b een represen ted a b o ve one has A ⊗ A µ / / D 1 I d 2 + I d 1 D 2 # # G G G G G G G G G A D   A whic h (as w e ha v e linear spaces a nd mappings) can b e b etter r epresen ted by A ⊗ A µ / / D ⊗ I d + I d ⊗ D   A D   A ⊗ A µ / / A A utomorphisms : An automorphism of A is an inv ertible linear mapping g : A − → A suc h that fo r a ll x, y ∈ A , o ne has g ( xy ) = g ( x ) g ( y ) , (14) whic h, in t he spirit of what precedes can b e represen ted by A ⊗ A µ / / g ⊗ g   A g   A ⊗ A µ / / A No w remark that, classically , group represen tations act as automorphisms and represen tations of Lie alg ebras act as deriv ations. This immediately prov ides a sc heme for constructing tensor pro ducts of tw o represen tations. 7 T ensor pr o duct of two r epr esentations (gr oups and Lie a lgebr as ) : First, take t w o represen tations of a group G , ρ i : G − → E nd ( V i ), i = 1 , 2. The action of g ∈ G on the tensor space V 1 ⊗ V 2 is giv en b y g ( v 1 ⊗ v 2 ) = g ( v 1 ) ⊗ g ( v 2 ) . (15) This means that the “tensor pro duct” o f t he t w o (group) represen tations ρ i , i = 1 , 2 is giv en by the fo llo wing data • Space : V 1 ⊗ V 2 • Action : ρ 1 × ρ 2 : g → ρ 1 ( g ) ⊗ ρ 2 ( g ) Lik ewise, if w e hav e t w o represen tations ρ i : G − → E nd ( V i ), i = 1 , 2 of the Lie algebra G the action of g ∈ G on a tensor pro duct V 1 ⊗ V 2 is giv en b y g ( v 1 ⊗ v 2 ) = g ( v 1 ) ⊗ v 2 + v 1 ⊗ g ( v 2 ) . (16) Again, the “tensor pro duct” of the tw o (Lie alg ebra) represen tations ρ i , i = 1 , 2 is g iven b y the following dat a • Space : V 1 ⊗ V 2 • Action : ρ 1 × ρ 2 : g → ρ 1 ( g ) ⊗ I d V 2 ( g ) + I d V 1 ( g ) ⊗ ρ 2 ( g ) Roughly sp eaking, in the ﬁrst case g acts by g ⊗ g and in the second o ne b y g ⊗ 1 + 1 ⊗ g . In view of tw o a b o ve cases it is con v enien t to construct linear mappings: A ∆ − → A ⊗ A , (17) suc h that, in eac h case, ρ 1 × ρ 2 = ( ρ 1 ⊗ ρ 2 ) ◦ ∆. In the ﬁrst case ( A = C [ G ]) one gets ∆ X g ∈ G α g g ! = X g ∈ G α g g ⊗ g . (18) In the second case, one has ﬁrst to construct the com ultiplication on the monomials g 1 ...g n ; g i ∈ G ) as they span ( A = U k ( G )). Then, using the rule ∆( g ) = g ⊗ 1 + 1 ⊗ g (for g ∈ G ) a nd the fact that ∆ is supp osed to be a morphism for the multiplication (the justiﬁcation of this rests on the fact that the constructed action m ust b e a represen tation see b elow around f o rm ula (36) and exercise (11.5) ), one ha s ∆( g 1 ...g n ) = ( g 1 ⊗ 1 + 1 ⊗ g 1 )( g 2 ⊗ 1 + 1 ⊗ g 2 ) ... ( g n ⊗ 1 + 1 ⊗ g n ) = X I + J =[1 ...n ] g [ I ] ⊗ g [ J ] . (19) Where, for I = { i 1 , i 2 , · · · , i k } (1 ≤ i 1 < i 2 < · · · < i k ≤ n ), g [ I ] stands for g i 1 g i 2 · · · g i k In each case (group algebra and en v elopping algebra) one again g ets a mapping ∆ : A 7→ A ⊗ A whic h will b e expresse d by ∆( a ) = n X i =1 a (1) i ⊗ a (2) i (20) 8 whic h is rephrased compactly by ∆( a ) = X (1)(2) a (1) ⊗ a (2) . (21) The action of a ∈ A on a tensor v 1 ⊗ v 2 is then, in b o th cases, giv en b y a. ( v 1 ⊗ v 2 ) = n X i =1 a (1) i .v 1 ⊗ a (2) i .v 2 = X (1)(2) a (1) .v 1 ⊗ a (2) .v 2 . (22) One can easily c hec k, in these t w o cases, that a.b. ( v 1 ⊗ v 2 ) = ( ab ) . ( v 1 ⊗ v 2 ) , (23) but in general (22) do es not g uaran tee (23) ; this p o int will b e discussed b elow in section (4) . Expression (21) is very con v enien t for pro ofs and computations and kno wn as Swee dler’s notation. Remarks 3.2 i) In every c ase, we have extr ac te d the “scheme o f action ” for building the tensor pr o duct of two r epr es e ntations. This sc h eme (a line ar map ping ∆ : A 7→ A ⊗ A ) is indep endent of the c onsider e d r epr esentations a n d, i n e ach c ase, ρ 1 × ρ 2 = ( ρ 1 ⊗ ρ 2 ) ◦ ∆ (24) ii) Swe e d ler’s no tation b e c omes tr ansp ar ent w hen one sp e aks the language of “s tructur e c onstants”. L et ∆ : C 7→ C ⊗ C b e a c omultiplic a tion and ( b i ) i ∈ I a (line ar) b asis of C . One has ∆( b i ) = X j,k ∈ I λ j,k i b j ⊗ b k . (25) the family ( λ j,k i ) i,j,k ∈ I is c al le d the “structur e c onstants” of the c omultiplic ation ∆ . Note the duality with the n otion the “structur e c onstants” o f a multiplic ation µ : A ⊗ A 7→ A : if ( b i ) i ∈ I is a (line ar) b asis of A , o n e has µ ( b i ⊗ b j ) = X k ∈ I λ k i,j b k . (26) F or ne c essary an d suﬃcient c onditions for a family to b e structur e c onstants (se e exer c i se (11.12) ). Then, the general construction for tensor pro ducts go e s a s follows. Deﬁnition 3.3 : L et A b e a ve ctor sp ac e , a c omultiplic ation ∆ on A is a line ar mappin g A ∆ − → A ⊗ A . Such a p air (ve ctor sp ac e, c omultiplic ation) without a n y pr escription ab out the line ar mapping “c omultiplic ation ” is c al le d a c o algebr a. 9 No w, imitating (24) , if A is an alg ebra and ρ 1 , ρ 2 are r epresen tations of A in V 1 , V 2 , f o r eac h a ∈ A , w e can construct an action o f a on V 1 ⊗ V 2 b y V 1 ⊗ V 2 ( ρ 1 ⊗ ρ 2 ) ◦ ∆( a ) − → V 1 ⊗ V 2 . (27) This means that if ∆( a ) = P (1)(2) a (1) ⊗ a (2) , t hen, ( a ) acts on the tensor pro duc t b y a. ( v 1 ⊗ v 2 ) = X (1)(2) a (1) .v 1 ⊗ a (2) .v 2 = X (1)(2) ρ 1 ( a (1) )[ v 1 ] ⊗ ρ 2 ( a (2) )[ v 2 ] . (28) But, at this stage, it is just an action and not (necessarily) a r epresen tation of A . W e shall later give the requiremen ts on ∆ for the construction of the tensor pro duct to b e reasonable (i. e. compatible with t he usual tensor prop erties). F or the momen t let us pause and consider some well kn own ex amples of com ultiplication. 3.2. Com binatorics of some c omultiplic ations The ﬁrst type of comultiplication is giv en b y dualit y . This means by a form ula o f t yp e h ∆( x ) | y ⊗ z i ⊗ 2 = h x | y ∗ z i (29) for a certain law of algebra V ⊗ V ∗ 7→ V , where h | i is a non degenerate scalar pro duct in V and h | i ⊗ 2 stands for its extension to V ⊗ V . In the case of w ords ∗ is the concatenation and h | i is given by h u | v i = δ u,v . The com ultiplication ∆ C auchy , dual to the concatenation, is given on a w ord w by ∆( w ) = X uv = w u ⊗ v . (30) In the same spirit, one can deﬁne a comu ltiplication on the a lgebra of a ﬁnite group by ∆( g ) = X g 1 g 2 = g g 1 ⊗ g 2 . (31) The second example is giv en b y the m ultiplication la w of elemen tary com ultiplications, that is, if fo r each letter x one has ∆( x ) = x ⊗ 1 + 1 ⊗ x , then ∆( w ) = ∆( a 1 ...a n ) = ∆( a 1 )∆( a 2 ) ... ∆( a n ) = X I + J =[1 ...n ] w [ I ] ⊗ w [ J ] (32) where w [ { i 1 , i 2 , · · · i k } ] = a i 1 a i 2 · · · a i k (for 1 ≤ i 1 < i 2 < · · · < i k ≤ n ). This com ultiplication is dual to (34) b elo w for q = 0 ( shuﬄe pro duct). Another example is a deformation (p erturbation for small q ) of the preceding. With ∆( a ) = a ⊗ 1 + 1 ⊗ a + q a ⊗ a , o ne has ∆( w ) = ∆( a 1 ...a n ) = ∆( a 1 )∆( a 2 ) ... ∆( a n ) = X I ∪ J =[1 ...n ] q | I ∩ J | w [ I ] ⊗ w [ J ] . (33) Note that this com ultiplication is dual (in the sense of (29) ) to the q -inﬁltratio n pro duct giv en by the recursiv e formula (f o r general q a nd with 1 A ∗ as the empt y w ord) 10 w ↑ 1 A ∗ = 1 A ∗ ↑ w = w au ↑ bv = a ( u ↑ bv ) + b ( au ↑ v ) + q δ a,b ( u ↑ v ) . (34) This product is an in terp olation betw een the sh uﬄe ( q = 0) and the (classical) inﬁltration ( q = 1) [22]. 4. Requirements for a reasona ble construction of tensor pro ducts W e hav e so far constructed an action of A on tensors, but nothing indicates that this is a represen tation (see exercise (11.5) ). So, the following question is natural. Q.1.) If A is an alg ebra a nd ∆ : A 7→ A ⊗ A , what do w e require on ∆ if w e w ant the construction ab o v e to b e a represen tation of A on t ensor pro duc ts ? F or a, b ∈ A , ρ i represen tations of A in V i , and v i ∈ V i for i = 1 , 2, w e mus t ha v e the follo wing identit y: a. ( b.v 1 ⊗ v 2 ) = ( ab ) .v 1 ⊗ v 2 ⇐ ⇒ ∆( ab ) .v 1 ⊗ v 2 = ∆( a ) . (∆( b ) .v 1 ⊗ v 2 ) . (35 ) One can pro v e that, if this is true iden t ically for all a, b ∈ A and all pairs of represen tations (see exercise (11.5) ), one has ∆( ab ) = ∆( a )∆( b ) . (36) and, of course, if the latter holds, (35) is true. This can b e rephrased b y sa ying that ∆ is a morphism A 7→ A ⊗ A . No w, one w ould lik e to k eep compatibilit y with the asso ciativit y of tensor pro ducts . This means that if w e wan t to tensor u ⊗ v with w it m ust giv e the same action as tensoring u with v ⊗ w . This means that w e hav e to address the followin g question. Q.2.) If A is an algebra and ∆ : A 7→ A ⊗ A is a morphism of a lg ebras, what do w e require on ∆ if w e w ant the construction ab ov e to b e a sso ciativ e ? More precisely , for three represen tations ρ i , i = 1 , 2 , 3 of A , w e w ant ρ 1 × ( ρ 2 × ρ 3 ) = ( ρ 1 × ρ 2 ) × ρ 3 (37) up to the iden tiﬁcations ( u ⊗ v ) ⊗ w = u ⊗ ( v ⊗ w ) = u ⊗ v ⊗ w (if one is not satisﬁed with this iden tiﬁcation, see exercise (11.6) ). Let us compute (up to the iden tiﬁcation ab o v e) a. [( u ⊗ v ) ⊗ w ] = ∆( a ) . ( u ⊗ v ) ⊗ w =  (∆ ⊗ I d ) ◦ ∆( a )  u ⊗ v ⊗ w  (38) on the other hand a. [ u ⊗ ( v ⊗ w )] = ∆( a ) .u ⊗ ( v ⊗ w ) =  ( I d ⊗ ∆) ◦ ∆( a )  u ⊗ v ⊗ w  . (39) 11 Again, one can prov e (see exercise (11.6) ) that this holds iden tically (i. e. f or ev ery a ∈ A and triple of represen tations) iﬀ ( I d ⊗ ∆) ◦ ∆ = ( ∆ ⊗ I d ) ◦ ∆, i . e . A ∆ / / ∆   A ⊗ A I d ⊗ ∆   A ⊗ A ∆ ⊗ I d / / A ⊗ A ⊗ A (40) Remark 4.1 The pr op erty (40) is c al le d c o-asso ciativity sinc e if o ne r everses the arr ows and r eplac es ∆ by µ , the multiplic ation in an algebr a, the diag r am expr esses asso ciativity (se e also exer cise (11.7) on duals of c o-algebr as). A ⊗ A ⊗ A µ ⊗ I d / / I d ⊗ µ   A ⊗ A µ   A ⊗ A µ / / A But the t ensor pro duct is not o nly associative , it has a “neutral” map whic h is “tensoring b y the ﬁeld of scalars”. This deriv es f r o m the fact that the cano nical mappings V ⊗ k k can r − → V can l ← − k ⊗ k V . (41) This can b e summarized b y the follo wing question. Q.3.) If A is an algebra and ∆ : A 7→ A ⊗ A a co-asso ciative morphism o f algebras, what do we require on ∆ if we w ant the construction ab ov e to admit “tensoring by the ﬁeld of scalars” as neutral ? More precisely , w e m ust hav e a represen tation of A in C (whic h means a morphism of algebras A ǫ − → C ) suc h that for a represen tation ρ of A , w e w ant ρ × ǫ = ǫ × ρ = ρ ( 4 2) up to the iden tiﬁcation u ⊗ 1 = 1 ⊗ u = u through the isomorphisms (4 1) (if o ne is not satisﬁed with this identiﬁc ation, see exercise (11.8) ). Hence, for all a ∈ A and ρ represen tation on V , w e should ha v e: can r ( a. ( v ⊗ 1)) = a.v , (43) and a. ( v ⊗ 1) =  X (1)(2) ρ ( a (1) ) ⊗ ǫ ( a (2) )  [ v ⊗ 1] = ( ρ ⊗ I d C ) ◦  X (1)(2) a (1) ⊗ ǫ ( a (2) )  [ v ⊗ 1] (44) = ( ρ ⊗ I d C ) ◦ ( I d ⊗ ǫ ) ◦ ∆( a )[ v ⊗ 1 ] , a.v = ρ ( a )[ v ] = can r ( ρ ( a )[ v ] ⊗ 1) = can r ( ρ ⊗ I d ( a )[ v ⊗ 1]) . (45) 12 Similar computations could b e made on the left, w e lea v e them to the reader as a n exercise. This means that one should require tha t A ∆ / / I d   A ⊗ A I d ⊗ ǫ   A A ⊗ C can r o o A ∆ / / I d   A ⊗ A ǫ ⊗ I d   A C ⊗ A can l o o Suc h a mapping ǫ : A − → C is called a c o-unit . Remark 4.2 A gain, one c an pr ove (se e exc er c i c e (11.7 ) for details) that ǫ is a c ounit for ( A , ∆) ⇐ ⇒ ǫ is a c ounit for ( A ∗ , ∗ ∆ ) . (46) 5. Bialgebras Motiv at ed b y t he preceding discussion, w e deﬁne a bialgebr a an algebra (a sso ciativ e with unit) endo w ed with a com ultiplication (co- a sso ciativ e with counit) whic h allow s for the t w o tensor prop erties o f asso ciativit y and unit (see discussion ab ov e). More precisely Deﬁnition 5.1 : ( A , · , 1 A , ∆ , ǫ ) is said to b e a b ialgebr a iﬀ (1) ( A , · , 1 A ) is an AAU, (2) ( A , ∆ , ǫ ) is a c o alge br a c o asso c i a tive wi th c ounit, (3) ∆ is a morphism o f AA U and ǫ is a morphism of AAU. The name bia lgebra comes from the fact that the sp ace A is endo w ed with t w o structures (one of AA U a nd one of co-AAU) with a certain compatibilit y b etw een the tw o. 5.1. Exam ples of bialgebr as F r e e algebr a (wor d version : no n c ommutative p olynomials) Let A b e a n alphab et (a set of v ariables) and A ∗ b e t he free monoid constructed o n A (see Basic Structures (12.2) ). F or any ﬁeld of scalars k (one may think of k = R or C ), w e call the algebra of noncommutativ e p olynomials k h A i (or free algebra), the algebra k [ A ∗ ] of the free monoid A ∗ constructed on A . This is the set of f unctions f : A ∗ 7→ k with ﬁnite supp ort endo w ed with the conv o lution pro duct f ∗ g ( w ) = X uv = w f ( u ) g ( v ) (47) Eac h w ord w ∈ A ∗ is iden tiﬁed with its characteristic f unction (i.e. the D irac function with v alue 1 at w and 0 elsewhere). These functions form a basis of k h A i a nd then, ev ery f ∈ k h A i can b e written uniquely as a ﬁnite sum f = P f ( w ) w . The inclusion mapping A ֒ → k h A i will b e denoted here by can A . Com ultiplications The free alg ebra k h A i admits man y comultiplications (ev en with the t w o requiremen ts of b eing a morphism and coasso ciativ e). As A ∗ is a basis of k h A i , 13 it is suﬃcien t to deﬁne it on the words (if w e require ∆ to b e a morphism it is enough to deﬁne it o n letters). Example 1 . — The ﬁrst example is the dual of the Cauc hy (or con v olution) pro duct ∆( w ) = X uv = w u ⊗ v (48) is not a morphism as ∆( ab ) = ab ⊗ 1 + a ⊗ b + 1 ⊗ ab and ∆( a )∆( b ) = ab ⊗ 1 + a ⊗ b + b ⊗ a + 1 ⊗ ab but it can b e c hec k ed that it is coasso ciativ e (se e also exercice (11 .7) for a quic k pro of of this fact). Example 2 . — A second example is give n, on the alphab et A = { a, b } b y ∆( a ) = a ⊗ b ; ∆( b ) = b ⊗ a then ∆( w ) = w ⊗ ¯ w where ¯ w stands for the w ord w with a (resp. b ) c hanged in b (resp. a ). This com ultiplication is a morphism but not coasso ciativ e a s ( I ⊗ ∆) ◦ ∆( a ) = a ⊗ b ⊗ a ; (∆ ⊗ I ) ◦ ∆( a ) = a ⊗ a ⊗ b Example 3 . — The thir d example is given o n the letters by ∆( a ) = a ⊗ 1 + 1 ⊗ a + q a ⊗ a where q ∈ k . One can prov e that ∆( w ) = ∆( a 1 ...a n ) = ∆( a 1 )∆( a 2 ) ... ∆( a n ) = X I ∪ J =[1 .. | w | ] q | I ∩ J | w [ I ] ⊗ w [ J ] (49) this com ultiplication is coasso ciativ e. F or q = 0 , one gets a com ultiplication giv en on t he letters b y ∆ s ( a ) = a ⊗ 1 + 1 ⊗ a . F or ev ery p olynomial P ∈ k h A i , set ǫ ( P ) = P (1 A ∗ ) (the constant term). T hen ( k h A i , ∗ , ∆ s , ǫ ) is a bialgebra. One has also a nother bia lg ebra structure with, for all a ∈ A ∆ h ( a ) = a ⊗ a ; ǫ aug ( a ) = 1 (50) this bialgebra ( k h A i , ∗ , ∆ h , ǫ aug ) is a substructure of the bialgebra of the free gr o up. A lgebr a of p olynomials (c ommutative p olynomials) W e con tin ue with the same alphab et A , but this time, w e tak e as alg ebra k [ A ]. The construction is similar but the mono mials, instead of w ords, ar e all the comm utativ e pro ducts o f letters i.e. a α 1 1 a α 2 2 · · · a α n n with n arbitrary and α i ∈ N . Denoting MON ( A ) the monoid of these monomials (comprising, a s neutral, t he empty one) and with ∆ s ( a ) = a ⊗ 1 + 1 ⊗ a , ǫ ( P ) = P (1 M O N ( A ) ), one can ag ain chec k that ( k [ A ] , ∗ , ∆ s , ǫ ) is a bialgebra. 14 A lgebr a of p artial ly c ommutative p olynomials F or t he detailed construction of a partially comm utative mono id, the r eader is referred to [18, 2 3 ]. These monoids generalize b ot h the free and free comm utativ e mono ids. T o a giv en graph (non-oriented and without lo ops) ϑ ⊂ A × A , one can assso ciate the mono id presen ted by generators and relations (see basic structures (1 2.2) and diagram (137) ) M ( A, ϑ ) = h A ; ( xy = y x ) ( x,y ) ∈ ϑ i Mon . (51) This is exactly t he monoid obtained as a quotien t structure of t he free mono id ( A ∗ ) b y the smallest equiv alence compatible with pro ducts (a congruence ‡ ) whic h con tains the pairs ( xy , y x ) ( x,y ) ∈ ϑ . A geometric mo del of this monoid using p i e c es was dev elopp ed by X. Viennot [48] where pieces are lo cated on “p ositions” (dra wn on a pla ne) t w o pieces “comm ute” iﬀ t hey are on p ositions whic h do not inters ect (see ﬁg 1 b elo w). a b c Fig 1 . — R epr esentation o f a p artial ly c ommutative mon omial by “h e aps of pie c es” [48]. Her e “ a ” c ommutes with “ c ” (they ar e on non-c oncurr ent p ositions ) a n d “ b ” c ommutes with no other pie c e. The monom ial r epr esente d c an b e written by sever al wor ds as acbc 3 ba 2 c = cabc 3 baca = cabc 3 bca 2 . The partially comm utativ e algebra k h A, ϑ i is the alg ebra k [ M ( A, ϑ )] [23]. Again, one can c hec k that ( k h A, ϑ i , ∗ , ∆ s , ǫ ) (constructed as ab o v e) is a bialgebra. A lgebr a of a gr oup Let G b e a gro up. The a lgebra under consideration is k [ G ]. W e deﬁne, for g ∈ G , ∆( g ) = g ⊗ g and ǫ ( g ) = 1, then, one can chec k t ha t ( k [ G ] , ., ∆ , ǫ ) is a bialgebra. 6. Antipo de and t he problem of duals Eac h vec tor space V comes with its dualΣ V ∗ = H om C ( V , C ) . (52) ‡ See exercise (11.4) and the presentation of monoids “by genera tors a nd relations in paragraph ”(12.2). Σ In general, for t wo k -vector spaces V and W , H om k ( V , W ) is the set of all linear ma ppings V 7→ W . 15 The spaces V ∗ and V are in duality by h p, ψ i = p ( ψ ) (53) No w, if one has a represen tation (on the left) of A on V , one gets a represen tation on the righ t on V ∗ b y h p.a, ψ i = h p, a.ψ i (54) If we w ant to ha v e the action of A on the left again, one should use an an ti-mo r phism α : A − → A tha t is α ∈ E nd k ( A ) suc h tha t, for all x, y ∈ A α ( xy ) = α ( y ) α ( x ) . (55) In the case of gro ups, g − → g − 1 do es the job; in the case of Lie alg ebras g − → − g (extended by rev erse pro ducts to the en v eloping algebra) w orks. On the o ther hand, in the classical textb o oks , the discussion of “complete reductibilit y” go es with the existence of an “ in v a rian t” scalar pro duct φ : V × V 7→ k on a space V . F or a g roup G , this reads ( ∀ g ∈ G )( ∀ x, y ∈ V )  φ ( g .x, g .y ) = φ ( x, y )  (56) (think of unitary represen tations). F or a L ie algebra G , this reads ( ∀ g ∈ G )( ∀ x, y ∈ V )  φ ( g .x, y ) + φ ( x, g .y ) = 0  (57) (think of the Killing form). No w, linearizing the s ituation b y Φ ( x ⊗ y ) = φ ( x, y ) a nd re mem b ering our unit represen tation, one can rephrase (5 6) and (57) in ( ∀ a ∈ A )( ∀ x, y ∈ V )  Φ( X (1)(2) a (1) .x ⊗ a (2) .y ) = ǫ ( a )Φ( x ⊗ y )  . (58) Lik ewise, we will say that a bilinear form Φ : U ⊗ V 7→ k is inv arian t is it satisﬁes ( ∀ a ∈ A )( ∀ x ∈ U, y ∈ V )  Φ( X (1)(2) a (1) .x ⊗ a (2) .y ) = ǫ ( a )Φ( x ⊗ y )  . (59) whic h means that Φ is A linear fo r t he structures b eing giv en resp ectiv ely by ρ 1 × ρ 2 on U ⊗ k V and ǫ on k . No w, supp ose that w e hav e constructed a represen tation of a certain algebra A o n U ∗ b y means of an antimorphis m α : A 7→ A . T o require that the natural contraction c : U ∗ ⊗ U 7→ k b e “in v arian t” means that ( ∀ a ∈ A )( ∀ f ∈ U ∗ , y ∈ U )  c ( X (1)(2) a (1) .f ⊗ a (2) .y ) = ǫ ( a ) c ( f ⊗ y ) = ǫ ( a ) f ( y )  ; ( ∀ a ∈ A )( ∀ f ∈ U ∗ , y ∈ U )  X (1)(2) f ( α ( a (1) ) a (2) .y ) = ǫ ( a ) f ( y )  . (60) 16 It is easy to c hec k (taking a basis ( e i ) i ∈ I and its dual fa mily ( e ∗ i ) i ∈ I for example) that (60) is equiv alent to ( ∀ a ∈ A )  ρ U ( X (1)(2) α ( a (1) ) a (2) ) = ǫ ( a ) I d U  . (61) Lik ewise, taking the natural contraction c : U ⊗ U ∗ 7→ k , one gets ( ∀ a ∈ A )  ρ U ( X (1)(2) a (1) α ( a (2) )) = ǫ ( a ) I d U  . (62) T aking U = A a nd for ρ U the left regular represen tation, one gets ( ∀ a ∈ A )  X (1)(2) a (1) α ( a (2) ) = X (1)(2) α ( a (1) ) a (2) ) = ǫ ( a )  . (63) Motiv at ed b y the preceding discussion, one can mak e the follow ing deﬁnition. Deﬁnition 6.1 L et ( A , · , 1 A , ∆ , ǫ ) b e a bialge b r a. A line ar m apping α : A 7→ A is c al le d an antip o de for A , if for al l a ∈ A X (1)(2) α ( a (1) ) a (2) = X (1)(2) a (1) α ( a (2) ) = 1 A ǫ ( a ) . (64) One can pro v e (see ex ercise (11.9) ), that this means that α is the inv ers e of I d A for a certain pro duct of an alg ebra (AA U) on E nd k ( A ) and this implies (see exercise (11.9) ) that (i) If α exists (as a solution of Eq. (64) ), it is unique. (ii) If α exists, it is a n a n timorphism. Deﬁnition 6.2 : (Hopf Algebr a) ( A , · , 1 A , ∆ , ǫ, S ) is said to b e a Hopf algebr a iﬀ (1) B = ( A , · , 1 A , ∆ , ǫ ) is a bialgebr a, (2) S is an antip o de (then unique) for B In many com binatorial cases (see exercise on lo cal ﬁniteness (11.1 0) ), one can compute the an tip o de by α ( d ) = ∞ X k =0 ( − 1) k +1 ( I + ) ( ∗ k ) ( d )) . (65) where I + is the pro jection on B + = k er ( ǫ ) suc h that I + (1 B ) = 0. 7. H opf algeb ras a nd partition functions 7.1. Partition F unction Inte gr and Consider the Partition F unction Z of a Quantum Stat istical Mec hanical System Z = T r exp ( − β H ) (66) 17 whose hamiltonian is H ( β ≡ 1 /k T , k = Boltzmann’s constant T = absolute temp erature). W e may ev a lua t e the t r ace o v er an y complete set of states; w e choose the (o v er-)complete set of coherent stat es | z i = e −| z | 2 / 2 X n ( z n / √ n !) a + n | 0 i (67) where a + is the b oson creation op e rato r satisfying [ a, a + ] = 1 and for whic h the completeness or resolution of unit y k prop erty is 1 π Z dz | z ih z | = I ≡ Z d ( z ) | z ih z | . (68) k Sometimes physicists write d 2 z to emphasize that the in tegra l is tw o dimensional (o ver R ) but here, the l. h. s. of (68) is the integration of the o p e r ator v alued function z → | z ih z | - see [10] Cha p. I I I Paragraph 3 - w.r .t. the Haar mesure of C which is dz . 18 The simplest, and generic, example is the f r ee single b oson hamiltonian H = ǫa + a for whic h the a ppro priate trace calculation is Z = 1 π Z dz h z | exp ( − β a + a ) | z i = = 1 π Z dz h z | : exp ( a + a ( e − β ǫ − 1) : | z i . (69) Here we hav e used the following w ell know n r elat io n for the forgetful normal ordering op erator : f ( a, a + ) : whic h means “normally order the creation and annihilation op erators in f forgetting the comm utation relation [ a, a + ] = 1” ¶ . W e ma y write the P artition F unction in general as Z ( x ) = Z F ( x, z ) d ( z ) (70) thereb y deﬁning the P artition F unction In tegrand (PFI) F ( x, z ). W e ha v e explicitly written the dep endence on x = − β , the in v erse temp erature, and ǫ , t he energy scale in the hamiltonian. 7.2. Com binatorial asp e cts: Bel l numb ers The generic free b oson example Eq. (7 0 ) ab o ve may b e rewritten to show the connection with certain w ell kno wn com binatorial num bers. W riting y = | z | 2 and x = − β ǫ , Eq. (70) b ecomes Z = Z ∞ 0 dy exp ( y ( e x − 1)) . (71) This is an in tegral o v er the class ical exponential generating function for the Bell p olynomials exp ( y ( e x − 1)) = ∞ X n =0 B n ( y ) x n n ! (72) where the Bell n um b er is B n (1) = B ( n ), the num ber of wa ys of putting n diﬀeren t ob jects in to n identical con tainers (some ma y b e left em pty). Related to t he Bell n um b ers are the Stirling n um b ers of the second kind S ( n, k ) , whic h are deﬁned as the n um b er of w a ys of putting n diﬀeren t ob jects in to k identical con tainers, leav ing none empt y . F rom the deﬁnition w e ha v e B ( n ) = P n k =1 S ( n, k ) (for n 6 = 0) . The foregoing giv es a combinatorial interpretation of the partit io n function integrand F ( x, y ) as the exp o nen tial generating function o f the Bell p olynomials. 7.3. Gr aphs W e now giv e a graphical represen tation of the Bell num bers. Consider lab elled lines whic h emanate from a white dot, the origin, and ﬁnish on a blac k dot, the vertex . W e shall allow only one line from each white dot but imp ose no limit on the n um b er of ¶ Of cours e, this pr o cedure may alter the v alue of the op er ator to which it is applied. 19 lines ending on a blac k dot. Clearly this sim ulates the deﬁnition of S ( n, k ) and B ( n ), with the white dots pla ying the role of the distinguishable ob jects, whence the lines are lab elled, and the black dots that of t he indistinguishable containers. The iden tiﬁcation of the graphs f or 1,2 and 3 lines is giv en in Fig ure 2. W e hav e concen trated on the Bell n um b er s equence and its a sso ciat ed gr a phs since, as w e shall sho w, there is a sense in whic h this sequence of graphs is generic. That is, we can represen t a ny com binatorial sequence b y the same sequence of graphs as in the F ig ure 2 Fig 2 . — Gr aphs for B(n), n = 1, 2, 3 . with suitable ve rtex m ultipliers (denoted by the V terms in the same ﬁgure). Consider a general partition function Z = T r exp ( − β H ) (73) where the Hamiltonian is given by H = ǫw ( a, a + ), with w a string (= sum of pro ducts of p ositiv e p ow ers) of b oson creation and annihilation op erators. The partition function in tegrand F for whic h we seek to giv e a gra phical expansion, is Z ( x ) = Z F ( x, z ) d ( z ) (74) where F ( x, z ) = h z | exp ( xw ) | z i = ( x = − β ǫ ) = ∞ X n =0 h z | w n | z i x n n ! = ∞ X n =0 W n ( z ) x n n ! = exp  ∞ X n =1 V n ( z ) x n n !  (75) with ob vious deﬁnitions o f W n and V n . The seq uences { W n } and { V n } ma y eac h b e recursiv ely obta ined fro m the other. This relates the se quence of m ultipliers { V n } of Figure 2 to the Hamiltonian of Eq. (73) . The lo w er limit 1 in the V n summation is a consequenc e of the normalization of the coheren t state | z i . 20 7.4. The Hopf A lgebr a L Bell W e brieﬂy describ e the Ho pf a lgebra L Bell whic h the diag rams of Figure 2 deﬁne. 1. Eac h distinct diagram is an individual basis elemen t of L Bell ; thus the dimension is inﬁnite. (Visualise eac h diagra m in a “b ox”.) The sum of t w o diagr a ms is simply the t w o b oxes con taining the diagrams. Scala r m ultiples are f ormal; for example, they ma y b e prov ided b y the V co eﬃcien ts. Precisely , as a v ector space, L Bell is the space freely generated b y the diagrams of F igure 2 (see APPENDIX : F unction Spaces). 2. The iden tity elemen t e is the empt y diagram (an empt y b ox ). 3. Multiplication is the juxtap osition of t w o diagrams within the same “b ox ”. L Bell is generated b y the connected diagrams; this is a consequence of the Connected Graph Theorem [34]. Since w e ha v e not here sp eciﬁed an order for the juxtap osition, m ultipli cation is commu tative. 4. The comu ltiplication ∆ : L Bell 7→ L Bell ⊗ L Bell is deﬁned by ∆( e ) = e ⊗ e (unit e , the empty b o x) ∆( x ) = x ⊗ e + e ⊗ x (generator x ) ∆( AB ) = ∆( A )∆( B ) otherwise so that ∆ is a n alg ebra homomorphism. (76) 8. The case of tw o mo des Let us consider a n hamilto nian on tw o mo des H ( a, a + , b, b + ), with [ a, a + ] = 1 [ b, b + ] = 1 [ a ǫ 1 , b ǫ 2 ] = 0 , ǫ i b eing + or empt y (4 relat io ns) (77) Supp ose that o ne can express H as H ( a, a + , b, b + ) = H 1 ( a, a + ) + H 2 ( b, b + ) . (78) and that, exp ( λH 1 ) and exp ( λH 2 ) ( λ = − β ) are solv ed i.e. that we hav e expressions F ( λ ) = exp ( λH 1 ) = ∞ X n =0 λ n n ! H ( n ) 1 ( a, a + ) ; G ( λ ) = exp ( λH 2 ) = ∞ X n =0 λ n n ! H ( n ) 2 ( b, b + ) (79) It is not diﬃcult to c hec k that exp ( λH ) = exp  λ ( H 1 + H 2 )  = F ( λ d dx ) G ( x )    x =0 (80) This leads us to deﬁne, in general, the “ Hadamard exp onential pro duct”. Let F ( z ) = X n ≥ 0 a n z n n ! , G ( z ) = X n ≥ 0 b n z n n ! (81) and deﬁne their pro duct (the “Hadamar d exp onen tial pro duct”) by 21 H ( F , G ) := X n ≥ 0 a n b n z n n ! = H ( F , G ) = F  z d dx  G ( x )     x =0 . (82) When F (0) and G (0 ) are not zero one can normalize the functions in this bilinear pro duct so that F (0) = G (0) = 1. W e w ould lik e to obta in compact and generic formu las. If w e write the functions as F ( z ) = exp ∞ X n =1 L n z n n ! ! and G ( z ) = exp ∞ X n =1 V n z n n ! ! . (83) that is, as free exp o nen tials, then by using Bell p o lynomials in the sets of v ariables L , V (see [21, 26] for details), w e o bta in H ( F , G ) = X n ≥ 0 z n n ! X P 1 ,P 2 ∈ U P n L T y pe ( P 1 ) V T y pe ( P 2 ) (84) where U P n is the set of unordered partitions of [1 · · · n ]. An unordered partitio n P o f a set X is a collection of (nonempt y) subsets of X , m utually disjoin t and co v ering X ( i.e. the union of all the subsets is X , see [28] for details). The t yp e of P ∈ U P n (denoted ab ov e b y T y pe ( P )) is the m ulti-index ( α i ) i ∈ N + suc h that α k is the n umber of k - blo c ks, that is the num b er of mem b ers of P with cardinality k . A t this p oin t the formu la entangles a nd the diagrams o f the theory arise. Note particularly that • the monomial L T y pe ( P 1 ) V T y pe ( P 2 ) needs m uc h less information than that whic h is con tained in the individual partitions P 1 , P 2 (for example, one can relab el the elemen ts without c hanging the monomial), • t w o partitions hav e an incidence mat r ix fr om whi ch it is stil l p ossible to r e c over the typ es of the p artitions. The construction now pro ceeds as follows . (i) T a k e tw o uno r dered partitions of [1 · · · n ], sa y P 1 , P 2 . (ii) W rite do wn their incidence matrix ( card( Y ∩ Z )) ( Y ,Z ) ∈ P 1 × P 2 . (iii) Construct the diagram represen ting the multiplicitie s of the incidence matrix : for eac h blo c k of P 1 dra w a black sp o t (resp. for each blo c k of P 2 dra w a white sp ot). (iv) Draw lines b et w een the blac k sp ot Y ∈ P 1 and the white s p o t Z ∈ P 2 ; there are card( Y ∩ Z ) suc h. (v) Remo v e the informatio n of the blo c ks Y , Z , · · · . In so doing, one obtains a bipartite graph with p (= card( P 1 )) black sp ots, q (= card( P 2 )) white sp ots, no isolated v ertex a nd in teger m ultiplicities. W e denote the set of suc h diagrams by diag . 22 ♠ ♠ ♠ ♠ ⑥ ⑥ ⑥ { 1 } { 2 , 3 , 4 } { 5 , 6 , 7 , 8 , 9 } { 10 , 11 } { 2 , 3 , 5 } { 1 , 4 , 6 , 7 , 8 } { 9 , 10 , 11 } Fig 3 . — Diagr am f r om P 1 , P 2 (set p art itions of [1 · · · 11] ). P 1 = {{ 2 , 3 , 5 } , { 1 , 4 , 6 , 7 , 8 } , { 9 , 10 , 1 1 }} and P 2 = {{ 1 } , { 2 , 3 , 4 } , { 5 , 6 , 7 , 8 , 9 } , { 10 , 11 }} (r esp e ctively black sp ots for P 1 and white sp ots for P 2 ). The inci denc e matrix c orr esp onding to the diagr am (as dr awn) or these p artitions is   0 2 1 0 1 1 3 0 0 0 1 2   . But, due to the f act that the deﬁning p artitions ar e u nor der e d, o ne c an p ermute the sp ots (black and white, b etwe en themselves) and, so, the lines and c olumns of this matrix c an b e p ermute d. Thus, the diagr am c ould b e r epr esente d b y the matrix   0 0 1 2 0 2 1 0 1 0 3 1   as wel l. The pro duct formula now reads H ( F , G ) = X n ≥ 0 z n n ! X d ∈ diag | d | = n mul t ( d ) L α ( d ) V β ( d ) (85) where α ( d ) ( r esp. β ( d )) is the “white sp ot ty p e” (resp. the “black sp ot type”) i.e. the m ulti-index ( α i ) i ∈ N + (resp. ( β i ) i ∈ N + ) suc h that α i (resp. β i ) is the num ber o f white sp ots (resp. black spo ts) of degree i ( i lines connected to the spo t) and mul t ( d ) is the n um b er of pairs of unordered partitions of [1 · · · | d | ] (here | d | = | α ( d ) | = | β ( d ) | is the num ber of lines of d ) with a sso ciated diagra m d . Note 8.1 The dia gr ams as wel l as the pr o duct formula wer e intr o duc e d in [ 3]. 8.1. Di a gr ams One can design a ( g raphically) natural m ultiplicativ e structure o n diag suc h that the arro w d 7→ L α ( d ) V β ( d ) . (86) 23 b e a mor phism. This is provided b y the concatenation of the diagrams (the result, i.e. the diagram obtained in pla cing d 2 at the r igh t of d 1 , will b e denoted by [ d 1 | d 2 ] D ). One m ust c hec k that this pro duct is compatible with the equiv alence of the p ermutation of white and black sp ots among t hemselv es, whic h is rather straigh tforward (see [21 , 28]). W e hav e Prop osition 8.2 [ 2 8] L et diag b e the set of diagr ams ( i n cluding the empty one). i) The law ( d 1 , d 2 ) 7→ [ d 1 | d 2 ] D endows diag w ith the structur e of a c ommutative monoid with the empty diagr am as neutr al element(this diagr am wil l, ther ef or e, b e de note d by 1 diag ). ii) The arr ow d 7→ L α ( d ) V β ( d ) is a morphism o f monoids, the c o dom ain of this arr ow b eing the mo noid of (c ommutative) mono m ials in the alphab et L ∪ V i.e. MON ( L ∪ V ) = { L α V β } α,β ∈ ( N + ) ( N ) = [ n,m ≥ 1 { L α 1 1 L α 2 2 · · · L α n n V β 1 1 V β 2 2 · · · V β m m } α i ,β j ∈ N . (87 ) iii) T he m onoid ( diag , [ −|− ] D , 1 diag ) is a fr e e c ommutative m onoid. The set on which it is built is the set of the c onne cte d (non-em pty) diag r ams. Remark 8.3 The r e ader w ho is not fa miliar with the algebr aic structur e of MON ( X ) c an ﬁnd rigor o us deﬁ n itions in p ar agr a p h (12.2 ) . W e denote φ mon,diag the arrow diag 7→ MON ( L ∪ V ). 8.2. L ab el le d diagr ams W e ha v e seen the diagrams ( o f diag ) are in one-to-o ne corresp ondence with classes of matrices a s in Fig. 3 . In order to ﬁx one represen tative of this class, w e hav e to n umber the black (resp. white) sp o t s from 1 to , say p (resp. q ). Doing so, one obtains a p acke d matrix [2 5] that is, a matrix of integers with no row nor column consisting en tirely of zero es. In this w a y , w e deﬁne the lab el le d diagr ams . Deﬁnition 8.4 A l a b el le d diagr am of size p × q is a bi-c olour e d (vertic es ar e p black and q w hite sp ots) gr aph • with no iso l a te d vertex • every black s p ot is joine d to a white sp ot by an arbitr ary quantity (but a p ositive inte ger) of lin es • the blac k (r esp. w hite) sp ots ar e numb er e d fr om 1 to p (r esp. fr om 1 to q ). As in para g raph (8.1) , one can concatenate the lab elled diagrams, the result, i.e. the diagram obtained in pla cing D 2 at t he right of D 1 , will b e denoted by [ D 1 | D 2 ] L . This time we need not c hec k compatibilit y with classes. W e hav e a structure of free mono id (but not comm utative this time) 24 Prop osition 8.5 [ 2 8] L et ldiag b e the set of lab ele d diagr ams (including the empty one). i) The la w ( d 1 , d 2 ) 7→ [ d 1 | d 2 ] L endows ldiag with the structur e of a nonc ommutative monoid with the empty diagr am ( p = q = 0 ) as neutr al element (wh ich wil l, ther efor e, b e den ote d by 1 ldiag ). ii) The arr ow fr om ldia g to dia g , which implies “for getting the lab els o f the vertic es” is a morphism of mon oids. iii) The monoid ( ldiag , [ −|− ] L , 1 ldiag ) is a fr e e (nonc ommutative) monoid which is c onstructe d on the set of irr e ducible diagr ams which ar e diagr am s d 6 = 1 ldiag which c annot b e written d = [ d 1 | d 2 ] L with d i 6 = 1 ldiag . Remark 8.6 i) In a gener al monoid ( M , ⋆, 1 M ) , the irr e ducible elements ar e the elements x 6 = 1 M such that x = y ⋆ z = ⇒ 1 M ∈ { y , z } . ii) It c an happ en that an irr e ducible of ldiag has a n image in diag which sp lits (i . e. is r e ducible), as shown by the simple exam p le o f the cross deﬁne d by the incid e nc e matrix  0 1 1 0  . 8.3. Hopf algebr as DIAG and LDIAG Let us ﬁrst construct the Hopf algebra on the lab elled diagrams (details can b e found in [28]). In o r der to deﬁne the com ultiplication, we need the no tion o f “restriction of a lab elled diagram”. Consider d ∈ ldiag of size p × q . F or any subset I ⊂ [1 ..p ], w e deﬁne a lab elle d diagram d [ I ] (of size k × l , k = card( I )) by taking the k = | I | blac k sp o t s n um b ered in I and the edges (resp. white sp o ts) that are connected to them. W e ta ke this subgraph and r elab el the black (resp. white) sp ot s in increasing order. The construction of the Hopf algebra LDIAG go e s as follow s : (i) the algebra structure is that of algebra of the mono id ldiag s o that the elemen ts of LDIAG are x = X d ∈ ldiag α d d (88) (the sum is ﬁnitely supp o rted) (ii) the comultiplic atio n is g iven, on a lab elled dia gram d ∈ ldiag of size p × q , b y ∆ L ( d ) = X I + J =[1 ..p ] d [ I ] × d [ J ] (89) (iii) the counit is “taking the co eﬃcien t of t he v oid diagram”, that is, fo r x as in Eq. (88) , ǫ L ( x ) = α 1 ldiag . (90) One can c hec k that ( LDIA G , [ −|− ] L , 1 ldiag , ∆ L , ǫ L ) is a bialgebra (for pro o f s see [28]). No w o ne can c hec k t ha t we satisfy t he conditions of exercise (11.10) question 3 and the an tip o de S L can b e computed b y formula (1 24) of the same exercise. 25 W e hav e so far constructed the Hopf algebra ( LDI A G , [ −|− ] L , 1 ldiag , ∆ , ǫ, S L ). The constructions a b o v e are compatible with the arrow φ DIA G , LDIAG : LDIAG 7→ D I A G deduced from the class-map φ diag , ldi ag : ld iag 7→ diag (a dia g ram is a class of lab elle d diagrams under p ermutations of blac k and white sp ots among t hemselv es ). So that, one can deduce “b y taking quotients” a structure of Hopf algebra o n t he algebra o f diag . Denoting this algebra b y DI AG , one has a natura l Hopf algebra structure ( DIAG , [ −|− ] D , 1 diag , ∆ D , ǫ D , S D ) and one can pro v e that this is the unique Hopf algebra structure suc h that φ DIA G , LDIAG is a morphism for the algebra and coalgebra structures. 9. Link b et w een LDIAG and other Hopf alge bras 9.1. The de f o rme d c ase One can construct a three-para meter Hopf algebra deformation of LDIAG (see [28]) , denoted LDIAG ( q c , q s , q t ) suc h that LDIAG (0 , 0 , 0) = LDIA G and LDIA G (1 , 1 , 1) = MQSym the alg ebra of Matrix Quasi Symmetric F unctions [25]). On the ot her hand, it was pro v ed by L. F o issy [32, 3 3] that one of the planar decorated trees Hopf algebra is isomorphic MQSym and ev en t o LDIA G (1 , q s , t ) for ev ery q s and t ∈ { 0 , 1 } . The complete picture is giv en b elo w. 10. Duals of Hopf algebras The question of dualizing a Hopf algebra (i.e. endo wing the dual - o r a subspace of it - with a structure of Hopf algebra) is solv ed, in complete generalit y , b y the ma chinery 26 of Sw eedler’s duals. The pro cedure consists in ta king the “ represen tativ e” linear forms (instead of all the linear forms) and dualize w.r.t. the f ollo wing table com ultiplication → m ultiplication counit → unit m ultiplication → com ultiplication unit → counit an tip o de → trsnspo se of the antipo de. In the case when the Hopf a lgebra is free as an algebra (whic h is o ften the case with noncomm utativ e Hopf algebras of com binatorial phys ics), one can use rational expres- sions of Automata Theory to get a g enuine calculus within this dual (see [29]). 27 11. E X ERCISES Exercise 11.1 R epr esentative functions on R (Complex value d) . A function R f − → C is said to b e r epr esen tative if t her e exist ( f (1) i ) n i =1 and ( f (2) i ) n i =1 such that, for al l x, y ∈ R o ne has f ( x + y ) = n X i =1 f (1) i ( x ) f (2) i ( y ) . (91) 1) S h ow that cos , cos 2 , sin , exp and a → a n ar e r epr esentative. Pr ovid e minim al s ums of typ e Eq. ( 9 1). 2) Show that the fol lowing ar e e quivalent i) f is r epr esentative. ii) Ther e exists a gr oup r epr esentation ( R , +) ρ − → C n × n , a r ow ve ctor λ ∈ C 1 × n , a c olumn ve ctor γ ∈ C n × 1 such that f ( x ) = λρ ( x ) γ . iii) ( f t ) t ∈ R is of ﬁnite r ank in C R (her e f t , the shift of f by t , is the function x − → f ( x + t ) ). 3) Show that the minimal n such t hat formula Eq.(9 1 ) holds is also the r ank ( o ver C ) of ( f t ) t ∈ R . 4) a) I f f is c ontinuous then ρ c an b e chosen so and ρ ( x ) = e xT for a c ertain matrix T ∈ C n × n . b) In this c ase show that r epr esentative functions ar e line ar c ombinations of pr o ducts of p olynomials an d ex p onentials. 5) f ∈ C R is r epr esentative iﬀ R e ( f ) = ( f + ¯ f ) / 2 and I m ( f ) = ( f − ¯ f ) / 2 i ar e r epr esentative in R R . 6) Show that the s et of r epr esentative functions of C R is a C -ve ctor sp ac e. This sp ac e wil l b e denote d R ep C ( R ) . 7) Show that the functions ϕ n,λ = x n e λx ar e a b asis of Rep C ( R ) ∩ C o ( R ; C ) (i.e. c ontinuous c omplex value d r epr esentative functions on R ). 8) De duc e fr om (7) that the fol lowing statemen t is false: “If a entir e function f : R 7→ C is such that ( f t ) t ∈ Z is of ﬁnite r ank, then it is r epr esentative”. Hint : Consider exp ( exp (2 i π x )) . Exercise 11.2 R epr esentative functions in gener al (se e also [1, 20]) L et M b e a monoid (sem igr oup with unit) and k a ﬁeld (one c an ﬁrst thin k of k = R , C ). F or a function M f − → k one de ﬁ nes the shifts: f z : x − → f ( z x ) , y f : x − → f ( xy ) , y f z : x − → f ( z xy ) . (92) 1)a) Che ck the fol lowing formulas ( f y 1 ) y 2 = f y 1 y 2 , y 2 ( y 1 f ) = y 2 y 1 f ( x f ) y = x ( f y ) = x f y . (93) 28 As for gr oups, if M is a monoid, a M - m o dule structur e on a v e ctor sp ac e V is deﬁne d by a morphi s m or an anti-morphism M 7→ E nd k ( V ) . b) F r om Eqs.(93) d e ﬁ ne two c anonic al M -mo dule structur e s of k M . 2)a) Show that the fol lowing ar e e quivalent i) ( f z ) z ∈ M is o f ﬁnite r ank in k M . ii) ( y f ) y ∈ M is o f ﬁnite r ank in k M . iii) ( y f z ) y , z ∈ M is of ﬁnite r ank in k M . iv) Ther e exist two families ( f (1) i ) n i =1 and ( f (2) i ) n i =1 such that f ( xy ) = n X i =1 f (1) i ( x ) f (2) i ( y ) . (94) v) Ther e exists a r epr esentation of M ρ : M − → k n × n , a r ow ve ctor λ ∈ k 1 × n , a c olumn ve ctor γ ∈ k n × 1 such that f ( x ) = λρ ( x ) γ for al l x ∈ M . b) Using (v) ab ove, show that the (p ointwise) pr o duct of two r epr esentative functions is r epr esentative. One denotes Rep k ( M ) the sp ac e of ( k -v a lue d ) r epr esentative functions on M . 3) a) R e c al l brieﬂy wh y the mapping k M ⊗ k M 7→ k M × M (95) deﬁne d by n X i =1 f (1) i ⊗ f (2) i →  ( x, y ) → n X i =1 f (1) i ( x ) f (2) i ( y )  is inje ctive. b) Show that, if n is minim al in (94) , the f a milies of functions f (1) i and f (2) i ar e r epr esentative and that the mapping Rep k ( M ) 7→ Rep k ( M ) ⊗ Rep k ( M ) d e ﬁne d by f → n X i =1 f (1) i ⊗ f (2) i (96) deﬁnes a structur e of c o asso ci a tive c o alg ebr a on R ep k ( M ) Has it a c o - uni t ? We denote by ∆ the c ompultiplic ation c ontructe d ab ove. 4) S h ow that ( Rep k ( M ) , ., 1 , ∆ , ǫ 1 M ) (with 1 the c onstant-value d function e qual to 1 o n M and ǫ the Di r ac me asur e f → f (1 M ) ) is a bial g e br a. Exercise 11.3 (Examp l e of a monoid c oming fr om dissip ation the ory [47]). L e t ( M n , × ) b e the mo n oid of n × n c omplex squar e ma tric e s endow e d with the usual pr o duct ( M n , . ) = ( C n × n , × ) We deﬁne a state (V on Neumann) as a p ositive semi-d e ﬁnite hermitian matrix of tr ac e one ( = 1 ) i.e. a matrix ρ such that (i) ρ = ρ ∗ (ii) ( ∀ x ∈ C n × 1 )( x ∗ ρx ≥ 0) 29 (iii) T r ( ρ ) = 1 . The set of such states w i l l b e denote d by S n . 1) (Structur e) a) Show that S n is c onvex. b) Show that S n is c omp act. Hin t : Consider the set of p ossible sp e ctr a i.e. t he s implex S = { ( λ 1 , λ 2 , · · · , λ n ) ∈ ( R + ) n | n X i =1 λ i = 1 } and s how that S n is the ra nge of the c ontinuous m apping φ : U ( n ) × S 7→ M n given by the formula φ ( u, s ) = udu ∗ ; with d =      λ 1 0 · · · 0 0 λ 2 · · · 0 . . . 0 · · · 0 λ n      (97) and wher e s = ( λ 1 , λ 2 , · · · , λ n ) . c) Show that the extr emal elements [9] of the c omp act S n is the set of ortho gonal pr oje ctions of r ank one and that this set is c onne cte d by ar cs. 2) (KS c ondition) We say that a ﬁnite family ( k i ) i ∈ I ( I is ﬁnite) of elemen ts in M n fulﬁls the KS c ondition iﬀ P i ∈ I k ∗ i k i = I n ( I n is the unity m atrix, the unit of the mon o id M n ). On the other han d , g iven two ﬁni te fa m ilies A = ( k i ) i ∈ I and B = ( l j ) j ∈ J , we deﬁn e A ∗ B as the family A ∗ B = ( k i l j ) ( i,j ) ∈ I × J (98) a) Show that, if A a n d B fulﬁl the KS c ondition then s o to o do es A ∗ B . T o every (ﬁn i te) family A = ( k i ) i ∈ I which fulﬁls the KS c ondition, we attach a tr an s f o rmation φ A : M n 7→ M n , given by the formula φ A ( M ) = X i ∈ I k i M k ∗ i (99) b) Show that φ A is line ar a n d pr eserves S n (i.e. φ A ( S n ) ⊂ S n ). c) Show that if A and B fulﬁl the KS c ondition, on e ha s φ A ◦ φ B = φ A ∗ B (100) Conclude that the φ A form a semigr oup of tr ans f o rmations (with unit). d) (Exampl e ) L et ( E i,j ) i,j ∈{ 1 , 2 } b e the set of the four matrix units in M 2 . Show that the fol lowing families fulﬁl KS c ondition A = ( E 11 , 1 p (2) E 12 , 1 p (2) E 22 ) B = ( 1 p (2) E 11 , 1 p (2) E 21 , E 22 ) (101) Compute A ∗ B . 3) (Description of the semigr oup at the level of multisets). In or der to pul l-b ack the formula (100) at the level of multisets, we r emark that the or der or the lab el ling o f the 30 elements k i is i rr elevant; al l that c ounts is their multiplicities. a) (Exam p le showing that the “set” structu r e is to o we a k). L e t ( E i,j ) i,j ∈{ 1 , 2 } b e the set of the four matrix units in M 2 and set A = ( 1 p (2) E 11 , 1 p (2) E 12 , 1 p (2) E 21 , 1 p (2) E 22 ) (102) c ompute A ∗ A and che ck that it has (n o n-zer o) r ep e ate d elements and thus c orr esp on ds to a multiset (se e app endix). b) Show that the multisets of M n ar e exactly the elements P M ∈M n λ M [ M ] (her e we note [ M ] the im age of M ∈ M n in the algebr a R [ M n ] in or der to forbid matrix ad d ition) of the algebr a R [ M n ] such that ( ∀ M ∈ M n )( λ M ∈ N ) (103) the set of elemen ts fulﬁl ling (103) w il l b e den o te d N [ M n ] . b) T o every ﬁnite family of matric es (in M n ) A = ( M i ) i ∈ I one may as so ciate its sum (in R [ M n ] ) S ( A ) = P i ∈ I M i , c h e ck that it is an element of N [ M n ] and that every ele m ent of N [ M n ] is obtaine d so. c) S h ow that S ( A ∗ B ) = S ( A ) .S ( B ) ∈ N [ M n ] ⊂ R [ M n ] and de duc e that ( N [ M n ] , . ) is a monoid. d) T o every multiset of matric e s (i n M n ) A = P M ∈M n α ( M )[ M ] , o ne asso ciates T ( A ) = X M ∈M n α ( M )[ M ∗ M ] show that, if A = ( M i ) i ∈ I is a ﬁnite set of ma tric es, one has X i ∈ I M ∗ i M i = T ( S ( A )) (104) e) Show that, if T ( A ) = I n , T ( A.B ) = T ( B ) . We denote N [ M n ] K S the set of A ∈ N [ M n ] such that T ( A ) = I n . f ) Che ck that the mapping φ A deﬁne d pr eviously dep ends only on S ( A ) and denote, for A ∈ N [ M n ] K S the ma pping Φ A de duc e d fr om this pr op erty. g) Pr ove that the map p ing ( N [ M n ] K S , . ) 7→ ( E nd C ( C n × n ) , ◦ ) is a morphism of semigr oups (pr eserving the units). 4) (Invertible ele ments) T o every A = P M ∈M n α ( M )[ M ] ∈ N [ M n ] K S one ass o ciates ǫ ( A ) = P M ∈M n α ( M ) ∈ N . a) Pr ove that ǫ ( A.B ) = ǫ ( A ) ǫ ( B ) (105) b) Pr ove that ǫ ( A ) = 1 = ⇒ | supp ( α ) | = 1 (106) and, thus, in this c o ndition, A = [ M ] (a single matrix). c) De duc e fr om (a ) and (b) that the set of invertible elements of N [ M n ] K S is exactly the unitary gr oup U ( n ) . 31 Exercise 11.4 L et ( M , ∗ , 1 M ) b e a m o noid, ≡ a n e quivalenc e r elation on M and C l : M 7→ M / ≡ the class function (which, to every eleme n t of M asso ciates its c l a ss C l ( x ) ). a) S upp ose that ther e is an (internal) law ⊥ on M / ≡ such that C l is a morphism i.e . one has ( ∀ x, y ∈ M )( C l ( x ∗ y ) = C l ( x ) ⊥ C l ( y )) (107) then pr ove that ≡ is c omp atible with the right and le f t “tr anslations” of the monoid this me ans that ( ∀ x, y , t, s ∈ S )( x ≡ y = ⇒ [ s ∗ x ∗ t ≡ s ∗ y ∗ t ]) (108) b) Conversely, we supp ose that ≡ is an e quivalenc e on M satisfying c ondition (108) , show that the r esult C l ( x ′ ∗ y ′ ) do es no t dep end on the ch o ic e of x ′ ∈ C l ( x ); y ′ ∈ C l ( x ) and ther efor e c onstruct a law ⊥ on M / ≡ such that the class function is a morphism (i.e. (107) ). c) Show mor e over that, in the pr e c e ding c onditions,  M / ≡ , ⊥ , C l (1 M )  is a mo n oid. Exercise 11.5 L et A b e an AA U and ∆ : A 7→ A ⊗ A a c omultiplic ation. We build (tensor) pr o ducts of two r epr esen tations by the formula (24) , mor e pr e cisely by can ◦ ( ρ 1 ⊗ ρ 2 ) ◦ ∆ (109) wher e can : E nd ( V 1 ) ⊗ E nd ( V 2 ) 7→ E nd ( V 1 ⊗ V 2 ) is the c a n onic al mapping. a) Pr ove that, i f ∆ is a morphi sm of a lgebr as, then the line ar mappi n g ρ 1 × ρ 2 : A 7→ E nd ( V 1 ⊗ V 2 ) (110) deﬁne d by the c omp osition (109) is a morphism of AAU (and henc e a r epr esentation). b) Pr ove that, if ρ 1 × ρ 2 is a r epr e s entation for any p air ρ 1 , ρ 2 of r epr esentations of A , then ∆ is a morphism of AA U (use ρ 1 = ρ 2 , the - r e gular - r e pr esentation of A on itself by multiplic ations on the le f t). Exercise 11.6 We c onsider the c anonic al isomorph isms can 1 | 23 : V 1 ⊗ ( V 2 ⊗ V 3 ) 7→ V 1 ⊗ V 2 ⊗ V 3 can 12 | 3 : ( V 1 ⊗ V 2 ) ⊗ V 3 7→ V 1 ⊗ V 2 ⊗ V 3 (111) Sho w that, in order to ha v e (37) fo r e ve ry triple ρ i , i = 1 , 2 , 3 of represen tations it is necessary and suﬃcien t that can 12 | 3 ◦ (∆ ⊗ I d A ) ◦ ∆ = can 1 | 23 ◦ ( I d A ⊗ ∆) ◦ ∆ (112) (for the necessary condition, consider aga in the left regular represen tations). Exercise 11.7 L et ( A , ∆) b e a c o algebr a ( ∆ is an arbitr a ry - but ﬁxe d - line ar mapping) and ( A ∗ , ∗ ∆ ) b e its dual algebr a. Explicitely, for f , g ∈ A ∗ and x ∈ A (fo r c onvenienc e, the law is written i n in ﬁx denotation) h f ∗ ∆ g | x i = h f ⊗ g | ∆( x ) i (113) 32 Pr o ve the fol lowing e quivalenc e s ∆ is c o-asso ciative ⇐ ⇒ ∗ ∆ is asso ciative (114) ( ∀ ǫ ∈ A ∗ )  ǫ is a unity fo r ( A ∗ , ∗ ∆ ) ⇐ ⇒ ǫ is a c o-unity for ( A , ∆)  (115) Exercise 11.8 The m a ppings can l , can r ar e as in (41) . Pr ove that, in or der that for any r epr ese n tation ρ o f A , one has can l ◦ ( ǫ ⊗ ρ ) ◦ ∆ = c an r ◦ ( ρ ⊗ ǫ ) ◦ ∆ (1 1 6) it is ne c essary and suﬃcient that ǫ b e a c o unit. Exercise 11.9 L et ( C , ∆) b e a c o algebr a and ( A , µ ) b a a n algebr a on the same (c ommutative) ﬁeld of sc alars k . We deﬁ n e a m ultiplic ation (c al le d c onvo l ution) o n H o m k ( C , A ) by f ∗ g = µ ◦ ( f ⊗ g ) ◦ ∆ (117) so that, if ∆( x ) = P (1)(2) x (1) x (2) , f ∗ g ( x ) = X (1)(2) f ( x (1) ) g ( x (2) ) . 1) If A is asso ciative and C c o asso ciative, show that the algebr a ( H om k ( C , A ) , ∗ ) is asso ciative. 2) We supp ose mor e over that C admits a c ounit ǫ : C 7→ k and A a unit 1 A (identiﬁe d with the line ar mapping k 7→ A given by λ → λ 1 A ). Show that 1 A ◦ ǫ (tr aditionnal ly denote d 1 A ǫ ) is the unit of the algebr a ( H om k ( C , A ) , ∗ ) . 3) L et ( B , ∗ , 1 , ∆ , ǫ ) b e a bia l g ebr a. The c onvolution under c onsider ation wil l b e that c onstructe d b etwe en the c o algebr a ( B , ∆ , ǫ ) a n d the algebr a ( B , ∗ , 1) . a) L et S ∈ E nd ( B ) . Show that the fol lowing a r e e quivalent i) S is an antip o d e fo r B ii) S is the inve rs e of I d B in ( E nd k ( B ) , ∗ ) . b) De duc e fr om (b) that the antip o d e, i f it e xists, is unique. c) Pr ove that the bial g ebr a ( k h A i , ∗ , ∆ h , ǫ aug ) deﬁne d ar ound e quation (50) admits no antip o de (if the alpha b et A is not empty). 4) L et ( C , ∆ , ǫ ) b e a c o alge br a c o asso cia tive with c ounit. We de ﬁ ne ∆ 2 by T 2 , 3 ◦ ∆ ⊗ ∆ wher e T 2 , 3 : C ⊗ 4 7→ C ⊗ 4 is the ﬂip b etwe en the 2nd and the 3r d c omp one n t T 2 , 3 ( x 1 ⊗ x 2 ⊗ x 3 ⊗ x 4 ) = x 1 ⊗ x 3 ⊗ x 2 ⊗ x 4 (118) a) Show that ( C ⊗ C , ∆ 2 , ǫ ⊗ ǫ ) (with ǫ ⊗ ǫ ( x ⊗ y ) = ǫ ( x ) ǫ ( y ) ) is c o asso cia tive c o algebr a with c ounit. L e t ( H , µ, 1 , ∆ , ǫ, S ) b e a Hopf algebr a. The c onvolution ∗ her e w i l l b e that c onstructe d b etwe en the c o algebr a ( H ⊗ H , ∆ 2 , ǫ ⊗ ǫ ) and the alg e br a ( H , µ, 1) . We c onsider the two elements ν i ∈ ( H om k ( H ⊗ H , H ) , ∗ ) deﬁne d by ν 1 = S ◦ µ and ν 2 ( x ⊗ y ) = S ( y ) S ( x ) . b) Show that the elements ν i ar e the c onvolutional inverses of µ . De duc e fr om this that S : H 7→ H is an antimorphism of algebr as. 33 Exercise 11.10 1) L et ( B , ∗ , 1 , ∆ , ǫ ) b e a bialge b r a, we denote by B + the kernel of ǫ . a) Pr ove that B = B + ⊕ k . 1 B . We denote I + the pr oje ction B 7→ B + with r esp e ct to the pr e c e ding de c omp osition. b) Pr ove that, f o r every x ∈ B + , one c an write ∆( x ) = x ⊗ 1 + 1 ⊗ x + X (1)(2) x (1) ⊗ x (2) with x ( i ) ∈ B + (119) 2) Deﬁne for x ∈ B + , ∆ + ( x ) = ∆( x ) − ( x ⊗ 1 + 1 ⊗ x ) = X (1)(2) x (1) ⊗ x (2) (120) a) Che ck that ( B + , ∆ + ) is a c o asso cia tive c o algebr a . Deﬁne ( B ∗ ) + = { f ∈ B ∗ | f (1) = 0 } (121) b) Pr ove that ( B ∗ ) + is a sub algebr a of ( B , ∗ ∆ ) and that its law is dual of ∆ + . c) Pr ove that the algebr a ( B ∗ , ∗ ∆ ) is obtaine d fr om  ( B ∗ ) + , ∗ ∆ +  by adjunction of the unity ǫ . 3) The bialgeb r a is c al le d lo c al l y ﬁnite if ( ∀ x ∈ B )( ∃ k ∈ N ∗ )(∆ +( k ) ( x ) = 0) . (122) The pr oje ction I + b eing as ab o ve, show that, in c ase B is lo c al ly ﬁnite, ( ∀ x ∈ B )( ∃ N ∈ N ∗ )( ∀ k ≥ N )(( I + ) k ( x ) = 0) (123) and that S = X n ∈ N ( − I + ) n (124) is an antip o d e fo r B . Exercise 11.11 1) L et G b e a gr oup and H = ( C [ G ] , ., 1 G , ∆ , ǫ, S ) b e the Hop f algebr a of G . a) Show that { ( g − 1) } g ∈ G −{ 1 } is a b asis of H + (deﬁne d as a b ove) and that ∆ + ( g − 1) = ( g − 1) ⊗ ( g − 1) . b) Show that, if G 6 = { 1 } , H + is n ot lo c al ly ﬁnite, but H admits an antip o de. 2) Pr ove that, if the c opr o duct of H is gr ade d (i.e. ther e exists a d e c omp osition H = ⊕ n ∈ N H n with ∆( H n ) ⊂ P a + b = n H a ⊗ H b ) and H 0 = k . 1 H , then the c omultiplic ation is lo c a l ly ﬁnite. 3) Deﬁne the de gr e e of a lab el le d diagr am as its numb er of e dg es and LDIA G n as the v e ctor sp ac e gener ate d by the dia g r a m s of de gr e e n and che ck that we satisfy the c onditions of exer cise (11.10) question 5. Exercise 11.12 1) S h ow that, in or der that a family ( λ k i,j ) i,j,k ∈ I b e the fa mily of structur e c onstants of s o me algebr a it is ne c essary and suﬃcient that ( ∀ ( i, j ) ∈ I 2 )  ( λ k i,j ) k ∈ I is ﬁ nitely supp orte d  (125) 34 2) Similarly sho w that in or d e r that a family ( λ j,k i ) i,j,k ∈ I b e the family of structur e c onstants of som e c o algebr a it is ne c essary and suﬃcie nt that ( ∀ i ∈ I )  ( λ j,k i ) ( j,k ) ∈ I 2 is ﬁ nitely supp orte d  (126) 3) Give examples of map pings λ : I 3 7→ k such that the c orr esp onding fa milies satisfy i) (125) and (126 ) ii) (125) and not (126 ) iii) (126) and not (125 ) iv) none of (125) a nd ( 126) 4) Give further examp l e s such as those i n 3) i-iii b ut now deﬁning asso ciative ( r esp. c o asso ciative) multiplic ations (r esp. c omultiplic ations). 12. A P PENDIX 12.1. F unction sp ac es Throughout the text, w e use the basic cons tructions of set theory and algebra (see [6, 7]). The set of mappings b et w een tw o sets X and Y is denoted by Y X . Th us if k is a ﬁeld k X = { f : X − → k } (127) the v ector space of all functions deﬁned on X with v alues in k . F or eac h function f ∈ k X , w e call the “ supp o rt of f ” the set of p oints x ∈ X suc h tha t f ( x ) is not zero + . supp ( f ) = { x ∈ X : f ( x ) 6 = 0 } (128) the set of functions with ﬁnite supp ort is a vec tor subspace o f k X whic h is denoted by k ( X ) . An interes ting extension of this notio n to other sets of co eﬃcien ts is the combinatorial notion of (ﬁnite) m ultisets. Recall that a m ultiset is a (set with rep etitions) [40]. F or example, the ﬁrst multisets with elemen ts from { a, b } are {} , { a } , { b } , { a, a } , { a, b } , { b, b } , { a, a, a } , { a, a, b } , { a, b, b } , { b, b, b } , { a, a, a, a } , { a, a, a, b } , { a, a, b, b } , { a, b, b, b } , { b, b, b, b } , · · · . (129) A m ultiset with elemen ts in X is then describ ed equiv alently b y a m ultiplicit y function α : X 7→ N with ﬁnite supp ort . The supp ort of suc h a mapping is deﬁned as in (128) a nd the set o f ﬁnite m ultiplicit y f unctions will b e denoted b y N ( X ) (see b elow the free comm utativ e monoid (12.4) ). F or example, t he m ultiplicit y functions { a, b } 7→ N corresp onding to the m ultisets giv en in (130) are, in the same order (w e characterize α b y the pair ( α ( a ) , α ( b ))) (0 , 0) , (1 , 0) , (0 , 1) , (2 , 0) , (1 , 1) , (0 , 2) , (3 , 0) , (2 , 1 ) , (1 , 2) , (0 , 3) , (4 , 0) , (3 , 1) , (2 , 2) , (1 , 3) , (0 , 4) , · · · (130) + In integration theory , the supp ort of a function is the clo sure o f what we deﬁne as the (algebr aic) suppo rt. 35 12.2. B asic structur es Deﬁnition 12.1 (Sem i g r oup) A semigr oup ( S, ∗ ) is a set S end owe d with a close d binary op er ation ∗ satisfying an asso ciative law, this m e ans that, for al l x, y , z ∈ S one has x ∗ ( y ∗ z ) = ( x ∗ y ) ∗ z . http://en.w ikipedia.org/wiki/Semigroup Deﬁnition 12.2 (Monoid) A monoid ( M , ∗ ) is a s emigr oup w hich p o ssesses a neutr al element, i.e. a n element e ∈ M such that, fo r al l x ∈ M : e ∗ x = x ∗ e = x. (131) Such an element, if it exists i s unique. The neutr al element is often denote d 1 M . http://en.w ikipedia.org/wiki/Monoid Deﬁnition 12.3 (F r e e Monoid) The fr e e mon o id of alpha b et X is the set of strings x 1 x 2 · · · x n with letters x i ∈ X (c om prising the empty s tring) . T h is set is denote d X ∗ , its law is the c onc atenation and its neutr al el e ment i s the e m pty string. It is easily seen that this mo no id is free in the follo wing sense. F or an y “ set-theoretical” mapping φ : X 7→ M , where ( M , ∗ ) is a monoid, φ can b e extended to strings so that X φ / / can ! ! C C C C C C C C M X ∗ ¯ φ O O (132) Deﬁnition 12.4 (F r e e C ommutative Monoid) The fr e e c ommutative mon o id of the alphab et X is the se t of monom ials X α ( α ∈ N ( X ) ). This set i s denote d by MON ( X ) and its law is the multiplic ation of mo n omials X α X β = X α + β (133) It is easily seen that this mo no id is free in the follo wing sense. F or an y “ set-theoretical” mapping φ : X 7→ M , where ( M , ∗ ) is a comm utativ e monoid, φ can b e extended to monomials so that X φ / / can % % J J J J J J J J J J M MON ( X ) ¯ φ O O (134) An in teresting application of the free monoid is the explic it construction of a monoid deﬁned “by generators and relations”. Let X b e a set (of generator s) and R = ( u i , v i ) i ∈ I a family of pairs of words, then one can construct explicitely the smallest (i.e. the in tersection o f ) congruence ≡ R for whic h ( ∀ i ∈ I )( u i ≡ v i ) . 36 Let say that t w o w ords U, V ∈ X ∗ are “related” b y ≡ R if there is a chain of replacemen ts of the ty p e pu i s → pv i s o r pv i s → pu i s p, s ∈ X ∗ leading from U to V . F ormally , there exists a chain U = U 0 , U 1 , · · · U n = V (135) suc h that for eac h j < n U j = p j Aq j ; U j +1 = p j B q j with ( A, B ) = ( u i , v i ) o r ( B , A ) = ( u i , v i ) for some i (dep e nding on j . One can sho w tha t the constructed ≡ R is a congruence (see exercise (108) ) and w e deﬁne h X ; R i Mon (136) as the quotient X ∗ / ≡ R . This monoid has the following prop ert y : if φ X ∗ 7→ M is a morphism (of monoids) suc h that, for all i ∈ I one has φ ( u i ) = φ ( v i ), t hen φ factorises uniquely through h X ; R i Mon X φ / / can % % K K K K K K K K K K M h X ; R i Mon ¯ φ O O (137) Deﬁnition 12.5 (Gr oup) A gr oup ( G, ∗ ) is a monoid such that for e ach x ∈ G ther e exists y such that x ∗ y = y ∗ x = e. (138) F or ﬁxe d x such an element is unique an d is usual ly denote d by x − 1 and c al le d the inverse of x . Deﬁnition 12.6 (Algebr a of a mon o id) L et k b e a ﬁeld (sc alars, for example k = R or C ). Th e algebr a k [ M ] of a mon o id M (with c o eﬃcients in k ) is the set of mappings k ( M ) endowe d with the c onvolution pr o d uct f ∗ g ( w ) = X uv = w f ( u ) g ( v ) . (139) The algebr a ( k [ M ] , ∗ ) is an AAU. Eac h m ∈ M may b e iden tiﬁed with its c haracteristic f unction (i.e. the Dirac function δ m with v alue 1 at m and 0 elsewhere). These functions for m a basis o f k [ M ] and then, ev ery f ∈ k [ M ] can b e written as a ﬁnite sum f = P w f ( w ) w . Through this iden tiﬁcation the unit y of M and k [ M ] coincide. The algebra of a monoid solv es the following univ eral pro blem. Let M b e a monoid a nd j : M 7→ k [ M ] the em b edding describ ed ab o v e. F or an y AA U A and an y morphis m of monoids φ : M 7→ A (( A , . ) is a monoid), one has an unique factorization M φ / / j   A k [ M ] ¯ φ < < z z z z z z z z 37 where ¯ φ is a morphism of AAU. Lik ewise, the env eloping algebra U k ( G ) of a Lie algebra G with the canonical mapping can : U k ( G ) 7→ G is the solution of a univ ersal pro blem. The sp eciﬁcations ar e the follo wing (i) U k ( G ) is an AAU (ii) can is a morphism of Lie algebras (f or this, U k ( G ) is endo w ed of the structure of Lie algebra giv en b y the brac k et [ X , Y ] = X Y − Y X ) . 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