Lectures on Tensor Categories and Modular Functors

Bakalov, Bojko; Kirillov, Alexander

doi:10.1090/ulect/021

Cited by 614 publications

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“…By [BK,3.1.21] and [ENO1,§10] we see that Lemma 3.7. The entries of the matrix S(M a , F ) are cyclotomic integers.…”

Section: The Matrices S(m a F )mentioning

confidence: 90%

Modular Categories, Crossed S-matrices, and Shintani Descent

Deshpande

2016

Int Math Res Notices

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Let C be a modular tensor category over an algebraically closed field k of characteristic 0. Then there is the ubiquitous notion of the S-matrix S(C ) associated with the modular category. The matrix S(C ) is a symmetric matrix, its entries are cyclotomic integers and the matrix (dim C ) − 1 2 · S(C ) is a unitary matrix. Here dim C ∈ k denotes the categorical dimension of C and it is a totally positive cyclotomic integer. Now suppose that we also have a modular autoequivalence F : C → C . In this paper, we will define and study the notion of a crossed S-matrix associated with the modular autoequivalence F . We will see that the crossed S-matrix occurs as a submatrix of the usual S-matrix of some "bigger" modular category and hence the entries of a crossed S-matrix are also cyclotomic integers. We will prove that the crossed S-matrix (normalized by the factor (dim C ) − 1 2 ) associated with any modular autoequivalence is a unitary matrix. We will also prove that the crossed S-matrix is essentially the "character table" of a certain semisimple commutative Frobenius k-algebra associated with the modular autoequivalence F . The motivation for most of our results comes from the theory of character sheaves on algebraic groups, where we expect that the transition matrices between irreducible characters and character sheaves can be obtained as certain crossed S-matrices. In the character theory of algebraic groups defined over finite fields, there is the notion of Shintani descent of Frobenius stable characters. We will define and study a categorical analogue of this notion of Shintani descent in the setting of modular categories.

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“…By [BK,3.1.21] and [ENO1,§10] we see that Lemma 3.7. The entries of the matrix S(M a , F ) are cyclotomic integers.…”

Section: The Matrices S(m a F )mentioning

confidence: 90%

Modular Categories, Crossed S-matrices, and Shintani Descent

Deshpande

2016

Int Math Res Notices

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“…By the same arguments as in [BK01], we have Proposition 23. The category C G is rigid, if and only if a V = 0 for all simple objects V .…”

Section: Lemma 22mentioning

confidence: 67%

“…The Yoneda lemma implies that this natural transformation comes from a natural isomorphism α A,B,C : (A⊗B)⊗C ∼ = → A⊗(B ⊗C). The arguments of [KP08,BK01] then imply that these isomorphisms satisfy the pentagon axiom.…”

Section: The Associativity Constraintsmentioning

confidence: 99%

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A geometric construction for permutation equivariant categories from modular functors

Barmeier

Schweigert

2011

Transformation Groups

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Let G be a finite group. Given a finite G-set X and a modular tensor category C, we construct a weak G-equivariant fusion category C X , called the permutation equivariant tensor category. The construction is geometric and uses the formalism of modular functors. As an application, we concretely work out a complete set of structure morphisms for Z/2-permutation equivariant categories, finishing thereby a program we initiated in [BFRS10].

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“…What is clear is that the basic interactions of the anyons can only be expressed in the categorical language. One needs a rather rich kind of categorical structure called a modular tensor category [46]. An expository account of this subject is given in [47].…”

Section: Topological Quantum Computingmentioning

confidence: 99%