Search

The commutative Hopf monoid of set compositions is a fundamental Hopf monoid internal to vector species, having undecorated bosonic Fock space the combinatorial Hopf algebra of quasisymmetric functions. We construct a geometric realization of this Hopf monoid over the adjoint of the (essentialized) braid hyperplane arrangement, which identifies the monomial basis with signed characteristic functions of the interiors of permutohedral tangent cones. We show that the indecomposable quotient Lie coalgebra is obtained by restricting functions to chambers of the adjoint arrangement, i.e., by quotienting out the higher codimensions. The resulting functions are characterized by the Steinmann relations of axiomatic quantum field theory, demonstrating an equivalence between the Steinmann relations, tangent cones to (generalized) permutohedra, and having algebraic structure internal to species. Our results give a new interpretation of a construction appearing in the mathematically rigorous formulation of renormalization by Epstein-Glaser, called causal perturbation theory. In particular, we show that operator products of time-ordered products correspond to the H-basis of the cocommutative Hopf monoid of set compositions, and generalized retarded products correspond to a spanning set of its primitive part Lie algebra. [ABSTRACT FROM AUTHOR]

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem. [ABSTRACT FROM AUTHOR]

Families of vector-like deformed relativistic quantum phase spaces and corresponding realizations are analyzed. A method for a general construction of the star product is presented. The corresponding twist, expressed in terms of phase space coordinates, in the Hopf algebroid sense is presented. General linear realizations are considered and corresponding twists, in terms of momenta and Poincaré-Weyl generators or $$\mathfrak {gl}(n)$$ generators are constructed and R-matrix is discussed. A classification of linear realizations leading to vector-like deformed phase spaces is given. There are three types of spaces: (i) commutative spaces, (ii) $$\kappa $$ -Minkowski spaces and (iii) $$\kappa $$ -Snyder spaces. The corresponding star products are (i) associative and commutative (but non-local), (ii) associative and non-commutative and (iii) non-associative and non-commutative, respectively. Twisted symmetry algebras are considered. Transposed twists and left-right dual algebras are presented. Finally, some physical applications are discussed. [ABSTRACT FROM AUTHOR]

We introduce Hopf polyads in order to unify Hopf monads and group actions on monoidal categories. A polyad is a lax functor from a small category (its source) to the bicategory of categories, and a Hopf polyad is a comonoidal polyad whose fusion operators are invertible. The main result states that the normalization of a Hopf polyad is a strong (co)monoidal action-type polyad (or strong monoidal pseudofunctor). The normalization of a polyad is a new polyad having simpler structure but the same category of modules. We show that, under certain assumptions, a Hopf polyad can be 'wrapped up' into a Hopf monad. This generalizes the fact that finite group actions on tensor categories can be seen as Hopf monads. Hopf categories in the sense of Batista, Caenepeel and Vercruysse can be viewed as Hopf polyads in a braided setting via the notion of Hopf polyalgebras. As a special case of the main theorem, we generalize a description of the center of graded fusion category due to Turaev and Virelizier to tensor categories: if $\mathcal {C}$ is a G-graded (locally bounded) tensor category, then G acts on the relative center of $\mathcal {C}$ with respect to the degree one part $\mathcal {C}_{1}$ , and the equivariantization of this action is the center of $\mathcal {C}$ . [ABSTRACT FROM AUTHOR]

We introduce a family of braided Hopf algebras that (in characteristic zero) generalizes the rank 1 Hopf algebras introduced by Krop and Radford and we study its cleft extensions. [ABSTRACT FROM PUBLISHER]

We characterize the module categories of suitably finite Hopf algebroids (more precisely, ×R-bialgebras in the sense of Takeuchi (1977) that are Hopf and finite in the sense of a work by the author (2000)) as those k-linear abelian monoidal categories that are module categories of some algebra, and admit dual objects for "sufficiently many" of their objects. Then we proceed to show that in many situations the Hopf algebroid can be chosen to be self-dual, in a sense to be made precise. This generalizes a result of Pfeiffer for pivotal fusion categories and the weak Hopf algebras associated to them. [ABSTRACT FROM AUTHOR]

We discuss some axioms that ensure that a Hopf algebra has its simple comodules classified using an analogue of the Borel-Weil construction. More precisely we show that a Hopf algebra having a dense big cell satisfies the above requirement. This method has its roots in the work of Parshall and Wang in the case of q-deformed quantum groups GL and SL. Here we examine the example of universal cosovereign Hopf algebras, for which the weight group is the free group on two generators. [ABSTRACT FROM AUTHOR]

We give a criterion for an open book to contain an n-times iterated Hopf plumbing summand. As an application, we show that fibre surfaces of positive braid knots admit a trefoil plumbing structure. [ABSTRACT FROM AUTHOR]

We provide a Hopf algebra structure on the space of superclass functions on the unipotent upper triangular group of type D over a finite field based on a supercharacter theory constructed by André and Neto in J. Algebra 305, 394-429 () and 322 (). Also, we make further comments with respect to types B and C. Type A was explored by M. Aguiar et al. in Adv. Math. 229 (); thus this paper is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types. [ABSTRACT FROM AUTHOR]

In this paper, we establish the invariance of cyclic (co)homology of left Hopf algebroids under the change of Morita equivalent base algebras. The classical result on Morita invariance for cyclic homology of associative algebras appears as a special example of this theory. In our main application we consider the Morita equivalence between the algebra of complex-valued smooth functions on the classical 2-torus and the coordinate algebra of the noncommutative 2-torus with rational parameter. We then construct a Morita base change left Hopf algebroid over this noncommutative 2-torus and show that its cyclic (co)homology can be computed by means of the homology of the Lie algebroid of vector fields on the classical 2-torus. [ABSTRACT FROM AUTHOR]

In this paper, a super version of the Hopf quiver theory is developed. The notion of Hopf superquivers is introduced. It is shown that only the path supercoalgebras of Hopf superquivers admit graded Hopf superalgebra structures. A complete classification of such graded Hopf superalgebras is given. A superquiver setting for general pointed Hopf superalgebras is also built up. In particular, a super version of the Gabriel type theorem and the Cartier-Gabriel decomposition theorem is given. [ABSTRACT FROM AUTHOR]