A Family of New Borel Subalgebras of Quantum Groups

Lentner, Simon; Vocke, Karolina

doi:10.1007/s10468-020-09956-y

Cited by 2 publications

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References 17 publications

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“…However there are more examples of right coideal subalgebras, in particular [LV21] studies those which retain the properties that all simple modules are 1-dimensional, and it can be checked which of these give rise to (co)commutative (co)algebras. For example the Example 2.12 in loc.…”

Section: Examples From Quantum Groupsmentioning

confidence: 99%

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

Creutzig

Ridout

Rupert

2023

Commun. Math. Phys.

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Let U be a braided tensor category, typically unknown, complicated and in particular non-semisimple. We characterize U under the assumption that there exists a commutative algebra A in U with certain properties: Let C be the category of local A-modules in U and B the category of A-modules in U, which are in our set-up usually much simpler categories than U. Then we can characterize U as a relative Drinfeld center Z C (B) and B as representations of a certain Hopf algebra inside C.In particular this allows us to reduce braided tensor equivalences to the knowledge of abelian equivalences, e.g. if we already know that U is abelian equivalent to the category of modules of some quantum group U q or some generalization thereof, and if C is braided equivalent to a category of graded vector spaces, and if A has a certain form, then we already obtain a braided tensor equivalence between U and Rep(U q ).A main application of our theory is to prove logarithmic Kazhdan-Lusztig correspondences, that is, equivalences of braided tensor categories of representations of vertex algebras and of quantum groups. Here, the algebra A and the corresponding category C are a-priori given by a free-field realization of the vertex algebra and by a Nichols algebra. We illustrate this in those examples where the representation theory of the vertex algebra is well enough understood. In particular we prove the conjectured correspondences between the singlet vertex algebra M(p) and the unrolled small quantum groups of sl 2 at 2p-th root of unity. Another new example is the Kazhdan-Lusztig correspondence between the affine vertex algebra of gl 1|1 and an unrolled quantum group of gl 1|1 . * this should be q = exp πi r ∨ (k+h ∨ ) with h ∨ the dual Coxeter number of g and r ∨ the lacing number of the even subalgebra.

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Section: Examples From Quantum Groupsmentioning

confidence: 99%

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

Creutzig

Ridout

Rupert

2023

Commun. Math. Phys.

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A Family of New Borel Subalgebras of Quantum Groups

Lentner

Vocke

2020

Algebr Represent Theor

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For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra.Besides the positive part of the quantum group and its reflections, we find new unfamiliar Borel subalgebras, for example ones containing copies of the quantum Weyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modules and prove among others that they have all irreducible finite-dimensional modules as quotients. We then give structural results using the graded algebra, which in particular leads to a conjectural formula for all triangular Borel subalgebras, which we partly prove. As examples, we determine all Borel subalgebras of U q (sl 2 ) and U q (sl 3 ) and discuss the induced modules.

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A Family of New Borel Subalgebras of Quantum Groups

Cited by 2 publications

References 17 publications

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

A Family of New Borel Subalgebras of Quantum Groups

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