arXiv · 2603.25141
These notes are an introduction to the theory of quantum symmetries of finite and infinite sets, graphs, and locally compact spaces.
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Christian Voigt. 2026-03-26. An introduction to quantum symmetries. https://arxiv.org/abs/2603.25141
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We introduce a new family of graded 2-categories generalizing the 2-quantum groups introduced by Khovanov, Lauda and Rouquier. We use them to categorify quasi-split iquantum groups in all symmetric types.
We give complete conjugacy classes of modular group SL(2,Z). Particularly, the conjugacy classes of hyperbolic elements are decided by the proper equivalence classes of indefinite forms, and we give an example. Finally, we describe the coset representatives of centralizer of S, ST, T and hyperbolic elements of SL(2,Z) by regular continued fraction. In conclusion, most Nichols algebras over modular group are infinite-dimensional except Proposition 4.10.
We develop a bornological version of Costello and Gwilliam's procedure for extracting vertex algebras from suitable prefactorization algebras on the complex plane. Using bornological complex analysis, we remove the discreteness condition imposed in their extraction theorem. We then construct, from a suitable Lie conformal algebra, a prefactorization algebra to which this extraction procedure applies, and prove that the resulting vertex algebra is isomorphic to the enveloping vertex algebra of the original Lie conformal algebra. Our construction uses a factorization envelope and extends the construction of Costello--Gwilliam in the affine vertex algebra case and that of Williams in the Virasoro vertex algebra case. Moreover, a super analogue yields new prefactorization algebras corresponding to vertex superalgebras, such as the Neveu--Schwarz vertex superalgebra, the $N=2$ vertex superalgebra, and the $N=4$ vertex superalgebra.