On a unified description of non-abelian charges, monopoles and dyons Kampmeijer, L.

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1 UvA-DARE (Digital Academic Repository) On a unified description of non-abelian charges, monopoles and dyons Kampmeijer, L. Link to publication Citation for published version (APA): Kampmeijer, L. (2009). On a unified description of non-abelian charges, monopoles and dyons. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam ( Download date: 18 Apr 2019

2 Appendix E Generalised transformation group algebras Irreducible representations of a semi-direct product are labelled by an orbit and a centraliser representation. This property is not unique. The irreducible representations of generalised transformations group algebras have a similar classification. This makes it interesting to consider a generalised transformation group algebra that reproduces the dyonic charge sectors as the skeleton group does. We shall see that such an generalised transformation group algebra also has the same fusion rules as the skeleton group. One thus might interpret such an algebra as a subalgebra of a unified electric-magnetic symmetry. A possible objection is that this transformation group algebra is not group and can thus not be interpreted as an invertible symmetry acting on states. On the other hand a transformation group algebra does contain a group algebra, which we can take to be the group algebra of an electric group. E.1 Irreducible representations The definition of a general transformation group algebra can be found in [102], see also [103]. In the definition we will be using is a finite group acting on a finite set Λ. Nonetheless, there is a valid generalisation for a locally compact group with a aar measure and Λ a discrete set. F (Λ ), the set of functions on Λ, is called a transformation group algebra if it is equipped with a multiplication given by f 1 f 2 (λ, h) f 1 (λ, g)f 2 (g 1 λ,g 1 h)dg. (E.1) 133

3 Appendix E. Generalised transformation group algebras The unit of this algebra is given by a δ-function: 1:(λ, h) Λ δ e,h C, (E.2) where e is the unit of. If Λ is also a group then F (Λ ) can be extended to a opf algebra. The coproduct, counit and antipode are defined by Δ(f)(λ 1,h 1 ; λ 2,h 2 ) f(λ 1 λ 2,h 1 )δ h1,h 2 (E.3) ɛ(f) f(e Λ,h)dh (E.4) S(f)(λ, h) f(h 1 λ 1,h 1 ). (E.5) At least if both and Λ are discrete sets there is a convenient basis for F (Λ ) defined by the projection functions P λ h :(, g) Λ δ λ, δ h,g. (E.6) In terms of these projection operators the opf algebra is given by P λ1 h 1 P λ2 h 2 δ λ1,h 1 λ 2 P λ1 h 1 h 2 (E.7) Δ(P λ h) P λ 1h P h (E.8) Λ 1 P λ h (E.9) ɛ(p λ h) δ eλ,λ (E.10) S(P λ h) P h 1 λ 1h 1. (E.11) An intersting set of functions is given by Ph :(λ, g) Λ δ h,g C. (E.12) One can check that these function generate a subalgebra in F (Λ ) which is isomorphic to the group algebra of. Before we review the the irreducible representations we introduce the ilbert spaces upon which the representations will act. Let N be a subgroup of, and a unitary representation of N acting on the space V. We define a ilbert space by the set of maps from to V that respect the action of N: F (, V ) { φ : V φ (hn) (n 1 ) φ (h), h, n N }. (E.13) The irreducible representations of F (Λ ) are described as follows. Let [λ] be an orbit in Λ under the action of and let be the centraliser of the representant λ Λ. Then, 134

4 E.2. Matrix elements and characters for each pair ([λ],) of an orbit [λ] and an irreducible representation of,wehavean irreducible unitary representation Π [λ] of F (Λ ) on F (, V ) is given by Π [λ] (f) φ (h) : f(h λ,g) φ (g 1 h)dg. (E.14) Moreover, all unitary irreducible representations of F (Λ ) are of this form and Π [λ] and Π [] α are equivalent if and only if [λ] [] and α. E.2 Matrix elements and characters For an irreducible representation Π [λ] we denote its carrier space F (, V ) by V [λ]. The dimension of V [λ] is equal to the product d [λ] of the number of elements in [λ] and d the dimension of V. To see this note that the functions φ V [λ] are completely determined once their value on one element in each coset h of / is chosen. The number of cosets equals d [λ] while φ(h) has d components. ence dimv [λ] d [λ] d. To define a basis for V [λ] we choose a basis { e i } for V and h for each [λ] such that h λ. The basis elements { ; e i } are given by the map The action of Π [λ] Π [λ] (f) ; e i (h ν) The matrix elements of Π [λ] ; e i : h νn δ ν (n 1 ) e i V. (E.15) (f) can be found by evaluating equation (E.14) on h ν for each ν [λ]. f(h ν λ,g) ; e i (g 1 h ν )dg ρ ) ; e i (h ρn 1 )dn ρ [λ] ρ [λ] Π [λ] (f),ν ij ρ )δ ρ(n) e i dn )(n) e i dn )(n) ij e j dn )(n) ij dn ν; e j (h ν). (E.16) with respect to the basis { ; e i } are thus given by )(n) ij dn. (E.17) 135

5 Appendix E. Generalised transformation group algebras In particular for f P σ h we have Π [λ] (P σh),ν ij P σ h(ν, h ν nh 1 δ σ,ν δ h,hνnh 1 )(n) ijdn (n) ijdn δ σ,ν δ h,ν (h 1 ν hh ) ij. (E.18) Consequently the character χ [λ] of Π [λ] χ [λ] (f) [λ] is defined by f(, h nh 1 )χ (n)dn, (E.19) where χ denotes the character of. For the projection operators this gives χ [λ] (P νh) P ν h(, h nh 1 )χ (n)dn [λ] δ ν δ h,hnh 1χ (n)dn [λ] δ,ν δ h ν,ν χ (h 1 ν hh ν ). [λ] One may now define an inner product by the formula χ 1,χ 2 χ 1 (P λ h)χ 2(P λ h)dh. (E.20) (E.21) One may check that the characters of the irreducible representation are orthogonal with respect to this inner product: χ [ρ],χ [σ] α δ,λ δ h λ,λ χ (h 1 λ hh λ) δ ν,λ δ h λ,λ χ α(h 1 λ hh λ)dh [ρ] ν [σ] δ [ρ][σ] δ h, χ (h 1 hh )χ α (h 1 hh )dh [ρ] δ [ρ][σ] χ (n)χ α (n)dn [ρ] N ρ δ [ρ][σ] δ α dn [ρ] N ρ δ [ρ][σ] δ α dim(). (E.22) If F (Λ ) can be equipped with a co-algebra structure one can use the characters to obtain the fusion rules. We shall consider this in more detail in the next section. 136

6 E.3. Fusion rules E.3 Fusion rules In special case that Λ is a group, F (Λ ) becomes a opf algebra with a co-multiplication defined by (E.8). The co-multiplication defines the tensor product of two representations and thus also the character of the tensor product. Since the characters of the irreducible representations are orthogonal one can compute the decomposition of the tensor product into irreducible representations by calculating the inner product of the characters of the irreducible representations with the character of the tensor product. In this way one obtains the fusion rules for the generalised transformation group algebra. We shall show that for Λ equal to the set of characters of an abelian group N the fusion rules of F (Λ ) equal the fusion rules of N. From equation (E.8) one find for the character χ a b of the representation a b χ a b (P λ h)χ a χ b (Δ(P λ h)) κ Λ χ a (P λκ 1h)χ b (P κ k). (E.23) If we take the irreducible representations a Π α [σ], b Π [η] β and c Π[ρ] we get from equations (E.20) and (E.21): χ c,χ a b χ c (P λ h)χ a b (P λh)dh χ c (P λ h) χ a(p λκ 1h)χ b(p κ k)dh κ Λ δ,λ δ h λ,λ χ (h 1 λ hh λ) κ Λ ν [σ] ζ [η] [ρ] [ρ] δ ν,λκ 1δ h (λκ 1 ),λκ 1χ α (h 1 λκ 1 hh λκ 1) δ ζ,κ δ h κ,κ χ β (h 1 κ hh κ)dh δ h, χ (h 1 hh ) δ ν,ζ 1δ h ν,ν χ α(h 1 ν hh ν ) δ h ζ,ζ χ β(h 1 ζ hh ζ)dh ν [σ] ζ [η] δ,νζ δ h, δ h ν,ν δ h ζ,ζ [ρ] ν [σ] ζ [η] (E.24) χ (h 1 hh )χ α(h 1 ν hh ν )χ β(h 1 hh ζ)dh. Comparing this with equation (4.43), which only differs with an irrelevant constant factor, we conclude that this does indeed give the fusion rules of N. ζ 137

7 Appendix E. Generalised transformation group algebras 138