On braided tensor categories of type<i>BCD</i>

Abstract. Let U q (g) be the quantum supergroup of gl m|n or the modified quantum supergroup of osp m|2n over the field of rational functions in q, and let V q be the natural module for U q (g). There exists a unique tensor functor, associated with V q , from the category of ribbon graphs to the category of finite dimensional representations of U q (g), which preserves ribbon category structures. We show that this functor is full in the cases g = gl m|n or osp 2ℓ+1|2n . For g = osp 2ℓ|2n , we show that the space Hom Uq(g) (V ⊗r q , V ⊗s q ) is spanned by images of ribbon graphs if r + s < 2ℓ(2n + 1). The proofs involve an equivalence of module categories for two versions of the quantisation of U(g).

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“…The theorem was already implicitly present in the study of braided tensor categories of type BCD in [48].…”

Section: 4mentioning

confidence: 91%

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Lehrer

Zhang

2016

Commun. Math. Phys.

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“…This gives Gr(O) a much simpler structure: the N k ij are totally symmetric in the i, j and k. Lemma 2.4 has a stronger consequence in the self-dual case (see [TuW2]): Corollary 2.9. Suppose X ⊗2 ∼ = i X i in a self-dual semisimple ribbon category O, and we have a basis of mutually annihilating idempotents p j ∈ End(X ⊗2 ) so that p j X ⊗2 ∼ = X j and X 1 ∼ = 1 1.…”

Section: Self-dual Categoriesmentioning

confidence: 98%

“…The key theorem we will use is the following special case of the main result in [TuW2] (Theorem 9.5):…”

Section: Dim V (X) = [−2k]mentioning

confidence: 99%

On a family of non-unitarizable ribbon categories

Rowell

2005

Math. Z.

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We consider several families of categories. The first are quotients of H. Andersen's tilting module categories for quantum groups of Lie type B at odd roots of unity. The second consists of categories of type BC constructed from idempotents in BMW -algebras. Our main result is to show that these families coincide as braided tensor categories using a recent theorem of Tuba and Wenzl. By appealing to similar results of Blanchet and Beliakova we obtain another interesting equivalence with these two families of categories and the quantum group categories of Lie type C at odd roots of unity. The morphism spaces in these categories can be equipped with a Hermitian form, and we are able to show that these categories are never unitary, and no braided tensor category sharing the Grothendieck semiring common to these families is unitarizable.

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“…. , are constructed using the well-known iterative relations (see, e.g., Lemma 7.2 in [33] or Sec. 2.3 in [8])…”

Section: Theorem 2 We Consider a Hecke-type Qm Algebra M(r F ) Genmentioning

confidence: 99%

Quantum matrix algebras of the GL(m|n) type: The structure and spectral parameterization of the characteristic subalgebra

2006

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We continue the study of quantum matrix algebras of the GL(m|n) type. We find three alternative forms of the Cayley-Hamilton identity; most importantly, this identity can be represented in a factored form. The factorization allows naturally dividing the spectrum of a quantum supermatrix into subsets of "even" and "odd" eigenvalues. This division leads to a parameterization of the characteristic subalgebra (the subalgebra of spectral invariants) in terms of supersymmetric polynomials in the eigenvalues of the quantum matrix. Our construction is based on two auxiliary results, which are independently interesting. First, we derive the multiplication rule for Schur functions s λ (M ) that form a linear basis of the characteristic subalgebra of a Hecke-type quantum matrix algebra; the structure constants in this basis coincide with the Littlewood-Richardson coefficients. Second, we prove a number of bilinear relations in the graded ring Λ of symmetric functions of countably many variables.

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On braided tensor categories of typeBCD

Cited by 60 publications

References 35 publications

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

On a family of non-unitarizable ribbon categories

Quantum matrix algebras of the GL(m|n) type: The structure and spectral parameterization of the characteristic subalgebra

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