In deze scriptie staan representaties van de Temperley-Lieb algebra centraal. Er zijn namelijk nog veel meer representaties te bedenken, maar die zijn vaak een stuk moeilijker. Het blijkt in veel gevallen mogeljk te zijn om “bouwstenen” te vinden waar alle representaties van gemaakt zijn.
Ook bekijken we variaties op te Temperley-Lieb algebra en representaties daarvan. Tot slot laten we zien hoe de Temperley-Lieb algebra en haar representaties gebruikt worden in (theoretische) natuurkunde.
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Chapter A
Miscellaneous
A.1
The central element Jn
A.1 Proposition. The element Jn acts on Vn,p as the identity times gn,p = (−1)n(q(n−2p+1)+
q−(n−2p+1)).
We remark that gn,p = (−1)nfn,p, where fn,p denotes the action of Fn on Vn,p given in
proposition A.2 of [26].
Proof. Any central element acts as a scalar times the identity onVn,p by proposition 4.6. So
we may compute the action of Jn on any(n, p)-link state. Let zp denote the element inVn,p
with simple links at 1, 3, . . . , 2p− 1. Then by using the Reidermeister moves we find
Jnzp= 1 2 2p− 1 2p =
Figure A.1: Using Reidermeister 2 on Jnzp.
Now let us concentrate on the lower n− 2p levels. We focus on the diagram left in figure ... and evaluate the top right tile
Jnzp= 2p+ 1 2p+ 2 n− 1 n = c ⋅ + c−1⋅
Figure A.2: The last n− 2p levels.
Focus on the first diagram on the right-hand side of figure A.2 and consider the crossing below the one we just changed (the right one on level 2p+ 2). Filling in the tile results in an extra link (use Reidermeister 2 again) thus giving coefficient 0 inVn,p. Hence only the tile
results in a nonzero coefficient. We obtain the equality in figure A.3. We can recycle the argument to fill in all of the right column.
c⋅ = c2⋅ = cn−2p⋅
Figure A.3: Expanding the right column of the diagram.
Now have a look at the left column. In order to not close defects (making the diagram 0), there is one possibility for the top left tile, and, using the same argument, for all tiles except the lowest one in the column. This is depicted in figure A.4
cn−2p+1⋅ = c2(n−2p)−1⋅ = c2(n−2p)⋅ + c2(n−2p)−2⋅
Figure A.4: Expanding the left column of the diagram.
We end with the factor βc2n−4p+ c2n−4p−2. Recall c2= −q and c2+ c−2= −β to find
βc2(n−2p)+ c2(n−2p)−2= (−c2− c−2)c2(n−2p)+ c2(n−2p)−2= −c2⋅ c2(n−2p)= (−1)nqn−2p+1. In a similar way, we may treat the right diagram on the right-hand side of figure A.1 to find a factor(−1)nq−(n−2p+1). Thus the total factor equals g
n,p= (−1)n(qn−2p+1+ q−(n−2p+1)), as
desired.
A.2
Preliminary representation theory
A.2 Definitions. Recall that for an associative algebra A a complex vector space V is called a
representationof A or A-module if there exists a homomorphism of algebras ρ∶ A → End(V ), i.e. a linear map ρ with ρ(1) = id, the identity map on V , and ρ(ab) = ρ(a)ρ(b). The element ρ(a)(v) ∈ V is often denoted ρ(a)v or av for short.
A subspace W⊆ V is called a subrepresentation if it is invariant under the operators ρ(a) ∶ V → V . A representation is said to be irreducible if it has no proper subrepresentations and
indecomposibleif it is not isomorphic to the direct sum of two nonzero subrepresentations. A semisimple or completely reducible representation is a representation that is the direct sum of irreducible representations.
A homomorphism or intertwining operator (or intertwiner for short) of representations is a linear map φ between A-modules V and W such that aφ(v) = φ(av) for all a ∈ A, v ∈ V . It is called an isomorphism if in addition φ is bijective.
A.2. Preliminary representation theory 93
A.3 Theorem(Wedderburn). Let A be a complex, finite-dimensional, semisimple, associative algebra with a complete set of irreduciblesL1, . . .Lr. Then
A≅
r
⊕
i=1(dim L i)Li.
On the other hand,
A.4 Proposition. Let A be a complex, finite-dimensional, associative algebra with complete set of
irreduciblesL1, . . .Lr. View A as module over itself and suppose
A≅ r ⊕ i=1(dim L i)Li. Then A is semisimple.
A.5 Theorem. Let A be a complex, finite-dimensional, associative algebra. Let {Pi} be the set of
indecomposables and{Li} be the set of irreducibles. There is a bijective correspondence between the set.
LetLi bet the irreducible quotient ofPi. If r is the common cardinality of these sets, then the regular
representation decomposes as A≅ r ⊕ i=0(dim L i)Pi.
A.6 Proposition. (Frobenius reciprocity). Let B ⊂ A be two finite-dimensional associative algebras
over C. Let M be a B-module and N be an A-module. Then, the following isomorphism between vector spaces of module homomorphisms holds:
HomA(M↑, N) ≅ HomB(M, N↓).
A.7 Proposition. SupposeM1,M2,M3are B-modules and
0 M1 M2 M3 0
is a short exact sequence. Let A be an algebra such that B⊂ A and Mi↑ = A ⊗BMi, then
M1↑ M2↑ M3↑ 0