We study projective unitary (co)representations of compact quantum groups and the associated second cohomology theory. We introduce left/right/bi/strongly projective corepresentations and study them in details. In particular, we prove that given any compact quantum group $\q$, there are compact quantum groups $\tilde{\q_l}, \tilde{\q_r}, {\tilde \q}_{bi}, {\tilde \q}_{stp}$, each of which contains $\q$ as a Woronowicz subalgebra and every left/right/bi/strongly projective unitary corepresentation of $\q$ lifts to a linear corepresentation of these quantum groups respectively. We observe that the strongly projective corepresentations are associated with the second invariant ($S^1$-valued) cohomology $H^2_{uinv}(\cdot)$ of the quantum group. We define a suitable analogue of normalizer of a compact quantum group in a bigger compact quantum group and using this, associate a canonical discrete group $Γ_\q$ to a compact quantum group $\q$ which is an alternative generalization of the second group cohomology and we show by an example that $Γ_\q$ in general may be different from $H^2_{uinv}(\q,S^1) $.
Symmetry plays a major role in both mathematics and physics. This triggered a flurry of research in the theory of groups and their representations, both in analytic and algebraic frameworks. In the formalism of quantum mechanics in terms of operators on Hilbert spaces, it is often natural to consider symmetry as a map on the level of rays, that is on the set of unit vectors up to scalar multiplication. This leads to consideration of projective representation of the symmetry group. Mathematically, the theory of projective representation of group is closely related with group extension and cohomology theory [Bro94].
Quantum groups and more generally the theory of rigid tensor categories have generalized the classical group symmetry in mathematical physiscs. Beginning with the algebraic formulation due to Drinfield and Jimbo ([Dri89], [Dri85], [Jim85]) which was motivated by questions in physics related to the solution of quantum Yang-Baxter equations, the theory of quantum groups and Hopf algebras have traversed a long way, with various analytic formalism due to a number of mathematician, most notably Woronowicz [Wor87], Vaes-Kusterman [KV99], [KV00], [KV03] Van Daele [VD94], [VDVK94], [VD96], [MVD98], [VDW96].
In physics, there is a lot of interaction between the mathematical theory of quantum groups and tensor categories with the emerging field of topological states of matter and quantum computation [CGLW13,Che16].
This makes it a natural question whether one can extend the classical theory of projective representation of group to the realm of quantum groups. Substantial work in this direction have already been done by Kenny De Commer([DCMN24], [DC11b], [DC11a], [DC09], [DCY15], [DCY12]), Sergey Neshveyev([NT11a]), [NT12]), [NY16]), Lars Tuset( [NT11b]), Makoto Yamashita( [NY16]). The goal of present article is to contribute to understanding of projective (co)representation of compact quantum groups and study the associated cohomology. Some of the main results obtained by us concern extension of projective corepresentations of a given (compact) quantum group to (linear) corepresentations of a bigger quantum group . Indeed, using Tannaka-Krein reconstruction theorem, for a given compact quantum group, we prove existence of a possibly larger compact quantum group such that any unitary projective corepresentation of the original quantum group lifts to a unitary corepresentation of the bigger quantum group. In fact, we carry out such enveloping construction for various types (left, right,bi) of projective corepresentation defined by us. For strongly projective corepresentations defined by us, which in a sense are the closet to the classical case, we can relaize the dual of the original quantum group as a normal quantum subgroup of (dual of) the corresponding envelope. This leads to connection with the the second invariant cohomology of quantum groups in the sense of ( [NT13]).We define a suitable analogue of normalizer of a compact quantum group in a bigger compact quantum group and using this, associate a canonical discrete group Γ Q to a compact quantum group Q which is an alternative generalization of the second group cohomology and we show by an example that Γ Q in general may be different from H 2 uinv (Q, S 1 ). ’ Let us briefly discuss some possible applications of our results to the domain of physics, more precisely topological phases. Some quantum systems permit a gapped spectrum with a single or degenerate vacuum state. In order to specify conditions for the latter, a natural definition of symmetry involves of maps that preserve the transition probability of rays. Changing the point of view from rays to specific states, the symmetry action becomes projective. Specifically, symmetries are operators on the Hilbert space that are either linear and unitary or anti-linear and anti-unitary. The first case can be interpreted as a projective representation of groups. The vacuum state degeneracy is then contigent to the projective phase, whenever the symmetry is preserved by the Hamiltonian in question. The issue, whether a vacuum is degenerate, is then solved by answering whether a phase redefinition on the states can eliminate the projective phase. This problem is related to the field of group cohomology. Different vacuum state are parametrized by different equivalence classes in the second group cohomology, H 2 (G, U (1)) of the symmetry group G. Symmetry protected topological phase are found to be exactly those vacuum states with non-trivial elements in H 2 (G, U (1)). When one replaces group symmetry on quantum mechanical system by a natural quantum group symmetry, it is clear that projective corepresentation and cohomology of quantum groups will play a crucial role to label and understand topologiacl phase.
2 Preliminaries Definition 2.1. Let Q be a unital C * -algebra and ∆ be a unital C * -homomorphism from Q to Q ⊗ Q. Then (Q, ∆) is said to be a compact quantum group if it satisfying the following
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