The B∞-Structure on the Derived Endomorphism Algebra of the Unit in a Monoidal Category

Consider a monoidal category that is at the same time abelian with enough projectives and such that projectives are flat on the right. We show that there is a $B_{\infty }$-algebra that is $A_{\infty }$-quasi-isomorphic to the derived endomorphism algebra of the tensor unit. This $B_{\infty }$-algebra is obtained as the co-Hochschild complex of a projective resolution of the tensor unit, endowed with a lifted $A_{\infty }$-coalgebra structure. We show that in the classical situation of the category of bimodules over an algebra, this newly defined $B_{\infty }$-algebra is isomorphic to the Hochschild complex of the algebra in the homotopy category of $B_{\infty }$-algebras.

Session Coalgebras: A Coalgebraic View on Session Types and Communication Protocols

Compositional methods are central to the development and verification of software systems. They allow breaking down large systems into smaller components, while enabling reasoning about the behaviour of the composed system. For concurrent and communicating systems, compositional techniques based on behavioural type systems have received much attention. By abstracting communication protocols as types, these type systems can statically check that programs interact with channels according to a certain protocol, whether the intended messages are exchanged in a certain order. In this paper, we put on our coalgebraic spectacles to investigate session types, a widely studied class of behavioural type systems. We provide a syntax-free description of session-based concurrency as states of coalgebras. As a result, we rediscover type equivalence, duality, and subtyping relations in terms of canonical coinductive presentations. In turn, this coinductive presentation makes it possible to elegantly derive a decidable type system with subtyping for $$\pi $$ π -calculus processes, in which the states of a coalgebra will serve as channel protocols. Going full circle, we exhibit a coalgebra structure on an existing session type system, and show that the relations and type system resulting from our coalgebraic perspective agree with the existing ones.

Twisted deformations vs. cocycle deformations for quantum groups

In this paper, we study two deformation procedures for quantum groups: deformations by twists, that we call “comultiplication twisting”, as they modify the coalgebra structure, while keeping the algebra one — and deformations by [Formula: see text]-cocycle, that we call “multiplication twisting”, as they deform the algebra structure, but save the coalgebra one. We deal with quantized universal enveloping algebras (in short QUEAs), for which we accordingly consider those arising from twisted deformations (in short TwQUEAs) and those arising from [Formula: see text]-cocycle deformations, usually called multiparameter QUEAs (in short MpQUEAs). Up to technicalities, we show that the two deformation methods are equivalent, in that they eventually provide isomorphic outputs, which are deformations (of either kinds) of the “canonical”, well-known one-parameter QUEA by Jimbo and Lusztig. It follows that the two notions of TwQUEAs and of MpQUEAs — which, in Hopf algebra theoretical terms are naturally dual to each other — actually coincide; thus, that there exists in fact only one type of “pluriparametric deformation” for QUEAs. In particular, the link between the realization of any such QUEA as a MpQUEA and that as a TwQUEA is just a (very simple, and rather explicit) change of presentation.

An A ∞-coalgebra structure on a closed compact surface

Let P be an n-gon with {n\geq 3} . There is a formal combinatorial {A_{\infty}} -coalgebra structure on cellular chains {C_{*}(P)} with non-vanishing higher order structure when {n\geq 5} . If {X_{g}} is a closed compact surface of genus {g\geq 2} and {P_{g}} is a polygonal decomposition, the quotient map {q\colon P_{g}\to X_{g}} projects the formal {A_{\infty}} -coalgebra structure on {C_{*}(P_{g})} to a quotient structure on {C_{*}(X_{g})} , which persists to homology {H_{\ast}(X_{g};\mathbb{Z}_{2})} , whose operations are determined by the quotient map q, and whose higher order structure is non-trivial if and only if {X_{g}} is orientable with {g\geq 2} or unorientable with {g\geq 3} . But whether or not the {A_{\infty}} -coalgebra structure on homology observed here is topologically invariant is an open question.

THE CURVED A∞-COALGEBRA OF THE KOSZUL CODUAL OF A FILTERED DG ALGEBRA

The goal of this article is to study the coaugmented curved A∞-coalgebra structure of the Koszul codual of a filtered dg algebra over a field k. More precisely, we first extend one result of B. Keller that allowed to compute the A∞-coalgebra structure of the Koszul codual of a nonnegatively graded connected algebra to the case of any unitary dg algebra provided with a nonnegative increasing filtration whose zeroth term is k. We then show how to compute the coaugmented curved A∞-coalgebra structure of the Koszul codual of a Poincaré-Birkhoff-Witt (PBW) deformation of an N-Koszul algebra.

Genetic coalgebras and their cubic stochastic matrices

We study the structure of genetic coalgebras and prove the existence of a decomposition as the direct sum of indecomposable subcoalgebras with genetic realization. To obtain such a decomposition, we first define a new multiplication for their related cubic stochastic matrices of type (1,2) and then, considering these matrices as cubic non-negative, we show how this new multiplication induces an index classification which can be used to study the genetic coalgebra structure. Genetically, the given coalgebra decomposition can be understood as a genetic clustering of the different types for the genetic trait defining the genetic coalgebra, which allows us to identify isolated ancestral populations.

Algebraic and topological structures on rational tangles

<p>In this paper we present the construction of a group Hopf algebra on the class of rational tangles. A locally finite partial order on this class is introduced and a topology is generated. An interval coalgebra structure associated with the locally finite partial order is specified. Irrational and real tangles are introduced and their relation with rational tangles are studied. The existence of the maximal real tangle is described in detail.</p>

Twists, realizations and Hopf algebroid structure of κ-deformed phase space

The quantum phase space described by Heisenberg algebra possesses undeformed Hopf algebroid structure. The κ-deformed phase space with noncommutative coordinates is realized in terms of undeformed quantum phase space. There are infinitely many such realizations related by similarity transformations. For a given realization, we construct corresponding coproducts of commutative coordinates and momenta (bialgebroid structure). The κ-deformed phase space has twisted Hopf algebroid structure. General method for the construction of twist operator (satisfying cocycle and normalization condition) corresponding to deformed coalgebra structure is presented. Specially, twist for natural realization (classical basis) of κ-Minkowski space–time is presented. The cocycle condition, κ-Poincaré algebra and R-matrix are discussed. Twist operators in arbitrary realizations are constructed from the twist in the given realization using similarity transformations. Some examples are presented. The important physical applications of twists, realizations, R-matrix and Hopf algebroid structure are discussed.

Partial categorification of Hopf algebras and representation theory of towers of \mathcalJ-trivial monoids

International audience This paper considers the representation theory of towers of algebras of $\mathcal{J} -trivial$ monoids. Using a very general lemma on induction, we derive a combinatorial description of the algebra and coalgebra structure on the Grothendieck rings $G_0$ and $K_0$. We then apply our theory to some examples. We first retrieve the classical Krob-Thibon's categorification of the pair of Hopf algebras QSym$/NCSF$ as representation theory of the tower of 0-Hecke algebras. Considering the towers of semilattices given by the permutohedron, associahedron, and Boolean lattices, we categorify the algebra and the coalgebra structure of the Hopf algebras $FQSym , PBT$ , and $NCSF$ respectively. Lastly we completely describe the representation theory of the tower of the monoids of Non Decreasing Parking Functions.