Skein construction of idempotents in Birman-Murakami-Wenzl algebras

Beliakova, Anna; Blanchet, Christian

doi:10.1007/s002080100233

Cited by 35 publications

(80 citation statements)

References 20 publications

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“…If c has coordinates (i, j) ( i -th row, and j -th column), then the corresponding point in D 2 is j+i √ −1 n+1 . In [2] we have constructed minimal idempotentsỹ λ ∈ K λ . Let us recall their main properties in the generic case (i.e.…”

Section: Idempotentsmentioning

confidence: 99%

“…A representative set of simple objects is the infinite set Γ(B n ) , so that the category is not pre-modular. We have the following specialized formula for the quantum dimensions (see [2,Prop. 7.6] …”

Section: The Odd Orthogonal Casementioning

confidence: 99%

“…Our method is based on the skein-theoretical construction of idempotents in the Birman-Murakami-Wenzl (BMW) algebras given in [2]. This work follows the program of Turaev and Wenzl [19,20].…”

Section: Introductionmentioning

confidence: 99%

“…This includes Bruguières' modularization criterion, and an explicit description of a modularization functor for a modularizable pre-modular category whose transparent simple objects are invertible. In the second section we recall the main definitions and properties of the minimal idempotents in the BMW algebras constructed in [2]. In the third section we construct the completed BMW category and use it in order to define series of pre-modular categories.…”

Section: Introductionmentioning

confidence: 99%

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Modular categories of types B, C and D

Beliakova¹,

Blanchet²

2001

Comment. Math. Helv.

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Abstract.We construct four series of modular categories from the two-variable Kauffman polynomial, without use of the representation theory of quantum groups at roots of unity. The specializations of this polynomial corresponding to quantum groups of types B, C and D produce series of pre-modular categories. One of them turns out to be modular and three others satisfy Bruguières' modularization criterion. For these four series we compute the Verlinde formulas, and discuss spin and cohomological refinements. Mathematics Subject Classification (2000). 57M25, 57R56.

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Section: Idempotentsmentioning

confidence: 99%

Section: The Odd Orthogonal Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Modular categories of types B, C and D

Beliakova¹,

Blanchet²

2001

Comment. Math. Helv.

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“…Here whereỹ λ is the 0-graded minimal idempotent defined in [5],P + is the idempotent p Analogously, the trace of (id V ⊗P + )(ỹ (2) ⊗ id S ) is nonzero. Therefore, p λS µ is a composition of non-negligible morphisms.…”

Section: Remark 22mentioning

confidence: 99%

Geometric construction of spinors in orthogonal modular categories

Beliakova

2003

Algebr. Geom. Topol.

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A geometric construction of Z 2 -graded odd and even orthogonal modular categories is given. Their 0-graded parts coincide with categories previously obtained by Blanchet and the author from the category of tangles modulo the Kauffman skein relations. Quantum dimensions and twist coefficients of 1-graded simple objects (spinors) are calculated. We show that invariants coming from our odd and even orthogonal modular categories admit spin and Z 2 -cohomological refinements, respectively. The relation with the quantum group approach is discussed.

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Affine and degenerate affine BMW algebras: actions on tensor space

Daugherty

Ram

Virk

2012

Sel. Math. New Ser.

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The degenerate affine and affine BMW algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper the theory is unified by treating the orthogonal and symplectic cases simultaneously; we make an exact parallel between the degenerate affine and affine cases via a new algebra which takes the role of the affine braid group for the degenerate setting. A main result of this paper is an identification of the centers of the affine and degenerate affine BMW algebras in terms of rings of symmetric functions which satisfy a "cancellation property" or "wheel condition" (in the degenerate case, a reformulation of a result of Nazarov). Miraculously, these same rings also arise in Schubert calculus, as the cohomology and K-theory of isotropic Grassmanians and symplectic loop Grassmanians. We also establish new intertwiner-like identities which, when projected to the center, produce the recursions for central elements given previously by Nazarov for degenerate affine BMW algebras, and by Beliakova-Blanchet for affine BMW algebras.

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Skein construction of idempotents in Birman-Murakami-Wenzl algebras

Cited by 35 publications

References 20 publications

Modular categories of types B, C and D

Modular categories of types B, C and D

Geometric construction of spinors in orthogonal modular categories

Affine and degenerate affine BMW algebras: actions on tensor space

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