CALCULO DE COEFICIENTES DE TRANSMISIÓN DE CALOR

In this section we briefly introduce locally compact groups in the language of locally compact quantum groups and we discuss quantum semigroups for the C^˚-algebraic and von Neumann algebraic settings. We will also briefly discuss Hopf algebras and the diffi-culties in using them for locally compact quantum groups.

2.1.1 Locally Compact Groups as Quantum Groups

It is recommended that the reader use this as a motivation for what follows and as a reference for the properties of quantum groups in the special case where we have a group.

The results are easier to prove here because of the commutativity of C₀pGq, however we feel that it may give the reader some motivation as to why we are interested in the results in the more general case of quantum groups.

Example 2.1.1 LetG be a locally compact group, then we can consider the commutative C^˚-algebra of continuous functions vanishing at infinity C0pGq. Given the structure on the groupG we define the map ∆ : C0pGq Ñ C^bpG ˆ Gq given by p∆pfqqpx, yq ÞÑ fpxyq forf P C⁰pGq. We have that ∆pfq is bounded as we have

}∆pfq} “ sup

x,yPG|p∆pfqqpx, yq| “ sup

x,yPG|fpxyq| “ sup

xPG|fpxq| “ }f}.

We note that CbpG ˆ Gq –ⁱ MpC⁰pG ˆ Gqq –ⁱ MpC⁰pGq b^minC₀pGqq where M denotes the multiplier algebra (see SectionA.5) and so ∆ : C0pGq Ñ MpC⁰pGq b^minC₀pGqq.

We have forx, y, z P G and f P C⁰pGq that

fppxyqzq “ r∆pfqspxy, zq “ rp∆ b idqp∆pfqqspx, y, zq

and

fpxpyzqq “ r∆pfqspx, yzq “ rpid b ∆qp∆pfqqspx, y, zq

and so by associativity ofG we havep∆ b idq ˝ ∆ “ pid b ∆q ˝ ∆. We call this property coassociativity.

We also have a left Haar weightφ : C0pGq^` Ñ r0, 8s given by f ÞÑ ş

X f dµ. Sim-ilarly we have a right Haar weightψ given by the right Haar measure. We can consider L²pG, µq as the space of square integrable (up to almost everywhere) functions on G, that isf P L²pG, µq if and only ifş

Gf^˚f dµă 8. Then L²pG, µq is a Hilbert space with inner

productpf|gq “ φpfg^˚q.

ForωP C⁰pGq^˚_`we have a positive measureν P MpGq such that xf, ωy “ş

Xf dν. It then follows from Fubini’s theorem that for allf P C⁰pGq^`we have

φppω b idqp∆pfqqq “ ż

Grpω b idqp∆pfqqspyq dµpyq

“ ż

G

ˆż

G

fpxyq dνpxq

˙

dµpyq “ ż

G

ˆż

G

fpyq dµpyq

˙

dνpxq “ ωp1qφpfq

It follows that left invariance of the Haar measure µ is equivalent to having φppω b idqp∆pfqqq “ φpfqωp1q for all ω P C⁰pGq^˚<sub>`</sub>andf P C<sup>0</sup>pGq<sup>`</sup>. It also follows that

lin tpω b idq∆pfq | f P C⁰pGq, ω P C⁰pGq^˚u “ C⁰pGq

and similarly forω acting on the right.

We can define an isometry W : L²pG, µq b L²pG, µq Ñ L²pG, µq b L²pG, µq by WpF qpx, yq “ F px, x^´1yq for F P L²pG ˆ G, µ ˆ µq “ L²pG, µq b L²pG, µq and it follows thatW^˚pf b gq “ ∆pgqpf b 1q. We denote W¹² “ W b 1, W²³ “ 1 b W and W13 “ σ²³W12 whereσ denotes the flip map on L²pG, µq b L²pG, µq. Then we have for F P L²pG ˆ G ˆ G, µ ˆ µ ˆ µq

ppW¹²W13W23qpF qqpx, y, zq “ pW¹³W23pF qqpx, x^´1y, zq

“ pW²³pF qqpx, x^´1y, x^´1zq “ F px, x^´1y, y^´1zq

and

ppW²³W12qpF qqpx, y, zq “ pW¹²pF qqpx, y, y^´1zq

“ F px, x^´1y, y^´1zq “ .ppW¹²W13W23qpF qqpx, y, zq.

As this then holds for all suchF we have W12W13W23 “ W²³W12.

Example 2.1.2 LetG be a locally compact group and µ the left Haar measure on G. Then we have the von Neumann algebra of measurable essentially bounded functions L⁸pG, µq.

We also have a map ∆ : L⁸pG, µq Ñ L⁸pG, µq b L⁸pG, µq given by ∆pfqpx, yq “ fpxyq where we’ve identified L⁸pG, µq b L⁸pG, µq –ⁱ L⁸pG ˆ G, µ ˆ µq.

We define φ : L⁸pG, µq^` Ñ r0, 8s by f ÞÑ ş

Xf dµ and we have a n.s.f. weight on L⁸pG, µq. As µ is a Haar measure then we have

ż

X

fpyxq dµpxq “ ż

X

fpxq dµpxq

for allyP G.

2.1.2 Quantum Semigroups

We now define the coproduct on an algebra to give us a notion of a “quantum semigroup”

or a bialgebra. On the way to defining a locally compact quantum group we need to define a C^˚-algebraic quantum semigroup or von Neumann algebraic quantum semigroup in order to capture the “topology” at the quantum level.

Definition 2.1.3 LetA be an algebra and ∆ : A Ñ MpA b Aq a non-degenerate map, then we say ∆ is coassociative if we havep∆ b idq ˝ ∆ “ pid b ∆q ˝ ∆. A non-degenerate coassociative homomorphism ∆ : A Ñ MpA b Aq is called a coproduct on A where if A is unital we require ∆ to be unital and if A is a ˚-algebra we require ∆ to be a

˚-homomorphism.

A C^˚-algebraic quantum semigroup is a pair pA, ∆q where A is a C^˚-algebra and

∆ : A Ñ MpA b^minAq is a coproduct on A. A von Neumann algebraic quantum semigroup is a pairpM, ∆q where M is a von Neumann algebra and ∆ : M Ñ M b M is a normal coproduct onM .

In Example2.1.1we gave a coproduct ∆ : C₀pGq Ñ MpC⁰pGq b^minC₀pGqq –ⁱ C_bpG ˆ Gq on C⁰pGq for a locally compact group G. Similarly from Example 2.1.2we have a coproduct ∆ : L⁸pG, µq Ñ L⁸pG, µq b L⁸pG, µq on L⁸pG, µq.

2.1.3 Hopf Algebras

For a quantum group we need operations that are equivalent to the identity and inversion operations at a “quantum level”. We will discuss the “inversion” operation with Hopf algebras now. First we give the definition of a Hopf algebra, then we discuss this definition in terms of the algebra of polynomials over a finite group and finally we discuss some of the problems for defining locally compact quantum groups using Hopf algebras in this way.

Definition 2.1.4 A Hopf algebra is a unital algebraA with multiplication map m : Ab A Ñ A and unit given by η : C Ñ A such that m is associative (i.e. m ˝ pm b idq “ m˝ pid b mq and m ˝ pη b idq “ m ˝ pid b ηq). Furthermore we have the following:

(i) We have a unital, coassociative homomorphism ∆ : A Ñ A b A and a homomor-phismε : AÑ C called the counit such that pε b idq ˝ ∆ “ id “ pid b εq ˝ ∆;

(ii) There is an antihomomorphismS : AÑ A such that

m˝ pS b idq ˝ ∆ “ η ˝ ε “ m ˝ pid b Sq ˝ ∆

called the antipode.

A Hopf˚-algebra is a Hopf algebra A with involution such that A is a ˚-algebra and ∆ andε are˚-homomorphisms.

Now consider the algebra of polynomials A over a finite group G with multiplication m : Ad A Ñ A given by mpf, gq ÞÑ fg for all f, g P A (where pfgqpxq “ fpxqgpxq for all x P G) and identity map η : C Ñ A given by λ ÞÑ λ1 for λ P C and 1 the unit of A.

We let ∆ be the usual coproduct p∆pfqqpx, yq “ fpxyq for all f P A and x, y P G. We can define the counit ε : A Ñ C given by f ÞÑ fpeq for all f P A (where e is the unit element of G) and the antipode S : A Ñ A given by pSpfqqpxq “ fpx^´1q for all f P A and x P G. We can easily show that this gives us a Hopf algebra. Note how we have used the group product to define ∆, the group identity to define ε and the group inverse to define S.

Unfortunately for locally compact quantum groups it does not necessarily follow that the maps ε and S are bounded. In particular we will see that S is unbounded for the example of SUqp2q that we will give in Chapter5. As such it is difficult to define locally compact quantum groups using Hopf algebras, however the antipode in particular does still play an important role. We show in the next section an appropriate generalisation to what we now consider a locally compact quantum group.