Local representations of the vertex algebras on the state space

Next, we exhibit on the vector spaceHa naturalC_v-bimodule structure for each vertexv ∈Σ⁰, that is local in the sense that it acts non-trivially only on the local degrees of freedom in a neighborhood of the vertex v ∈ Σ⁰. This is analogous to the existence of local actions of the Drinfeld double D(H) on the state space in the ordinary Kitaev model without defects for a semisimple Hopf algebra H [BMCA, BK2]. In our construction, however, the algebras C_v are in general not Hopf algebras and we only obtain bimodule structures on H. (A Cv-bimodule structure is equivalent to a left (C_v ⊗C_v^op)-action, where C_v^op has the opposite multiplication of C_v. Whenever C_v is a Hopf algebra, such as D(H), any C_v-bimodule structure can be pulled back to a left Cv-action via the algebra map (id⊗S)◦∆ : Cv → Cv ⊗C_v^op, using the co-multiplication ∆ and the antipode S of the Hopf algebra.)

Letv ∈Σ⁰ be any vertex. Recall from Subsection 2.1.3 that the algebra C_v =H_Σ^∗sit

v =K_Σ^0.5_v 24

2.2 Construction of a Kitaev model with defects is a crossed product of H_Σ^∗sit

v and K_Σ^0.5_v and contains these as subalgebras, and that H_Σ^∗sit

v = N

p∈Σ^sit_v

H_p^∗

is the tensor product of the algebras H_p^∗ for each site p ∈ Σ^sit_v . A C_v-bimodule structure on H is therefore fully determined by aK_Σ^0.5_v -bimodule structure and H_p^∗-bimodule structures for each site p ∈ Σ^sit_v , provided that for each p ∈ Σ^sit_v the left and right actions of K_Σ^0.5_v and H_p^∗ each satisfy the straightening formula (2.3) of the crossed product algebra H_p^∗=K_Σ^0.5_v , which we prove in Theorem 13.

We start by exhibiting a K_Σ^0.5_v -bimodule structure on the vector space H. This is the anal-ogon of the action of the Hopf algebra H for every vertex in the ordinary Kitaev model for a semisimple Hopf algebra H.

Definition 11. Let v ∈Σ⁰. TheK_Σ^0.5

v -bimodule structure on H Ae_v :K_Σ^0.5_v ⊗K_Σ^op0.5

v ⊗H−→H,

is defined on the vector space of linear maps H = Hom_k(K_Σ¹, Z_Σ⁰) in the standard way by pre-composing with the left action on K_Σ¹ and post-composing with the left action on Z_Σ⁰, which are defined as follows:

• Firstly, the vector space K_Σ¹ becomes a leftK_Σ^0.5_v -module as follows. Restrict the regular K_Σ¹-bimodule structure of K_Σ¹, seen as a left (K_Σ¹ ⊗ K_Σ^op1)-action, to the subalgebra K_Σ^0.5_v ⊆K_Σ¹ ⊗K_Σ^op1.

• Secondly, the vector spaceZ_Σ⁰ becomes a leftK_Σ^0.5_v -module as follows. Restrict the given Cv-module structure on Zv to the subalgebra K_Σ^0.5_v ⊆ N

v∈Σ⁰(H_Σ^∗sit

v =K_Σ^0.5_v ) = Cv and extend the action trivially to the vector spaceZ_Σ⁰ =Z_v ⊗N

w∈Σ⁰\{v}Z_w.

Next we will exhibit, for any site p ∈ Σ^sit_v incident to a vertex v ∈ Σ⁰, an H_p^∗-bimodule structure on H.

Recall that Σ^sit_p denotes the set of incidences of a vertex with a given plaquette p (which we also call sites) and denote by Σ^1.5_p the set of incidences of an edge with the given plaquette p (which we call plaquette edges). We consider their union Σ^sit_p ∪Σ¹_p together with a cyclic order on it, given by the clockwise direction along the boundary of p with respect to the orientation of Σ, as illustrated in Figure 2.7

p Figure 2.7: Cyclic order on the set Σ^sit_p ∪Σ¹_p of sites and plaquette edges of a plaquettep

Furthermore, for any plaquette edgee∈Σ¹_p at the plaquettep, let the signε_p(e)∈ {+1,−1}

be positive if the plaquette edge e∈Σ¹_p is clockwise directed around the plaquette p:

2 Defects and boundaries in Kitaev models

p

e Figure 2.8: A plaquette edge e with sign ε_p(e) := +1

and negative ife∈Σ¹_p is directed counter-clockwise around p:

p

e Figure 2.9: A plaquette edge e with sign ε_p(e) :=−1

Recall that, attached to each plaquettep∈Σ², there is a Hopf algebraHp. Now, depending on choice of a site v ∈ Σ^sit_p at p, we define an H_p^∗-bimodule structure on the vector space H.

This is the analogon of the action of the dual Hopf algebra H^∗ for every site in the ordinary Kitaev model for a semisimple Hopf algebra H.

Definition 12. Letp∈Σ². We define, for each sitev ∈Σ^sit_p , the H_p^∗-bimodule structure on H, or left action of the enveloping algebra H_p^∗⊗(H_p^∗)^op,

Be_(p,v):H_p^∗⊗(H_p^∗)^op⊗H−→H, by the following left and right H_p^∗-actions on H.

• We start by declaring that H_p^∗ acts from the left on H = (N

e∈Σ¹K_e^∗)⊗(N

w∈Σ⁰Z_w) by the action of H_p^∗ ⊆ H_Σ^∗sit

v =K_Σ^0.5_v on the (H_Σ^∗sit

v =K_Σ^0.5_v )-module Zv and by acting as the identity on the remaining tensor factors of H.

• For the right action of H_p^∗ on H, we use the total order on the set (Σ^sit_p ∪Σ^1.5_p )\ {v}

starting right after v ∈Σ^sit_p inΣ^sit_p ∪Σ^1.5_p with respect to the cyclic order declared above, given by the clockwise direction around the plaquette p. We first exhibit individual right H_p^∗-actions on the tensor factors of (N

e∈Σ¹_pK_e^∗)⊗(N

w∈Σ⁰_p\{v}Z_w):

– For any e∈Σ^1.5_p , the vector space K_e^∗ becomes a right H_p^∗-module as follows. K_e is a right Hp^εp(e)-comodule and, hence, a left (H_p^∗)^εp(e)-module. Thus the vector space dual K_e^∗ becomes a right (H_p^∗)^εp(e)-module, and finally, by pulling back along the algebra isomorphism ?^hεp(e)i :H_p^∗ →H_p^∗εp(e), a right H_p^∗-module.

Recall that ?^{h+1i def}= id_Hp^∗ and ?^{h−1i def}= S, the antipode of H_p^∗. Explicitly, this right H_p^∗-action is given by

K_e^∗⊗H_p^∗ −→K_e^∗, ϕ⊗f 7−→

k 7→ϕ k₍₀₎f

k_(ε^hεp(e)i

p(e))

.

– For any w∈ Σ^sit_p \ {v}, the vector space Z_w becomes a right H_p^∗-module as follows.

The (H_Σ^∗2

w =K_Σ¹

w)-module Z_w comes with a left H_p^∗-action since H_p^∗ ⊆ H_Σ^∗2

w =K_Σ¹ is a subalgebra. We letH_p^∗ act on Z_w from the right by pulling back this left actionw

along the antipode?^h−1i=S :H_p^∗ →H_p^∗. 26

2.2 Construction of a Kitaev model with defects Then we declareH_p^∗to act from the right on the tensor product(N

e∈Σ¹_pK_e^∗)⊗(N

w∈Σ⁰_p\{v}Z_w) by applying the co-multiplication onH_p^∗ suitably many times and then acting individually on the tensor factors in the sequence given by the image of the clockwise linear order that we have prescribed on the set (Σ^sit_p ∪Σ^1.5_p )\ {v} under the map (Σ^sit_p ∪Σ^1.5_p )\ {v} → (Σ⁰_p ∪Σ¹_p)\ {v} that assigns to a site its underlying vertex and to a plaquette edge its underlying edge. Finally, this gives a right H_p^∗-action onH= (N

e∈Σ¹K_e^∗)⊗(N

w∈Σ⁰Z_w) by acting with the identity on all remaining tensor factors.

So far we have defined, in Definitions 11 and 12, on the vector space H an K_Σ^0.5_v -bimodule structure Ae_v for each vertex v ∈Σ⁰ and anH_p^∗-bimodule structure Be_(p,v) for each sitep∈Σ^sit_v . These are analogous to the actions of the Hopf algebra H and the dual Hopf algebra H^∗ defined for each site in the ordinary Kitaev model without defects. Just as the latter are shown to interact with each other non-trivially, giving a representation of the Drinfeld double D(H) at each site [BMCA], we will now proceed to study how the bimodule structures Ae_v andBe_(p,v⁰₎ of K_Σ^0.5_v and H_p^∗ for various v and (p, v⁰)interact with each other.

In order to simplify the proof we will make a certain regularity assumption on the cell decomposition of the surfaceΣ: We call a cell decompositionregular if it has no looping edges, i.e. there is no edge which has the same source vertex as target vertex and if the Poincaré-dual cell decomposition also has no looping edges, i.e. in the original cell decomposition there is no plaquette that has two incidences with one and the same edge (on its two sides).

Theorem 13. Let H be the vector space defined in Definition 9 for an oriented surface Σwith a labelled cell decomposition. Recall from Definitions 11 and 12 theK_Σ^0.5_v -bimodule structureAe_v on H for every vertex v ∈ Σ⁰, and the H_p^∗-bimodule structure Be_(p,v) on H for every plaquette p∈Σ² together with incident site v⁰ ∈Σ^sit_p . Then

• For any pair of vertices v₁ 6=v₂ ∈Σ⁰, the actions Ae_v1 and Ae_v2 commute with each other.

• For any pair of sites (p₁ ∈ Σ², v₁ ∈ Σ^sit_p1) and (p₂ ∈ Σ², v₂ ∈ Σ^sit_p2) such that p₁ 6= p₂, the actions Be_(p1,v1) and Be_(p2,v2) commute with each other.

• Assume that the cell decomposition of Σ is regular. For any site (p ∈ Σ², v ∈ Σ^sit_p ), the actions Ae_v and Be_(p,v) compose to give onH a bimodule structure over the crossed product algebra H_(p,v)^∗ =K_Σ^0.5_v ,

Be_(p,v)Ae_v :H_p^∗⊗K_Σ^0.5_v ⊗(H_p^∗⊗K_Σ^0.5_v )^op⊗H−→H,

f ⊗k⊗f⁰⊗k⁰⊗x7−→Be_(p,v)^f⊗f0Ae^k⊗k_v ⁰(x).

Proof.

• The left K_Σ^0.5_v

1- and K_Σ^0.5_v

2 -actions act as the identity on all tensor factors ofH except on Z_v1 and Z_v2, respectively. It is thus clear that they commute for v₁ 6=v₂.

The right K_Σ^0.5

v1- and K_Σ^0.5

v2-actions only have a common tensor factor on which they do not act by the identity for every edge e ∈ Σ¹ that joins the vertices v₁ and v₂. Such an edge is directed away from one of the vertices and directed towards the other. Hence, the action for one of the vertices comes from left multiplication ofK_e and the other one from right multiplication, so they commute.

2 Defects and boundaries in Kitaev models

• The left H_p^∗₁- andH_p^∗₂-actions act as the identity on all tensor factors ofH except on Z_v1 and Z_v2, respectively. It is thus clear that they commute for v₁ 6= v₂. In the remaining case v₁ =v₂ =:v, H_p^∗

1 and H_p^∗

2 are commuting subalgebras in C_v. Since their actions on Z_v are by Definition 12 the restrictions of the C_v-action that Z_v comes with, they must therefore commute.

The right H_p^∗

1- and H_p^∗

2-actions only have a common tensor factor on which they do not act by the identity for every vertex v ∈ Σ⁰ and for every edge e ∈ Σ¹ that lies in the boundaries of both plaquettes p₁ and p₂. For any such vertex v, the two actions come from the (H_Σ^∗sit

v =K_Σ^0.5

v )-action on Z_v restricted to the two subalgebras H_p^∗

1 and

H_p^∗₂, respectively. These subalgebras commute insideH_Σ^∗sit

v =K_Σ^0.5_v , therefore showing the claim.

• The left K_Σ^0.5_v - and H_p^∗-actions on H are simply the restrictions of the left C_v-action on Z_v toK_Σ^0.5

v andH_p^∗, respectively, and the identity on all other tensor factors ofH. Hence, by construction they satisfy the commutation relations of the crossed product algebra H_p^∗=K_Σ^0.5_v ⊆C_v, see also (2.5).

The rightK_Σ^0.5

v - andH_p^∗-actions onHare non-trivial only on the tensor factorsN

e∈Σ¹_vK_e^∗ and (N

e∈Σ¹_pK_e^∗)⊗(N

w∈Σ⁰_p\{v}Z_w), respectively. We can therefore restrict our attention to the vector space (N

e∈Σ¹_v∪Σ¹_pK_e^∗)⊗(N

w∈Σ⁰_p\{v}Z_w), on which K_Σ^0.5_v and H_p^∗ act from the right.

For convenience, for the remainder of the proof we now switch to the dual vector space (N

e∈Σ¹_v∪Σ¹_pK_e)⊗(N

w∈Σ⁰_p\{v}Z_w^∗), with the corresponding left actions of K_Σ^0.5_v and H_p^∗. With the notation of Subsection 2.1.2, let e_p, e⁰_p ∈ Σ^0.5_v be the half-edges at v on the two sides of the site p ∈ Σ^sit_v , with signs ε := ε(e_p) and ε⁰ := ε(e⁰_p). The K_Σ^0.5_v - and H_p^∗-actions only overlap on the tensor factors (K_e)e∈Σ¹_v∩Σ¹_p corresponding to the edges underlying the half-edges e_p, e⁰_p ∈ Σ^0.5_v . Due to our regularity assumption on the cell decomposition, the half-edges e_p and e⁰_p have distinct underlying edges. Then the action of K_Σ^0.5_v = (K_e^ε_p ⊗K_e^ε0⁰

p)⊗N

e∈Σ^0.5_v \{ep,e⁰_p}Ke^ε(e) on N

e∈Σ¹_vK_e, which is a tensor product of algebras, decomposes into a tensor product of the action of K_e^ε_p⊗K_e^ε0⁰

p on K_ep⊗K_e⁰_p and the action ofN

e∈Σ^0.5_v \{ep,e⁰_p}Ke^ε(e) onN

e∈Σ¹_v\{ep,e⁰_p}K_e. On the latter vector space,H_p^∗ does not act non-trivially by our regularity assumption on the cell decomposition. Hence, it remains to consider the interactions of the left actions of K_e^ε

p ⊗K_e^ε0⁰

p and H_p^∗ on the vector space K_ep ⊗K_e⁰_p ⊗(N

e∈Σ¹_p\{ep,e⁰_p}K_e)⊗(N

w∈Σ⁰_p\{v}Z_w^∗). We abbreviate by V :=

(N

e∈Σ¹_p\{ep,e⁰_p}K_e)⊗(N

w∈Σ⁰_p\{v}Z_w^∗)the tensor factor on which onlyH_p^∗ acts non-trivially.

Furthermore, without loss of generality, we write the left H_p^∗-action on V in terms of the Sweedler notation for the corresponding right H_p-coaction, V →V ⊗H_p, v 7→v₍₀₎⊗v₍₁₎:

H_p^∗⊗V −→V, v7−→f.v=:f(v₍₁₎)v₍₀₎. Finally, it is left to analyze the interaction between the H_p^∗-action

H_p^∗⊗K_ep⊗K_e⁰_p⊗V −→K_ep⊗K_e⁰_p⊗V,

f⊗x⊗x⁰⊗v 7−→f₍₃₎.x⊗f₍₁₎.x⁰⊗f₍₂₎.v

=f

x^0hε_(ε⁰0ⁱ)v₍₁₎x^h−εi_(−ε)

x₍₀₎⊗x⁰₍₀₎⊗v₍₀₎, 28

2.2 Construction of a Kitaev model with defects and the(K_e^ε_p⊗K_e^ε0⁰

p)-action (K_e^ε_p⊗K_e^ε00

p)⊗Kep⊗Ke⁰_p⊗V −→Kep⊗Ke⁰_p⊗V, a⊗a⁰⊗x⊗x⁰⊗v 7−→a.x⊗a⁰.x⁰ ⊗v

(a·_εx)⊗(a⁰·_ε⁰x⁰)⊗v, where ·ε and ·ε⁰ denote the multiplication in K_e^ε_p and K_e^ε0⁰

p, respectively, that is a·_εx:=

(ax, ε = +1, xa, ε =−1.

It remains to show that that these actions satisfy the straightening formula f(a^0h−_(ε0^ε)⁰ⁱ·?·a^hεi_(−ε)).(a₍₀₎⊗a⁰₍₀₎).(x⊗x⁰⊗v) = (a⊗a⁰).f.(x⊗x⁰⊗v), for all f ∈H_p^∗, a⊗a⁰ ∈K_e^ε_p⊗K_e^ε0⁰

p and x⊗x⁰⊗v ∈K_ep⊗K_e⁰_p⊗V. Indeed, the following calculation, which is analogous to the calculation in the proof of [BMCA, Theorem 1] but more general and at the same time shorter, verifies this.

f

a^0h−ε_(ε0)⁰ⁱ·?·a^hεi_(−ε)

.(a₍₀₎⊗a⁰₍₀₎).(x⊗x⁰⊗v)

=f

a^0h−ε_(ε0)⁰ⁱ·?·a^hεi_(−ε)

.((a₍₀₎·_εx)⊗(a⁰₍₀₎·_ε⁰ x⁰)⊗v)

=f

a^0h−_(2ε^ε0⁰)ⁱ·(a⁰₍₀₎·_ε⁰x⁰)^hε_(ε⁰0ⁱ)·v₍₁₎·(a₍₀₎·_εx)^h−εi_(−ε)·a^hεi_(−2ε) ((a₍₀₎·_εx)₍₀₎⊗(a⁰₍₀₎·_ε⁰ x⁰)₍₀₎⊗v₍₀₎)

=f

a^0h−ε_(2ε0⁰)ⁱ·a^0hε_(ε⁰0ⁱ)·x^0hε_(ε⁰0ⁱ)·v₍₁₎·x^h−εi_(−ε)·a^h−εi_(−ε)·a^hεi_(−2ε) ((a₍₀₎·_εx₍₀₎)⊗(a⁰₍₀₎·_ε⁰ x⁰₍₀₎)⊗v₍₀₎)

=f

x^0hε_(ε⁰0ⁱ)·v₍₁₎·x^h−εi_(−ε)

((a·_εx₍₀₎)⊗(a⁰·_ε⁰ x⁰₍₀₎)⊗v₍₀₎)

= (a⊗a⁰).

f

x^0hε_(ε⁰0ⁱ)·v₍₁₎·x^h−εi_(−ε)

(x₍₀₎⊗x⁰₍₀₎⊗v₍₀₎)

= (a⊗a⁰). f.(x⊗x⁰⊗v) . This proves thatH_p^∗ and K_e^ε_p⊗K_e^ε0⁰

p together give a representation of the crossed product algebraH_p^∗=(K_e^ε_p⊗K_e^ε0⁰

p), as claimed.

Remark 14. Taking all sites p∈ Σ^sit_v around a given vertex v ∈ Σ⁰ together, we thus get, due to Theorem 13, on H a bimodule structure over the vertex algebra C_v. It is remarkable that this makes the crossed product algebra structure on C_v show up naturally – analogous to the appearance of the algebra structure of the Drinfeld double in the commutation relation of the vertex and plaqette actions in the standard Kitaev model without defects.

2 Defects and boundaries in Kitaev models

2.2.2 Towards local projectors: Symmetric separability idempotents

Im Dokument Hopf-algebraic structures inspired by Kitaev models : defects, comodule algebras and idempotents for non-semisimple Hopfalgebras (Seite 34-40)