Showing papers on "Hopf algebra published in 2016"

Lectures on Yangian Symmetry

Abstract: In these introductory lectures we discuss the topic of Yangian symmetry from various perspectives. Forming the classical counterpart of the Yangian and an extension of ordinary Noether symmetries, first the concept of nonlocal charges in classical, two-dimensional field theory is reviewed. We then define the Yangian algebra following Drinfel’d's original motivation to construct solutions to the quantum Yang–Baxter equation. Different realizations of the Yangian and its mathematical role as a Hopf algebra and quantum group are discussed. We demonstrate how the Yangian algebra is implemented in quantum, two-dimensional field theories and how its generators are renormalized. Implications of Yangian symmetry on the two-dimensional scattering matrix are investigated. We furthermore consider the important case of discrete Yangian symmetry realized on integrable spin chains. Finally we give a brief introduction to Yangian symmetry in planar, four-dimensional super Yang–Mills theory and indicate its impact on the dilatation operator and tree-level scattering amplitudes. These lectures are illustrated by several examples, in particular the two-dimensional chiral Gross–Neveu model, the Heisenberg spin chain and ${ \mathcal N }=4$ superconformal Yang–Mills theory in four dimensions.

G. whitehead's generalization of h. hopf's invariant

Abstract: In the present paper', the generalized H. Hopf's invariant H: 7rdwn++(Sn+l) w7rd +n+i(Sqn+ ), due to G. Whitehead [10], is given a new definition which has some similarity with the original H. Hopf's definition [5]. The invariant H(f ) appears as depending in particular on the position in Sd+n+l of the inverse images M, Md byf: Sdln+l -_ Sn+1 of two regular values q, q' in Sn+1. Arnold Shapiro has defined the linking coefficient of two spheres Sp, Sq imbedded (without common point) in Em+i for p + q > m.2 In ? 5, the notion of linking coefficient is extended to (p + q m)-connected 7rmanifolds Mp, Mq in Em+i. It is an element of the stable homotopy group 7rW+N(SN), where r p + q m (7r-manifold = manifold which can be imbedded in some euclidean space with a trivial normal bundle). In the definition of H given in ? 3, M, and M' are 7r-manifolds but need not be (d n)-connected. As a consequence, H will in general also depend on the fields of normal vectors over M and M'. Therefore, it cannot be considered strictly as a linking coefficient which should be uniquely determined by the position in space of the two manifolds. It is an open question whether the method can be used to define the linking coefficient of (non-necessarily (p + q m)-connected) w-manifolds Mp, Mq in Em+i by going over to the quotient of 7Wp+qm,+N(SN) by some suitable subgroup. As an application of the new definition of H, it is proved that any regular imbedding (without self-intersection) of the d-sphere into euclidean (d + n)-space induces over Sd the trivial normal bundle provided that 2n > d + 1. A partial result in this direction was announced in [6].

A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra

Abstract: This is a set of working notes which give a second proof of the Shareshian--Wachs conjecture, the first (and recent) proof being by Brosnan and Chow in November 2015. The conjecture relates some symmetric functions constructed combinatorially out of unit interval graphs (their $q$-chromatic quasisymmetric functions), and some symmetric functions constructed algebro-geometrically out of Tymoczko's representation of the symmetric group on the equivariant cohomology ring of a family of subvarieties of the complex flag variety, called regular semisimple Hessenberg varieties. Brosnan and Chow's proof is based in part on the idea of deforming the Hessenberg varieties. The proof given here, in contrast, is based on the idea of recursively decomposing Hessenberg varieties, using a new Hopf algebra as the organizing principle for this recursion. We hope that taken together, each approach will shed some light on the other, since there are still many outstanding questions regarding the objects under study.

On finite GK-dimensional Nichols algebras over abelian groups

Abstract: We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, $\operatorname{GKdim}$ for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over $\mathbb Z$ with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite $\operatorname{GKdim}$ if and only if the size of the block is 2 and the eigenvalue is $\pm 1$; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite $\operatorname{GKdim}$ if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite $\operatorname{GKdim}$. Consequently we present several new examples of Nichols algebras with finite $\operatorname{GKdim}$, including two not in the class alluded to above. We determine which among these Nichols algebras are domains.

Quantum Binary Polyhedral Groups And Their Actions On Quantum Planes

Abstract: We classify quantum analogues of actions of finite subgroups G of SL_2(k) on commutative polynomial rings k[u,v]. More precisely, we produce a classification of pairs (H,R), where H is a finite dimensional Hopf algebra that acts inner faithfully and preserves the grading of an Artin-Schelter regular algebra R of global dimension two. Remarkably, the corresponding invariant rings R^H share similar regularity and Gorenstein properties as the invariant rings k[u,v]^G in the classic setting. We also present several questions and directions for expanding this work in noncommutative invariant theory.

Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras

Abstract: The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the “baby model” of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.

Interacting Frobenius Algebras are Hopf

Abstract: 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 [9] have shown that, given a suitable distribution 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 [9] 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, and recover the system of Bonchi et al as a subtheory in the prime power dimensional case. However the more general theory does not arise from a distributive law.

Nakayama automorphism and applications

Abstract: Nakayama automorphism is used to study group actions and Hopf algebra actions on Artin-Schelter regular algebras of global dimension three.

Quivers with loops and generalized crystals

Abstract: In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure richer than the usual Kashiwara crystals. Along with the geometric study of Nakajima quiver varieties, in the same context, this yields a notion of generalized crystals, coming with a tensor product. As an application, we define the semicanonical basis of the Hopf algebra generalizing quantum groups, which was already equipped with a canonical basis. The irreducible components of the Nakajima varieties provide the family of highest weight crystals associated to dominant weights, as in the classical case.

Hopfological algebra and categorification at a root of unity: The first steps

Abstract: Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable H, our construction could be a starting point in the program of categorifying quantum invariants of 3-manifolds.

Decomposition spaces in combinatorics

Abstract: A decomposition space (also called unital 2-Segal space) is a simplicial object satisfying an exactness condition weaker than the Segal condition: just as the Segal condition expresses (up to homotopy) composition, the new condition expresses decomposition. It is a general framework for incidence (co)algebras. In the present contribution, after establishing a formula for the section coefficients, we survey a large supply of examples, emphasising the notion's firm roots in classical combinatorics. The first batch of examples, similar to binomial posets, serves to illustrate two key points: (1) the incidence algebra in question is realised directly from a decomposition space, without a reduction step, and reductions are often given by CULF functors; (2) at the objective level, the convolution algebra is a monoidal structure of species. Specifically, we encounter the usual Cauchy product of species, the shuffle product of L-species, the Dirichlet product of arithmetic species, the Joyal-Street external product of q-species and the Morrison `Cauchy' product of q-species, and in each case a power series representation results from taking cardinality. The external product of q-species exemplifies the fact that Waldhausen's S-construction on an abelian category is a decomposition space, yielding Hall algebras. The next class of examples includes Schmitt's chromatic Hopf algebra, the Fa\`a di Bruno bialgebra, the Butcher-Connes-Kreimer Hopf algebra of trees and several variations from operad theory. Similar structures on posets and directed graphs exemplify a general construction of decomposition spaces from directed restriction species. We finish by computing the M\

Distinguished pre-nichols algebras

Abstract: We formally define and study the distinguished pre-Nichols algebra \( \tilde{B} \)(V) of a braided vector space of diagonal type V with finite-dimensional Nichols algebra B(V). The algebra \( \tilde{B} \)(V) is presented by fewer relations than B(V), so it is intermediate between the tensor algebra T(V) and B(V). Prominent examples of distinguished pre-Nichols algebras are the positive parts of quantized enveloping (super)algebras and their multiparametric versions. We prove that these algebras give rise to new examples of Noetherian pointed Hopf algebras of finite Gelfand-Kirillov dimension. We investigate the kernel (in the sense of Hopf algebras) of the projection from \( \tilde{B} \)(V) to B(V), generalizing results of De Concini and Procesi on quantum groups at roots of unity.

S -matrix algebra of the AdS 2 × S 2 superstring

Abstract: In this paper, we find the Yangian algebra responsible for the integrability of the ${\mathrm{AdS}}_{2}\ifmmode\times\else\texttimes\fi{}{S}^{2}\ifmmode\times\else\texttimes\fi{}{T}^{6}$ superstring in the planar limit. We demonstrate the symmetry of the corresponding exact $S$ matrix in the massive sector, including the presence of the secret symmetry. We give two alternative presentations of the Hopf algebra. The first takes the usual canonical form, which, as the relevant representations are long, leads to a Yangian representation that is not of evaluation type. After investigating the relationship between cocommutativity, evaluation representations and the shortening condition, we find an alternative realization of the Yangian whose representation is of the evaluation type. Finally, we explore two limits of the $S$ matrix. The first is the classical $r$ matrix, where we rediscover the need for a secret symmetry also in this context. The second is the simplifying zero-coupling limit. In this limit, taking the $S$ matrix as a generating $R$ matrix for the algebraic Bethe ansatz, we obtain an effective model of free fermions on a periodic spin-chain. This limit should provide hints to the one-loop anomalous dimension of the mysterious superconformal quantum mechanics dual to the superstring theory in this geometry.

Algebraic renormalisation of regularity structures

Abstract: We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms. This subgroup is sufficiently large to be able to implement a version of the BPHZ renormalisation prescription in this context. This is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms, which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations. Our construction is based on bialgebras of decorated coloured forests in cointeraction. More precisely, we have two Hopf algebras in cointeraction, coacting jointly on a vector space which represents the generalised functions of the theory. Two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic Birkhoff factorisation that arises naturally in perturbative quantum field theory.

Actions of some pointed Hopf algebras on path algebras of quivers

Abstract: We classify Hopf actions of Taft algebras T(n) on path algebras of quivers, in the setting where the quiver is loopless, finite, and Schurian. As a corollary, we see that every quiver admitting a faithful ℤn-action (by directed graph automorphisms) also admits inner faithful actions of a Taft algebra. Several examples for actions of the Sweedler algebra T(2) and for actions of T(3) are presented in detail. We then extend the results on Taft algebra actions on path algebras to actions of the Frobenius–Lusztig kernel uq(sl2), and to actions of the Drinfeld double of T(n).

A Combinatorial Perspective on Quantum Field Theory

Abstract: Part I Preliminaries -- Introduction -- Quantum field theory set up -- Combinatorial classes and rooted trees -- The Connes-Kreimer Hopf algebra -- Feynman graphs -- Part II Dyson-Schwinger equations -- Introduction to Dyson-Schwinger equations -- Sub-Hopf algebras from Dyson-Schwinger equations -- Tree factorial and leading log toys -- Chord diagram expansions -- Differential equations and the (next-to)m leading log expansion -- Part III Feynman periods -- Feynman integrals and Feynman periods -- Period preserving graph symmetries -- An invariant with these symmetries -- Weight -- The c2 invariant -- Combinatorial aspects of some integration algorithms -- Index.

Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type

Abstract: We present a generalization of the theory of quantum symmetric pairs as developed by Kolb and Letzter. We introduce a class of generalized Satake diagrams that give rise to (not necessarily involutive) automorphisms of the second kind of symmetrizable Kac-Moody algebras $\mathfrak{g}$. These lead to right coideal subalgebras $B_{\mathbf{c},\mathbf{s}}$ of quantized enveloping algebras $U_q(\mathfrak{g})$. In the case that $\mathfrak{g}$ is a twisted or untwisted affine Lie algebra of classical type Jimbo found intertwiners (equivariant maps) of the vector representation of $U_q(\mathfrak{g})$ yielding trigonometric solutions to the parameter-dependent quantum Yang-Baxter equation. In the present paper we compute intertwiners of the vector representation restricted to the subalgebras $B_{\mathbf{c},\mathbf{s}}$ when $\mathfrak{g}$ is of type ${\rm A}^{(1)}_n$, ${\rm B}^{(1)}_n$, ${\rm C}^{(1)}_n$ and ${\rm D}^{(1)}_n$. These intertwiners are matrix solutions to the parameter-dependent quantum reflection equation known as trigonometric reflection matrices. They are symmetric up to conjugation by a diagonal matrix and in many cases satisfy a certain sparseness condition: there are at most two nonzero entries in each row and column. Conjecturally, this classifies all such solutions in vector spaces carrying this representation. A group of Hopf algebra automorphisms of $U_q(\mathfrak{g})$ acts on these reflection matrices, allowing us to show that each reflection matrix found is equivalent to one with at most two additional free parameters. Additional characteristics of the reflection matrices such as eigendecompositions and affinization relations are also obtained. The eigendecompositions suggest that for all these matrices there should be a natural interpretation in terms of representations of Hecke-type algebras.

The Splitting Process in Free Probability Theory

Abstract: Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu’s theory of free probability. The relation between free moments and free cumulants is usually described in terms of M obius calculus over the lattice of non-crossing partitions. In this work we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf algebra structure on the linear span of non-crossing partitions. The generating series of free moments is seen as a character on this Hopf algebra. It is characterized by solving a linear fixed point equation that relates it to the generating series of free cumulants. These phenomena are explained through a process similar to (though different from) the arborification process familiar in the theory of dynamical systems, and originating in Cayley’s work.

Generalizations of the Sweedler Dual

Abstract: As left adjoint to the dual algebra functor, Sweedler’s finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler’s construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring R the left adjoint of the dual algebra functor on the category of R-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf R-algebras, provided that R is noetherian and absolutely flat.

Globalization of twisted partial hopf actions

Abstract: In this work, we review some properties of twisted partial actions of Hopf algebras on unital algebras and give necessary and sufficient conditions for a twisted partial action to have a globalization. We also elaborate a series of examples.

Formality Theorem for Quantizations of Lie Bialgebras

Abstract: Using the theory of props we prove a formality theorem associated with universal quantizations of Lie bialgebras.

Co-stability of Radicals and Its Applications to PI-Theory

Abstract: We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either charF = 0 or charF > dimA, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite-dimensional associative algebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite-dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur's conjecture for graded polynomial identities of finite-dimensional associative algebras over a field of characteristic 0, graded by any group. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.