Lifting Coalgebra Modalities and MELL\mathsf{MELL} Model Structure to Eilenberg-Moore Categories

A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic ($\mathsf{MELL}$), known as a \emph{linear category}, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by Blute and Scott's work on categories of modules of Hopf algebras as models of linear logic, we study categories of algebras of monads (also known as Eilenberg-Moore categories) as models of $\mathsf{MELL}$. We define a $\mathsf{MELL}$ lifting monad on a linear category as a Hopf monad -- in the Brugui{\`e}res, Lack, and Virelizier sense -- with a special kind of mixed distributive law over the monoidal coalgebra modality. As our main result, we show that the linear category structure lifts to the category of algebras of $\mathsf{MELL}$ lifting monads. We explain how groups in the category of coalgebras of the monoidal coalgebra modality induce $\mathsf{MELL}$ lifting monads and provide a source for such groups from enrichment over abelian groups. Along the way we also define mixed distributive laws of symmetric comonoidal monads over symmetric monoidal comonads and lifting differential category structure.Comment: An extend abstract version of this paper appears in the conference proceedings of the 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018), under the title "Lifting Coalgebra Modalities and $\mathsf{MELL}$ Model Structure to Eilenberg-Moore Categories

Interacting Frobenius Algebras are Hopf

Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object---the so-called phase group---and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.Comment: 32 pages; submitte

Interacting Hopf Algebras

We introduce the theory IH of interacting Hopf algebras, parametrised over a principal ideal domain R. The axioms of IH are derived using Lack's approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoid-comonoid pairs. This construction is instrumental in showing that IH is isomorphic to the PROP of linear relations (i.e. subspaces) over the field of fractions of R

Lifting Coalgebra Modalities and IMELL Model Structure to Eilenberg-Moore Categories

A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic (IMELL), known as a linear category, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by R. Blute and P. Scott\u27s work on categories of modules of Hopf algebras as models of linear logic, we study Eilenberg-Moore categories of monads as models of IMELL. We define an IMELL lifting monad on a linear category as a Hopf monad - in the Bruguieres, Lack, and Virelizier sense - with a mixed distributive law over the monoidal coalgebra modality. As our main result, we show that the linear category structure lifts to Eilenberg-Moore categories of IMELL lifting monads. We explain how monoids in the Eilenberg-Moore category of the monoidal coalgebra modality can induce IMELL lifting monads and provide sources for such monoids. Along the way, we also define mixed distributive laws of bimonads over coalgebra modalities and lifting differential category structure to Eilenberg-Moore categories of exponential lifting monads

Quantum Random Walks and Time Reversal

Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated to a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motivated from the theory of Kac algebras: If $H$ is any Hopf algebra, it may be realised in \Lin(H) in such a way that \Delta h=W(h\tens 1)W^{-1} for an operator $W$. This $W$ is interpreted as the time evolution operator for the system at time $t$ coupled quantum-mechanically to the system at time $t+\delta$. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a CTP-type theorem.Comment: 32 pages, LATEX, (DAMTP/92-20

Right Coideal Subalgebras of the Quantum Borel Algebra of type G2

In this paper we describe the right coideal subalgebras containing all group-like elements of the multiparameter quantum group Uq+(g), where g is a simple Lie algebra of type G2, while the main parameter of quantization q is not a root of 1. If the multiplicative order t of q is finite, t>4, t different from 6, then the same classification remains valid for homogeneous right coideal subalgebras of the positive part uq+(g) of the multiparameter version of the small Lusztig quantum group