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For [Formula: see text] a commutative group, we give a purely Hopf [Formula: see text]-coalgebra construction of [Formula: see text]-colored [Formula: see text]-manifolds invariants using the notion of modified integral. 

In [M. De Renzi, A. Gainutdinov, N. Geer, B. Patureau-Mirand and I. Runkel, 3-dimensional TQFTs from non-semisimple modular categories, preprint (2019), arXiv:1912.02063[math.GT]], we constructed 3-dimensional topological quantum field theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category [Formula: see text]. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of the above reference are equivalent to those obtained by Lyubashenko via generators and relations in [V. Lyubashenko, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172(3) (1995) 467–516, arXiv:hep-th/9405167]. Finally, we show that, when [Formula: see text] is the category of finite-dimensional representations of the small quantum group of [Formula: see text], the action of all Dehn twists for surfaces without marked points has infinite order. 

Abstract We develop the categorical context for defining Hermitian non‐semisimple topological quantum field theories (TQFTs). We prove that relative Hermitian modular categories give rise to modified Hermitian Witten–Reshetikhin–Turaev TQFTs and provide numerous examples of these structures coming from the representation theory of quantum groups and quantum superalgebras. The Hermitian theory developed here for the modified Turaev–Viro TQFT is applied to define new pseudo‐Hermitian topological phases that can be considered as non‐semisimple analogs of Levin–Wen models.