Representation theory of the two-boundary Temperley-Lieb algebra Zajj Daugherty (Joint work in progress with Arun Ram) September 10, 2014
Temperley-Lieb algebras The Temperley-Lieb algebra TL k ( q ) is the algebra of non-crossing pairings on 2 k vertices 1 2 3 4 k 1 2 3 4 k with multiplication given by stacking diagrams, subject to the relation = q + q − 1
Temperley-Lieb algebras The Temperley-Lieb algebra TL k ( q ) is the algebra of non-crossing pairings on 2 k vertices 1 2 3 4 k 1 2 3 4 k with multiplication given by stacking diagrams, subject to the relation = q + q − 1 Multiplication:
Temperley-Lieb algebras The Temperley-Lieb algebra TL k ( q ) is the algebra of non-crossing pairings on 2 k vertices 1 2 3 4 k 1 2 3 4 k with multiplication given by stacking diagrams, subject to the relation = q + q − 1 Multiplication:
Temperley-Lieb algebras The Temperley-Lieb algebra TL k ( q ) is the algebra of non-crossing pairings on 2 k vertices 1 2 3 4 k 1 2 3 4 k with multiplication given by stacking diagrams, subject to the relation = q + q − 1 Multiplication:
Temperley-Lieb algebras The Temperley-Lieb algebra TL k ( q ) is the algebra of non-crossing pairings on 2 k vertices 1 2 3 4 k 1 2 3 4 k with multiplication given by stacking diagrams, subject to the relation = q + q − 1 Multiplication: ∗ ( q + q − 1 ) 2 =
Temperley-Lieb algebras The one-boundary Temperley-Lieb algebra TL (1) k ( q, z 0 ) is the algebra of one-walled non-crossing pairings on 2 k vertices 1 2 3 4 k 1 2 3 4 k with multiplication given by stacking diagrams, subject to the relations = q + q − 1 and = 1 or = z 0 . if even # if odd # connections connections below below
Odd/even relations The algebra TL (1) k ( q, z 0 ) is generated by i 1 e i = and e 0 = i 1 for i = 1 , . . . , k − 1
Odd/even relations The algebra TL (1) k ( q, z 0 ) is generated by i 1 e i = and e 0 = i 1 for i = 1 , . . . , k − 1 , with relations e i e i ± 1 e i = e i for i ≥ 1 =
Odd/even relations The algebra TL (1) k ( q, z 0 ) is generated by i 1 e i = and e 0 = i 1 for i = 1 , . . . , k − 1 , with relations e i e i ± 1 e i = e i for i ≥ 1 = or =
Odd/even relations The algebra TL (1) k ( q, z 0 ) is generated by i 1 e i = and e 0 = i 1 for i = 1 , . . . , k − 1 , with relations e i e i ± 1 e i = e i for i ≥ 1 = or = e 2 = ( q + q − 1 ) i = ae i
Odd/even relations The algebra TL (1) k ( q, z 0 ) is generated by i 1 e i = and e 0 = i 1 for i = 1 , . . . , k − 1 , with relations e i e i ± 1 e i = e i for i ≥ 1 = or = e 2 = ( q + q − 1 ) i = ae i or = z 0
Odd/even relations The algebra TL (1) k ( q, z 0 ) is generated by i 1 e i = and e 0 = i 1 for i = 1 , . . . , k − 1 , with relations e i e i ± 1 e i = e i for i ≥ 1 = or = e 2 = ( q + q − 1 ) i = ae i or = z 0 Side loops are resolved with a 1 or a z 0 depending on whether there are an even or odd number of connections below their lowest point.
Temperley-Lieb algebras The one-boundary Temperley-Lieb algebra TL (1) k ( q, z 0 ) is the algebra of one-walled non-crossing pairings on 2 k vertices 1 2 3 4 k 1 2 3 4 k with multiplication given by stacking diagrams, subject to the relations = q + q − 1 and = 1 or = z 0 . if even # if odd # connections connections below below
Our main object: two-boundary Temperley-Lieb algebra Nienhuis, De Gier, Batchelor (2004): The two-boundary Temperley-Lieb algebra TL (2) k ( q, z 0 , z k ) = T k is the algebra of two-walled non-crossing pairings on 2 k vertices 1 2 3 4 k 1 2 3 4 k so that each wall always has an even number of connections, with multiplication given by stacking diagrams, subject to the relations = q + q − 1 and = = 1 or = z 0 , = z k . if even # if odd # connections connections below below
Our main object: two-boundary Temperley-Lieb algebra Nienhuis, De Gier, Batchelor (2004): The two-boundary Temperley-Lieb algebra TL (2) k ( q, z 0 , z k ) = T k is the algebra of two-walled non-crossing pairings on 2 k vertices 1 2 3 4 k 1 2 3 4 k so that each wall always has an even number of connections, with multiplication given by stacking diagrams, subject to the relations = q + q − 1 and = = 1 or = z 0 , = z k . if even # if odd # connections connections below below
Our main object: two-boundary Temperley-Lieb algebra Nienhuis, De Gier, Batchelor (2004): The two-boundary Temperley-Lieb algebra TL (2) k ( q, z 0 , z k ) = T k is the algebra of two-walled non-crossing pairings on 2 k vertices 1 2 3 4 k e i = e 0 = e k = 1 2 3 4 k so that each wall always has an even number of connections, with multiplication given by stacking diagrams, subject to the relations = q + q − 1 and = = 1 or = z 0 , = z k . if even # if odd # connections connections below below
Our main object: two-boundary Temperley-Lieb algebra TL k is finite-dimensional ( n th Catalan number)
Our main object: two-boundary Temperley-Lieb algebra TL k is finite-dimensional ( n th Catalan number) TL (1) is finite-dimensional k
Our main object: two-boundary Temperley-Lieb algebra TL k is finite-dimensional ( n th Catalan number) TL (1) is finite-dimensional k TL (2) = T k is infinite-dimensional! k 2 ℓ
Our main object: two-boundary Temperley-Lieb algebra TL k is finite-dimensional ( n th Catalan number) TL (1) is finite-dimensional k TL (2) = T k is infinite-dimensional! k 2 ℓ de Gier, Nichols (2008) : Explored representation theory of T k . 1 Take quotients giving = z to get finite-dimensional algebras. 2 Establish connection to the affine Hecke algebras of type A and C to facilitate calculations. 3 Use diagrammatics and an action on ( C 2 ) ⊗ k to help classify representations in quotient (most modules are 2 k dim’l; some split).
Our main object: two-boundary Temperley-Lieb algebra TL k is finite-dimensional ( n th Catalan number) SWD � TL (1) is finite-dimensional SWD � k TL (2) = T k is infinite-dimensional! k 2 ℓ de Gier, Nichols (2008) : Explored representation theory of T k . 1 Take quotients giving = z to get finite-dimensional algebras. SWD � � �� � 2 Establish connection to the affine Hecke algebras of type A and C to facilitate calculations. 3 Use diagrammatics and an action on ( C 2 ) ⊗ k to help classify representations in quotient (most modules are 2 k dim’l; some split).
Quantum groups and braids Fix q ∈ C ∗ . Let U = U q g be the Drinfel’d-Jimbo quantum group associated to a reductive Lie algebra g . Let V, M be U -modules. Then U ⊗ U has invertible R = � R R 1 ⊗ R 2 that yields a map M ⊗ V ˇ R V M : V ⊗ M − → M ⊗ V � v ⊗ m �− → R 1 m ⊗ R 2 v R V ⊗ M that (1) satisfies braid relations, and (2) commutes with the action of U q g .
Quantum groups and braids Fix q ∈ C ∗ . Let U = U q g be the Drinfel’d-Jimbo quantum group associated to a reductive Lie algebra g . Let V, M be U -modules. Then U ⊗ U has invertible R = � R R 1 ⊗ R 2 that yields a map M ⊗ V ˇ R V M : V ⊗ M − → M ⊗ V � v ⊗ m �− → R 1 m ⊗ R 2 v R V ⊗ M that (1) satisfies braid relations, and (2) commutes with the action of U q g . The braid group shares a commuting action with U q g on V ⊗ k : ⊗ ⊗ ⊗ ⊗ V V V V V ⊗ ⊗ ⊗ ⊗ V V V V V
Quantum groups and braids Fix q ∈ C ∗ . Let U = U q g be the Drinfel’d-Jimbo quantum group associated to a reductive Lie algebra g . Let V, M be U -modules. Then U ⊗ U has invertible R = � R R 1 ⊗ R 2 that yields a map M ⊗ V ˇ R V M : V ⊗ M − → M ⊗ V � v ⊗ m �− → R 1 m ⊗ R 2 v R V ⊗ M that (1) satisfies braid relations, and (2) commutes with the action of U q g . The one-boundary/affine braid group shares a commuting action with U q g on N ⊗ V ⊗ k : ⊗ ⊗ ⊗ ⊗ N ⊗ V V V V V Around the pole: N ⊗ V = ˇ R NV ˇ R V N N ⊗ V N ⊗ ⊗ ⊗ ⊗ ⊗ V V V V V
Quantum groups and braids Fix q ∈ C ∗ . Let U = U q g be the Drinfel’d-Jimbo quantum group associated to a reductive Lie algebra g . Let V, M be U -modules. Then U ⊗ U has invertible R = � R R 1 ⊗ R 2 that yields a map M ⊗ V ˇ R V M : V ⊗ M − → M ⊗ V � v ⊗ m �− → R 1 m ⊗ R 2 v R V ⊗ M that (1) satisfies braid relations, and (2) commutes with the action of U q g . The two-boundary braid group shares a commuting action with U q g on N ⊗ V ⊗ k ⊗ M : ⊗ ⊗ ⊗ ⊗ N ⊗ V V V V V ⊗ M Around the pole: N ⊗ V = ˇ R NV ˇ R V N N ⊗ V N ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ M V V V V V
Affine type C Hecke algebra and two-boundary braids k − 2 k − 1 0 1 2 3 4 k · · · Fix constants t 0 , t k , and t = t 1 = · · · = t k − 1 . The affine Hecke algebra of type C, H k , is generated by T 0 , T 1 , . . . , T k with relations j i 2 if i j T i T j . . . = T j T i . . . where m i,j = 3 if � �� � � �� � m i,j factors m i,j factors j i 4 if i = ( t 1 / 2 − t − 1 / 2 and T 2 ) T i + 1 . i i
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