We study categories associated to a decorated marked surface S_△, which is obtained from an unpunctured marked surface S by adding a set of decorating points. For any triangulation T of S_△, let Γ_T be the associated Ginzburg dg algebra. We show that there is a bijection between reachable open arcs in S_△ and the reachable rigid indecomposables in the perfect derived category perΓ_T. This is the dual of the bijection, between simple closed arcs in S_△ and reachable spherical objects in the 3-Calabi-Yau category D_{fd}(ΓT), constructed in the prequel (Qiu in Math Ann 365:595–633, 2016). Moreover, we show that Amiot’s quotient perΓ_T/D_{fd}(ΓT) that defines the generalized cluster categories corresponds to the forgetful map S_△→S (forgetting the decorating points) in a suitable sense.

The Matsumoto-Imai (MI) cryptosystem was the first multivariate public key cryptosystem proposed for practical use. Though MI is now considered insecure due to Patarin’s linearization attack, the core idea of MI has been used to construct many variants such as Sflash, which has recently been accepted for use in the New European Schemes for Signatures, Integrity, and Encryption project. Linearization attacks take advantage of the algebraic structure of MI to produce a set of equations that can be used to recover the plaintext from a given ciphertext. In our paper, we present a solution to the problem of finding the dimension of the space of linearization equations, a measure of how much work the attack will require.

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The construction of the Q-operator for twisted affine superalgebra Uq(C(2)(2)) is given. It is shown that the corresponding prefundamental representations give rise to evaluation modules some of which do not have a classical limit, which nevertheless appear to be a necessary part of fusion relations.