2,281 research outputs found
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
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
Two interacting Hopf algebras of trees
Hopf algebra structures on rooted trees are by now a well-studied object,
especially in the context of combinatorics. In this work we consider a Hopf
algebra H by introducing a coproduct on a (commutative) algebra of rooted
forests, considering each tree of the forest (which must contain at least one
edge) as a Feynman-like graph without loops. The primitive part of the graded
dual is endowed with a pre-Lie product defined in terms of insertion of a tree
inside another. We establish a surprising link between the Hopf algebra H
obtained this way and the well-known Connes-Kreimer Hopf algebra of rooted
trees by means of a natural H-bicomodule structure on the latter. This enables
us to recover recent results in the field of numerical methods for differential
equations due to Chartier, Hairer and Vilmart as well as Murua.Comment: Error in antipode formula (section 7) corrected. Erratum submitte
Quantum and braided ZX calculus*
We revisit the notion of interacting Frobenius Hopf algebras for ZX-calculus
in quantum computing, with focus on allowing the algebras to be noncommutative
and coalgebras to be noncocommutative. We introduce the notion of *-structures
in ZX-calculus at this algebraic level and construct examples based on the
quantum group u_q(sl_2) at a root of unity. We provide an abstract formulation
of the Hadamard gate at this level and clarify its relationship to Hopf algebra
self-duality. We then solve the problem of extending the notion of interacting
Hopf algebras and ZX-calculus to take place in a braided tensor category. In
the ribbon case, the Hadamard gate coming from braided self-duality obeys a
modular identity. We give the example of b_q(sl_2), the self-dual braided
version of u_q(sl_2).Comment: 29 pages AMSlatex, 7 figures; correction to prop 2.7 and minor
improvement
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
Refinement for Signal Flow Graphs
The symmetric monoidal theory of Interacting Hopf Algebras provides a sound and complete axiomatisation for linear relations over a given field. As is the case for ordinary relations, linear relations have a natural order that coincides with inclusion. In this paper, we give a presentation for this ordering by extending the theory of Interacting Hopf Algebras with a single additional inequation. We show that the extended theory gives rise to an abelian bicategory - a concept due to Carboni and Walters - and highlight similarities with the algebra of relations. Most importantly, the ordering leads to a well-behaved notion of refinement for signal flow graphs
Nonlocal, noncommutative diagrammatics and the linked cluster Theorems
Recent developments in quantum chemistry, perturbative quantum field theory,
statistical physics or stochastic differential equations require the
introduction of new families of Feynman-type diagrams. These new families arise
in various ways. In some generalizations of the classical diagrams, the notion
of Feynman propagator is extended to generalized propagators connecting more
than two vertices of the graphs. In some others (introduced in the present
article), the diagrams, associated to noncommuting product of operators inherit
from the noncommutativity of the products extra graphical properties. The
purpose of the present article is to introduce a general way of dealing with
such diagrams. We prove in particular a "universal" linked cluster theorem and
introduce, in the process, a Feynman-type "diagrammatics" that allows to handle
simultaneously nonlocal (Coulomb-type) interactions, the generalized diagrams
arising from the study of interacting systems (such as the ones where the
ground state is not the vacuum but e.g. a vacuum perturbed by a magnetic or
electric field, by impurities...) or Wightman fields (that is, expectation
values of products of interacting fields). Our diagrammatics seems to be the
first attempt to encode in a unified algebraic framework such a wide variety of
situations. In the process, we promote two ideas. First, Feynman-type
diagrammatics belong mathematically to the theory of linear forms on
combinatorial Hopf algebras. Second, linked cluster-type theorems rely
ultimately on M\"obius inversion on the partition lattice. The two theories
should therefore be introduced and presented accordingl
Hopf-Frobenius algebras and a simpler Drinfeld double
The ZX-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of -special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras, starting from a single Hopf algebra which is not necessarily commutative or cocommutative. We provide the necessary and sufficient condition for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in FVect is a Hopf-Frobenius algebra. Hopf-Frobenius algebras provide a notion of duality, and give us a "dual" Hopf algebra that is isomorphic to the usual dual Hopf algebra in a compact closed category. We use this isomorphism to construct a Hopf algebra isomorphic to the Drinfeld double that is defined on rather than
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