Search

In this paper, we study if, for a given simple module over a Hopf algebra, there exists a virtual module such that their tensor product is the regular module. This is related to a conjecture by Donald Knutson, later disproved and refined by Savitskii, stating that for every irreducible character of a finite group, there exists a virtual character such that their tensor product is the regular character. We also introduce the Knutson Index as a measure of Knutson's Conjecture failure, discuss its algebraic properties and present counter-examples to Savitskii's Conjecture. [ABSTRACT FROM AUTHOR]

Given a cocommutative Hopf algebra H over a commutative ring K and a symmetric partial action of H on a K -algebra A , we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product A # H , involving the Hochschild homology of A and the partial homology of H. An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of H under consideration is based on the category of the partial representations of H. A specific partial representation of H on a subalgebra B of the partial "Hopf" algebra H p a r is involved in the definition and we construct a projective resolution of B. [ABSTRACT FROM AUTHOR]

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups of odd prime degree could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of group-like elements isomorphic to PSL n (q) (n ≥ 4) , PSL 3 (q) (q > 2) , or PSp 2 n (q) (n ≥ 3) , is isomorphic to a group algebra. [ABSTRACT FROM AUTHOR]

This paper constructs a novel Hopf algebra cf (UT •) on the class functions of the unipotent upper triangular groups UT n (F q) over a finite field. This construction is representation theoretic in nature and uses the machinery of Hopf monoids in the category of vector species. In contrast with a similar known construction, this Hopf algebra has the property that induction to the finite general linear group induces a homomorphism to Zelevinsky's Hopf algebra of GL n (F q) class functions. Furthermore, cf (UT •) contains a Hopf subalgebra which is isomorphic to a known combinatorial Hopf algebra, previously used to prove a conjecture about chromatic quasisymmetric functions. Some additional Hopf algebraic properties are also established. [ABSTRACT FROM AUTHOR]

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F. Király and H. Oberhauser. In particular, we show how to recover the character property of the associated linear map on the tensor algebra by considering a deformed quasi-shuffle product of words. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties. [ABSTRACT FROM AUTHOR]

In this paper, we study cyclic cohomology theory on a regular multiplier Hopf algebra. Given a modular pair consisting of a group-like element, a character and in involution for a regular multiplier Hopf algebra, we associate a cyclic module. The approach is based on the Connes-Moscovici cyclic module of Hopf algebras and we obtain the same cyclic modules when our algebra has an identity. [ABSTRACT FROM AUTHOR]

We introduce two new bases of Q S y m , the reverse extended Schur functions and the row-strict reverse extended Schur functions, as well as their duals in NSym , the reverse shin functions and row-strict reverse shin functions. These bases are the images of the extended Schur basis and shin basis under the involutions ρ and ω , which generalize the classical involution ω on the symmetric functions. In addition, we prove a Jacobi-Trudi rule for certain shin functions using creation operators. We define skew extended Schur functions and skew-II extended Schur functions based on left and right actions of NSym and Q S y m respectively. We then use the involutions ρ and ω to translate these and other known results to our reverse and row-strict reverse bases. [ABSTRACT FROM AUTHOR]

In support variety theory, representations of a finite dimensional (Hopf) algebra A can be studied geometrically by associating any representation of A to an algebraic variety using the cohomology ring of A. An essential assumption in this theory is the finite generation condition for the cohomology ring of A and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra A , we show that the finite generation condition on A -modules can be replaced by a condition on any affine commutative A -module algebra R under the assumption that R is integral over its invariant subring R A. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if A is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra A ′ over a finite field, where A can be viewed as a deformation of A ′ , and prove that if the finite generation condition holds for A ′ , then the same condition holds for A. [ABSTRACT FROM AUTHOR]

Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, SupLat, of complete lattices and join-preserving morphisms. We connect the two methods by showing that any Hopf algebra, H in SupLat, has a corresponding group, R (H) , which we call its remnant and a co-quasitriangular structure on H induces a YBE solution on R (H) , which is compatible with its group structure. Conversely, any group with a compatible YBE solution can be realised in this way. Additionally, it is well-known that any such group has an induced secondary group structure, making it a skew left brace. By realising the group as the remnant of a co-quasitriangular Hopf algebra, H , this secondary group structure appears as the projection of the transmutation of H. Finally, for any YBE solution, we obtain a SupLat-FRT Hopf algebra in SupLat, whose remnant recovers the universal skew brace of the solution. [ABSTRACT FROM AUTHOR]

Let (M , Q) be a dg manifold. The space of vector fields with shifted degrees (X (M) [ − 1 ] , L Q) is a Lie algebra object in the homology category H ((C M ∞ , Q) − mod) of dg modules over (M , Q) , the Atiyah class α M being its Lie bracket. The triple (X (M) [ − 1 ] , L Q ; α M) is also a Lie algebra object in the Gabriel-Zisman homotopy category Π ((C M ∞ , Q) − mod). In this paper, we describe the universal enveloping algebra of (X (M) [ − 1 ] , L Q ; α M) and prove that it is a Hopf algebra object in Π ((C M ∞ , Q) − mod). As an application, we study Fedosov dg Lie algebroids and recover a result of Stiénon, Xu, and the second author on the Hopf algebra arising from a Lie pair. [ABSTRACT FROM AUTHOR]

By constructing explicit examples, we show that the core of a group-like element in a cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over the group ring of a finite abelian group is not always completely trivial. [ABSTRACT FROM AUTHOR]

The objects of study in this paper are Hopf algebras H which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra A by a finite dimensional Hopf algebra H ‾ : = H / A + H , where A + is the augmentation ideal of A. Basic structural and homological properties are recalled and classes of examples are listed. A structure theorem is proved when (⁎) H ‾ is semisimple and cosemisimple, showing that in this case the noncommutativity of H arises from the action of a finite group. For example, when (⁎) holds and H is prime and pointed, it is a crossed product of a smooth affine commutative domain by a finite group, and the simple H -modules are described by a type of Clifford's theorem. [ABSTRACT FROM AUTHOR]

We prove that in q -Weyl generators the multi-parameter quantizations of type A n + and B n + are quadratic-linear Koszul algebras. [ABSTRACT FROM AUTHOR]

We construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product Hopf algebras, namely double cross products A ↪ A ⋈ H ↩ H , double cross coproducts Image 1 , biproducts Image 2 and bicrossproducts Image 3 on the assumption that the factors have strongly bicovariant calculi Ω (A) , Ω (H) (or a braided version Ω (B)). We use super versions of each of the constructions. Moreover, the latter three quantum groups all coact canonically on one of their factors and we show that this coaction is differentiable. In the case of the Drinfeld double D (A , H) = A op ⋈ H (where A is dually paired to H), we show that its canonical actions on A , H are differentiable. Examples include a canonical Ω (C q [ G L 2 ⋉ C 2 ]) for the quantum group of affine transformations of the quantum plane and Ω (C λ [ Poin c 1 , 1 ]) for the bicrossproduct Poincaré quantum group in 2 dimensions. We also show that Ω (C q [ G L 2 ]) itself is uniquely determined by differentiability of the canonical coaction on the quantum plane and of the determinant subalgebra. [ABSTRACT FROM AUTHOR]

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by lifting method of Andruskiewitsch and Schneider. Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. In the present paper, all rank 4 Nichols algebras of diagonal type with a finite arithmetic root system over fields of arbitrary characteristic are classified. Our proof uses the classification of the finite arithmetic root systems of rank 4. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of a Rota-Baxter paired module to study Rota-Baxter modules without necessarily a Rota-Baxter operator. We obtain two characterizations of Rota-Baxter paired modules, and give some basic properties of Rota-Baxter paired modules. Beginning with the connection between the notion of integrals in the representations of Hopf algebra and of the notion of an integral algebra as a motivation and special case of Rota-Baxter algebra of weight zero, we obtain a large number of Rota-Baxter paired modules from Hopf related algebras and modules, including semisimple Hopf algebras, weak Hopf algebras, Long bialgebras, quasitriangular Hopf algebras, weak Hopf modules, dimodules and Doi-Hopf modules. Some of them give new examples of Rota-Baxter algebras. [ABSTRACT FROM AUTHOR]

Recently, M. Baker and N. Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields. [ABSTRACT FROM AUTHOR]

In this paper, we classify all prime Hopf algebras H of GK-dimension one satisfying the following two conditions: 1) H has a 1-dimensional representation of order PI.deg(H) and 2) the invariant components of H with respect to this 1-dimensional representation are domains (see Section 2 for related definitions). As consequences, 1) a number of new Hopf algebras of GK-dimension one are found and some of them are not pointed, 2) we give a partial answer to a question posed in [12] and 3) the question (1.1) is solved. [ABSTRACT FROM AUTHOR]

Let H be a cocommutative Hopf algebra acting on an algebra A. Assuming the base field to be algebraically closed and the H -action on A to be integral, that is, it is given by a coaction of some Hopf subalgebra of the finite dual H ∘ that is an integral domain, we stratify the prime spectrum Spec A in terms of the prime spectra of certain commutative algebras. For arbitrary H -actions in characteristic 0, we show that the largest H -stable ideal of A that is contained in a given semiprime ideal of A is semiprime as well. [ABSTRACT FROM AUTHOR]

We prove that the q-deformed unitary group, i.e., U q (N) , is the universal compact quantum group in the category of (compact) quantum groups which coact on the q-deformed odd sphere S q 2 N − 1 leaving the space spanned by the natural set of generators invariant and preserving the unique S U q (N) invariant functional on S q 2 N − 1. Using this, we identify U q (N) as the quantum group of orientation preserving isometries (in the sense of Bhowmick and Goswami [5]) for a natural spectral triple associated with S q 2 N − 1 constructed by Chakraborty and Pal [9]. [ABSTRACT FROM AUTHOR]

Let W be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras H q (W) as (left) coideal subalgebras. Our Hecke-Hopf algebras 1 H (W) have a number of applications. In particular they provide new solutions of quantum Yang-Baxter equation and lead to a construction of a new family of endo-functors of the category of H q (W) -modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras; for an arbitrary Coxeter group W the "Demazure" part of H (W) is being acted upon by generalized braided derivatives which generate the corresponding (generalized) Nichols algebra. [ABSTRACT FROM AUTHOR]

Abstract We study Hopf Galois structures on separable field extensions L / K of degree p n , p an odd prime, and denote by G the Galois group of the normal closure over K. We prove that if L / K has a Hopf Galois structure of cyclic type, then it has no structure of noncyclic type. For n = 3 , we characterize the transitive groups G such that L / K has cyclic Hopf Galois structures. We prove that if L / K has a nonabelian Hopf Galois structure of type N , then it has an abelian structure whose type has the same exponent as N. We obtain that, for p > 3 , the two abelian noncyclic Hopf Galois structures do not occur on the same extension. Finally, we list all possible sets of Hopf Galois structure types on a separable extension of degree p 3 , for p > 3 a prime. [ABSTRACT FROM AUTHOR]

Abstract We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra B in the category of comodules of a coquasitriangular Hopf algebra A has an associated coquasitriangular Hopf algebra Image 1. As an application we find new generators for c q [ S L 2 ] reduced at q a primitive odd root of unity with the remarkable property that their monomials are essentially a dual basis to the standard PBW basis of the reduced Drinfeld–Jimbo quantum enveloping algebra u q (s l 2). Our methods apply in principle for general c q [ G ] as we demonstrate for c q [ S L 3 ] at certain odd roots of unity. [ABSTRACT FROM AUTHOR]

Let C be a finite tensor category and M an exact left C -module category. We call M unimodular if the finite multitensor category Rex C (M) of right exact C -module endofunctors of M is unimodular. In this article, we provide various characterizations, properties, and examples of unimodular module categories. As our first application, we employ unimodular module categories to construct (commutative) Frobenius algebra objects in the Drinfeld center of any finite tensor category. When C is a pivotal category, and M is a unimodular, pivotal left C -module category, the Frobenius algebra objects are symmetric as well. Our second application is a classification of unimodular module categories over the category of finite dimensional representations of a finite dimensional Hopf algebra; this answers a question of Shimizu [40]. Using this, we provide an example of a finite tensor category whose categorical Morita equivalence class does not contain any unimodular tensor category. [ABSTRACT FROM AUTHOR]

We investigate some Hopf algebra structures over an Ore extension of a Hopf algebra. Necessary and sufficient conditions for an Ore extension of a Hopf algebra to have a Hopf algebra structure of a certain type are given. This construction, called a generalized Hopf–Ore extension, generalizes Hopf–Ore extensions. We describe the generalized Hopf–Ore extensions of the enveloping algebras of Lie algebras. For some Lie algebras g , the generalized Hopf–Ore extensions of U ( g ) are classified. [ABSTRACT FROM AUTHOR]

In this paper, we construct the weak version of peak quasisymmetric functions inside the Hopf algebra of weak composition quasisymmetric functions (WCQSym) defined by Guo, Thibon and Yu. Weak peak quasisymmetric functions (WPQSym) are studied in several aspects. First we find a natural basis of WPQSym lifting peak functions introduced by Stembridge. Then we confirm that WPQSym is a Hopf subalgebra of WCQSym by giving explicit multiplication, comultiplication and antipode formulas. By extending Stembridge's descent-to-peak maps, we also show that WPQSym is a Hopf quotient of WCQSym. On the other hand, we prove that WPQSym embeds as a Rota–Baxter subalgebra of WCQSym, thus of the free commutative Rota–Baxter algebra of weight 1 on one generator. Moreover, WPQSym can also be a Rota–Baxter quotient of WCQSym. [ABSTRACT FROM AUTHOR]

We define a new basis of quasisymmetric functions, the row-strict dual immaculate functions, as the generating function of a particular set of tableaux. We show that this definition gives a function that can also be obtained by applying the involution ψ to the dual immaculate functions of Berg, Bergeron, Saliola, Serrano, and Zabrocki (2014), and we establish numerous combinatorial properties for our functions. We give an equivalent formulation of our functions via Bernstein-like operators, in a similar fashion to Berg et al. (2014). We conclude the paper by defining skew dual immaculate functions and hook dual immaculate functions, and establishing combinatorial properties for them. [ABSTRACT FROM AUTHOR]

An internal coproduct is described, which is compatible with Hoffman's quasi-shuffle product. Hoffman's quasi-shuffle Hopf algebra, with deconcatenation coproduct, is a comodule-Hopf algebra over the bialgebra thus defined. The relation with Ecalle's mould calculus, i.e., mould composition and contracting arborification is precised. [ABSTRACT FROM AUTHOR]

Quasi-shuffle products, introduced by the first author, have been useful in studying multiple zeta values and some of their analogues and generalizations. The second author, together with Kajikawa, Ohno, and Okuda, significantly extended the definition of quasi-shuffle algebras so it could be applied to multiple q -zeta values. This article extends some of the algebraic machinery of the first author's original paper to the more general definition, and demonstrates how various algebraic formulas in the quasi-shuffle algebra can be obtained in a transparent way. Some applications to multiple zeta values, interpolated multiple zeta values, multiple q -zeta values, and multiple polylogarithms are given. [ABSTRACT FROM AUTHOR]

We introduce a graphical syntax for signal flow diagrams based on the language of symmetric monoidal categories. Using universal categorical constructions, we provide a stream semantics and a sound and complete axiomatisation. A certain class of diagrams captures the orthodox notion of signal flow graph used in control theory; we show that any diagram of our syntax can be realised, via rewriting in the equational theory, as a signal flow graph. [ABSTRACT FROM AUTHOR]

We introduce Nakayama functors for coalgebras and investigate their basic properties. These functors are expressed by certain (co)ends as in the finite case discussed by Fuchs, Schaumann, and Schweigert. This observation allows us to define Nakayama functors for Frobenius tensor categories in an intrinsic way. As applications, we establish the categorical Radford S 4 -formula for Frobenius tensor categories and obtain some related results. These are generalizations of works of Etingof, Nikshych, and Ostrik on finite tensor categories and some known facts on co-Frobenius Hopf algebras. [ABSTRACT FROM AUTHOR]

We investigate Frobenius algebras and symmetric algebras in the monoidal category of right comodules over a Hopf algebra H ; for the symmetric property H is assumed to be cosovereign. If H is finite dimensional and A is an H -comodule algebra, we uncover the connection between A and the smash product A # H ⁎ with respect to the Frobenius and symmetric properties. [ABSTRACT FROM AUTHOR]

Primitive cohomology of a Hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra. Among many of its applications, two classifications are presented. Firstly we classify all non-locally PI, pointed Hopf algebra domains of Gelfand–Kirillov dimension two; and secondly we classify all pointed Hopf algebras of rank one. The first classification extends some results of Brown, Goodearl and others in an ongoing project to understand all Hopf algebras of low Gelfand–Kirillov dimension. The second generalizes results of Krop–Radford and Wang–You–Chen which classified Hopf algebras of rank one under extra hypothesis. Properties and algebraic structures of the primitive cohomology are discussed. [ABSTRACT FROM AUTHOR]

Following Bachmann's recent work on bi-brackets and multiple Eisenstein series, Zudilin introduced the notion of multiple q -zeta brackets, which provides a q -analog of multiple zeta values possessing both shuffle as well as quasi-shuffle relations. The corresponding products are related in terms of duality. In this work we study Zudilin's duality construction in the context of classical multiple zeta values as well as various q -analogs of multiple zeta values. Regarding the former we identify the derivation relation of order two with a Hoffman–Ohno type relation. Then we describe relations between the Ohno–Okuda–Zudilin q -multiple zeta values and the Schlesinger–Zudilin q -multiple zeta values. [ABSTRACT FROM AUTHOR]

Let H be a semisimple finite dimensional Hopf algebra over a field F of zero characteristic. We prove three major theorems. 1. The Representability theorem which states that every H -module (associative) F -algebra W satisfying an ordinary PI, has the same H -identities as the Grassmann envelope of an H ⊗ ( F Z / 2 Z ) ⁎ -module algebra which is finite dimensional over a field extension of F . 2. The Specht problem for H -module (ordinary) PI algebras. That is, every H – T -ideal Γ which contains an ordinary PI contains H -polynomials f 1 , … , f s which generate Γ as an H – T -ideal. 3. Amitsur's conjecture for H -module algebras, saying that the exponent of the H -codimension sequence of an ordinary PI H -module algebra is an integer. [ABSTRACT FROM AUTHOR]

For a finite Galois extension K / k and an intermediate field F such that Gal ( K / F ) has a normal complement in Gal ( K / k ) , we construct and characterize Hopf Galois structures on K / k which are induced by a pair of Hopf Galois structures on K / F and F / k . [ABSTRACT FROM AUTHOR]

The classification of affine prime regular Hopf algebras of GK-dimension one is completed. As consequences, 1) we give a negative answer to Question 7.1 posed in [5] and 2) we show that there do exist prime regular Hopf algebras of GK-dimension one which are not pointed. [ABSTRACT FROM AUTHOR]

Quantum quasigroups and quantum loops are self-dual objects providing a general framework for the nonassociative extension of quantum group techniques. Bialgebra reducts of Hopf algebras are quantum loops, while sufficient conditions are given for quantum loop structure to augment to a Hopf algebra. The Moufang–Hopf algebras of Benkart et al., the Hopf quasigroups and coquasigroups of Klim–Majid, and the coassociative H -bialgebras of Pérez-Izquierdo (for instance, the universal enveloping algebras of Sabinin algebras), all form quantum loops. Other quantum quasigroups offer natural nonassociative extensions of Hopf algebra constructions: quasigroup algebras, dual quasigroup algebras, and quantum couples of groups with quasigroups. Further examples include an algebra of rooted binary trees, and an algebra of skein polynomials. [ABSTRACT FROM AUTHOR]

Let H be a connected Hopf k -algebra of finite Gel'fand–Kirillov dimension over an algebraically closed field k of characteristic 0. The objects of study in this paper are the left or right coideal subalgebras T of H . They are shown to be deformations of commutative polynomial k -algebras. A number of well-known homological and other properties follow immediately from this fact. Further properties are described, examples are considered, invariants are constructed and a number of open questions are listed. [ABSTRACT FROM AUTHOR]

Under suitable assumptions on the base field, we prove that a commutative semisimple Yetter–Drinfel'd Hopf algebra over a finite abelian group is trivial, i.e., is an ordinary Hopf algebra, if its dimension is relatively prime to the order of the finite abelian group. Furthermore, we prove that a finite-dimensional cocommutative cosemisimple Yetter–Drinfel'd Hopf algebra contains a trivial Yetter–Drinfel'd Hopf subalgebra of dimension greater than one, at least if the Yetter–Drinfel'd Hopf algebra itself has dimension greater than one. [ABSTRACT FROM AUTHOR]

We study Galois descents for categories of mixed Tate motives over O N [ 1 / N ] , for N ∈ { 2 , 3 , 4 , 8 } or O N for N = 6 , with O N the ring of integers of the N th cyclotomic field, and construct families of motivic iterated integrals with prescribed properties. In particular this gives a basis of multiple zeta values via multiple zeta values at roots of unity μ N . It also gives a new proof, via Goncharov's coproduct, of Deligne's results [9] : the category of mixed Tate motives over O k N [ 1 / N ] , for N ∈ { 2 , 3 , 4 , 8 } is spanned by the motivic fundamental groupoid of P 1 ∖ { 0 , μ N , ∞ } with an explicit basis. By applying the period map, we obtain a generating family for multiple zeta values relative to μ N . [ABSTRACT FROM AUTHOR]

Let ( H , R ) be a finite dimensional quasitriangular Hopf algebra over a field k , and M H the representation category of H . In this paper, we study the braided autoequivalences of the Drinfeld center YD H H trivializable on M H . We establish a group isomorphism between the group of those autoequivalences and the group of quantum commutative bi-Galois objects of the transmutation braided Hopf algebra H R . We then apply this isomorphism to obtain a categorical interpretation of the exact sequence of the equivariant Brauer group BM ( k , H , R ) in [18] . To this aim, we have to develop the braided bi-Galois theory established by Schauenburg in [14,15] , which generalizes the Hopf bi-Galois theory over usual Hopf algebras to the one over braided Hopf algebras in a braided monoidal category. [ABSTRACT FROM AUTHOR]

We show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective special linear group over a finite field, corresponding to non-semisimple orbits, have infinite dimension. We spell out a new criterium to show that a rack collapses. [ABSTRACT FROM AUTHOR]

We study generalised differential structures ( Ω 1 , d ) on an algebra A , where A ⊗ A → Ω 1 given by a ⊗ b → a d b need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf algebra left covariant case to pairs ( Λ 1 , ω ) where Λ 1 is a right module and ω a right module map, and the Hopf algebra bicovariant case corresponds to morphisms ω : A + → Λ 1 in the category of right crossed (or Drinfeld–Radford–Yetter) modules over A . When A = U ( g ) the generalised left covariant differential structures are classified by cocycles ω ∈ Z 1 ( g , Λ 1 ) . We then introduce and study the dual notion of a codifferential structure ( Ω 1 , i ) on a coalgebra and for Hopf algebras the self-dual notion of a strongly bicovariant differential graded algebra ( Ω , d ) augmented by a codifferential i of degree −1. Here Ω is a graded super-Hopf algebra extending the Hopf algebra Ω 0 = A and, where applicable, the dual super-Hopf algebra gives the same structure on the dual Hopf algebra. Accordingly, group 1-cocycles correspond precisely to codifferential structures on algebraic groups and function algebras. Among general constructions, we show that first order data ( Λ 1 , ω ) on a Hopf algebra A extends canonically to a strongly bicovariant differential graded algebra via the braided super-shuffle algebra. The theory is also applied to quantum groups with Ω 1 ( C q ( G ) ) dually paired to Ω 1 ( U q ( g ) ) . [ABSTRACT FROM AUTHOR]

We construct a new bigraded Hopf algebra whose bases are indexed by square matrices with entries in the alphabet { 0 , 1 , … , k } , k ⩾ 1 , without null rows or columns. This Hopf algebra generalizes the one of permutations of Malvenuto and Reutenauer, the one of k -colored permutations of Novelli and Thibon, and the one of uniform block permutations of Aguiar and Orellana. We study the algebraic structure of our Hopf algebra and show, by exhibiting multiplicative bases, that it is free. We moreover show that it is self-dual and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra, we obtain a new one indexed by alternating sign matrices. We study some of its properties and algebraic quotients defined through alternating sign matrices statistics. [ABSTRACT FROM AUTHOR]

Given a Hopf algebra H and a projection H → A to a Hopf subalgebra, we construct a Hopf algebra r ( H ) , called the partial dualization of H , with a projection to the Hopf algebra dual to A . This construction provides powerful techniques in the general setting of braided monoidal categories. The construction comprises in particular the reflections of generalized quantum groups [9] . We prove a braided equivalence between the Yetter–Drinfel'd modules over a Hopf algebra and its partial dualization. [ABSTRACT FROM AUTHOR]

The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the classical Heisenberg algebra. The aim of this paper is to study Heisenberg doubles arising from towers of algebras in this manner. First, we develop the basic representation theory of such Heisenberg doubles and show that if induction and restriction satisfy Mackey-like isomorphisms then the Fock space representation of the Heisenberg double has a natural categorification. This unifies the existing categorifications of the polynomial representation of the Weyl algebra and the Fock space representation of the Heisenberg algebra. Second, we develop in detail the theory applied to the tower of 0-Hecke algebras, obtaining new Heisenberg-like algebras that we call quasi-Heisenberg algebras . As an application of a generalized Stone–von Neumann Theorem, we give a new proof of the fact that the ring of quasisymmetric functions is free over the ring of symmetric functions. [ABSTRACT FROM AUTHOR]

Abstract: We study dualities between Lie algebras and Lie coalgebras, and their respective (co)representations. To allow a study of dualities in an infinite-dimensional setting, we introduce the notions of Lie monads and Lie comonads, as special cases of YB–Lie algebras and YB–Lie coalgebras in additive monoidal categories. We show that (strong) dualities between Lie algebras and Lie coalgebras are closely related to (iso)morphisms between associated Lie monads and Lie comonads. In the case of a duality between two Hopf algebras – in the sense of Takeuchi – we recover a duality between a Lie algebra and a Lie coalgebra – in the sense defined in this note – by computing the primitive and the indecomposable elements, respectively. [Copyright &y& Elsevier]