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Register a new userInduction of quantum group representations
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Authors
N. Ciccoli
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Induced representations for quantum groups are defined starting from coisotropic quantum subgroups and their main properties are proved. When the coisotropic quantum subgroup has a suitably defined section such representations can be realized on associated quantum bundles on general embeddable quantum homogeneous spaces.
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We propose an encoding for topological quantum computation utilizing quantum representations of mapping class groups. Leakage into a non-computational subspace seems to be unavoidable for universality in general. We are interested in the possible gate sets which can emerge in this setting. As a first step, we prove that for abelian anyons, all gates from these mapping class group representations are normalizer gates. Results of Van den Nest then allow us to conclude that for abelian anyons this quantum computing scheme can be simulated efficiently on a classical computer. With an eye toward more general anyon models we additionally show that for Fibonnaci anyons, quantum representations of mapping class groups give rise to gates which are not generalized Clifford gates.
We develop a theory of localization for braid group representations associated with objects in braided fusion categories and, more generally, to Yang-Baxter operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a tensor power of a fixed vector space in such a way that the braid group generators act locally. Although related to the notion of (quasi-)fiber functors for fusion categories, remarkably, such localizations can exist for representations associated with objects of non-integral dimension. We conjecture that such localizations exist precisely when the object in question has dimension the square-root of an integer and prove several key special cases of the conjecture.
We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation gl_N[t]-modules and the scheme-theoretic intersection of suitable Schubert varieties. Moreover, we prove that the multiplicity space as a module over the Bethe algebra is isomorphic to the coregular representation of the scheme-theoretic intersection. In particular, this result implies the simplicity of the spectrum of the Bethe algebra for real values of evaluation parameters and the transversality of the intersection of the corresponding Schubert varieties.
We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.
We define and study representations of quantum toroidal $gl_n$ with natural bases labeled by plane partitions with various conditions. As an application, we give an explicit description of a family of highest weight representations of quantum affine $gl_n$ with generic level.
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