In the last decennia two generalizations of the Hopf algebra of symmetric functions have appeared and shown themselves important, the Hopf algebra of noncommutative symmetric functions NSymm and the Hopf algebra of quasisymmetric functions QSymm. It has also become clear that it is important to understand the noncommutative versions of such important structures as Symm the Hopf algebra of symmetric functions. Not least because the right noncommmutative versions are often more beautiful than the commutaive ones (not all cluttered up with counting coefficients). NSymm and QSymm are not truly the full noncommutative generalizations. One is maximally noncommutative but cocommutative, the other is maximally non cocommutative but commutative. There is a common, selfdual generalization, the Hopf algebra of permutations of Malvenuto, Poirier, and Reutenauer (MPR). This one is, I feel, best understood as a Hopf algebra of endomorphisms. In any case, this point of view suggests vast generalizations leading to the Hopf algebras of endomorphisms and word Hopf algebras with which this paper is concerned. This point of view also sheds light on the somewhat mysterious formulas of MPR and on the question where all the extra structure (such as autoduality) comes from. The paper concludes with a few sections on the structure of MPR and the question of algebra retractions of the natural inclusion of Hopf algebras of NSymm into MPR and section of the naural projection of MPR onto QSymm.
Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is the selfdual Hopf algebra of permutations (MPR Hopf algebra). This latter Hopf algebra can be seen as a Hopf algebra of endomorphisms of a Hopf algebra. That turns out to be a fruitful way of looking at things and gives rise to wide ranging further generalizations such as the word Hopf algebra and the double word Hopf algebra (and many more).
We prove a variety results on tensor product factorizations of finite dimensional Hopf algebras (more generally Hopf algebras satisfying chain conditions in suitable braided categories). The results are analogs of well-known results on direct product factorizations of finite groups (or groups with chain conditions) such as Fitting's Lemma and the uniqueness of the Krull-Remak-Schmidt factorization. We analyze the notion of normal (and conormal) Hopf algebra endomorphisms, and the structure of endomorphisms and automorphisms of tensor products. The results are then applied to compute the automo…
In \cite{Kreimer1,Connes,Broadhurst,Kreimer2}, a commutative, non cocommutative Hopf algebra H_R of (decorated) rooted trees was introduced. It is related to the Hopf algebra H_CM introduced in \cite{Moscovici}. Its dual Hopf algebra is the enveloping algebra of the Lie algebra of rooted trees L^1. In this paper, we introduce a non commutative, non cocommutative Hopf algebra H_PR of decorated planar rooted trees. We first show that H_PR is self dual. We then use this result to construct a non degenerate bilinear form (,) on H_PR, which respects the Hopf algebra structure of H_PR. Moreover, we …
We show that the (co)endomorphism algebra of a sufficiently separable "fibre" functor into Vect_k, for k a field of characteristic 0, has the structure of what we call a "unital" von Neumann core in Vect_k. For Vect_k, this particular notion of algebra is weaker than that of a Hopf algebra, although the corresponding concept in Set is again that of a group.
The aim of this paper is an algebraic study of the Hopf algebra H_R of rooted trees, which was introduced in \cite{Kreimer1,Connes,Broadhurst,Kreimer2}. We first construct comodules over H_R from finite families of primitive elements. Furthermore, we classify these comodules by restricting the possible families of primitive elements, and taking the quotient by the action of certain groups. In the next section, we give a formula about primitive elements of the subalgebra of ladders, and construct a projection operator on the space of primitive elements. It allows us to obtain a basis of the pri…
We study the algebra of conformal endomorphisms $\Cend^{G,G}_n$ of a finitely generated free module $M_n$ over the coordinate Hopf algebra $H$ of a linear algebraic group $G$. It is shown that a conformal subalgebra of $\Cend_n$ acting irreducibly on $M_n$ generates an essential left ideal of $\Cend^{G,G}_n$ if enriched with operators of multiplication on elements of $H$. In particular, we describe such subalgebras for the case when $G$ is finite.
We continue our study of the McKay Correspondence for grading preserving actions of semisimple Hopf algebras H on (noncommutative) Artin-Schelter regular algebras A. Here, we establish correspondences between module categories over A^H, over A\#H, and over \End_{A^H} A. We also study homological properties of (endomorphism rings of) maximal Cohen-Macaulay modules over A^H.
Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity.
We show that endomorphisms of Weyl algebras over fields of characteristic zero are flat and that birational endomorphisms are automorphisms by reducing to positive characteristic. We also give examples showing that endomorphisms of Weyl algebras are not in general flat over fields of positive characteristic.
We give a definition of a noncommutative torsor by a subset of the axioms previously given by Grunspan. We show that noncommutative torsors are an equivalent description of Hopf-Galois objects (without specifying the Hopf algebra). In particular, this shows that the endomorphism of a torsor featuring in Grunspan's definition is redundant.
In this paper we introduce a trace-like invariant for the irreducible representations of a finite dimensional complex Hopf algebra H. We do so by considering the trace of the map induced by the antipode S on the endomorphisms End(V) of a self-dual module V. We also compute the values of this trace for the representations of two non-semisimple Hopf algebras: u_q(sl_2) and D(H_n(q)), the Drinfeld double of the Taft algebra.
The coset $G$-space of a finite group and a subgroup is a fundamental module of study of Schur and others around 1930; for example, its endomorphism algebra is a Hecke algebra of double cosets. We study and review its generalization $Q$ to Hopf subalgebras, especially the tensor powers and similarity as modules over a Hopf algebra, or what's the same, Morita equivalence of the endomorphism algebras. We prove that $Q$ has a nonzero integral if and only if the modular function restricts to the modular function of the Hopf subalgebra. We also study and organize knowledge of $Q$ and its tensor pow…
Many related products and coproducts (e.g. Hadamard, Cauchy, Kronecker, induction, internal, external, Solomon, composition, Malvenuto-Reutenauer, convolution, etc.) have been defined in the following objects : species, representations of the symmetric groups, symmetric functions, endomorphisms of graded connected Hopf algebras, permutations, non-commutative symmetric functions, quasi-symmetric functions, etc. With the purpose of simplifying and unifying this diversity we introduce yet, another -non graded- product the Heisenberg product, that for the highest and lowest degrees produces the cl…
We prove the formality theorem for the differential graded Lie algebra module of Hochschild chains for the algebra of endomorphisms of a smooth vector bundle. We discuss a possible application of this result to a version of the algebraic index theorem for Poisson manifolds.
We prove that every endomorphism of a simple quantum generalized Weyl algebra $A$ over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms of $A$. Our main result applies to minimal primitive factors of the quantized enveloping algebra of $U_q(\mathfrak{sl}_2)$ and certain minimal primitive quotients of the positive part of $U_q(\mathfrak{so}_5)$.
In this notes, we study some basic deformation of A-infinity algebra. It includes a two-dimensional rescaling deformation and the Maurer-Cartan element or bounding cochain deformation used in Lagrangian Floer Homology theory. We show that such deformation always can be derived from some (weakly) strict endomorphism equivalently.
Endomorphisms of Weyl algebras are studied using bimodules. Initially, for a Weyl algebra over a field of characteristic zero, Bernstein's inequality implies that holonomic bimodules finitely generated from the right or left form a monoidal category. The most important bimodule in this paper is the graph of an endomorphism. We prove that the graph of an endomorphism of a Weyl algebra over a field of characteristic zero is a simple bimodule. The simplicity of the tensor product of the dual graph and the graph is equivalent to the Dixmier conjecture. It is also shown how the graph construction l…
For a braided vector space $(V,σ)$ with braiding $σ$ of Hecke type, we introduce three associative algebra structures on the space $\oplus_{p=0}^{M}\mathrm{End}S_σ^p(V)$ of graded endomorphisms of the quantum symmetric algebra $S_σ(V)$. We use the second product to construct a new trace. This trace is an algebra morphism with respect to the third product. In particular, when $V$ is the fundamental representation of $\mathcal{U}_{q}\mathfrak{sl}_{N+1}$ and $σ$ is the action of the $R$-matrix, this trace is a scalar multiple of the quantum trace of type $A$.
We introduce a notion of Hopf-Lie-Rinehart algebra and show that the universal algebra of a Hopf-Lie-Rinehart algebra acquires an ordinary Hopf algebra structure.
We present an independent short proof of the main result of arXiv:0706.3725 that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. We derive this from the results of arXiv:0712.1183 about the shift of argument subalgebras.
Let H be a quasi-Hopf algebra, a weak Hopf algebra or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v:H\rightarrow B. Then we can define an object B^{co(H)} which is a left-left Yetter-Drinfeld module over H, having extra properties that allow to make a smash product B^{co(H)}# H which is an H-bicomodule algebra, isomorphic to B.
In this paper, we study the endomorphism properties of vertex operator algebras over an arbitrary field $\mathbb{F}$, with $\text{Char}(\mathbb{F}) \neq 2$. Let $V$ be a strongly finitely generated vertex operator algebra over $\mathbb{F}$, and $M$ be an irreducible admissible $V$-module. We prove that every element in $\text{End}_V(M)$ is algebraic over $\mathbb{F}$ and that $\text{End}_V(M)$ is also finite-dimensional. As an application, we prove Schur's lemma for strongly finitely generated vertex operator algebras over arbitrary algebraically closed fields, and we give a test for absolute …
A Hecke endomorphism algebra is a natural generalisation of the $q$-Schur algebra associated with the symmetric group to a Coxeter group. For Weyl groups, B. Parshall, L. Scott and the first author \cite{DPS,DPS4} investigated the stratification structure of these algebras in order to seek applications to representations of finite groups of Lie type. In this paper we investigate the presentation problem for Hecke endomorphism algebras associated with arbitrary Coxeter groups. Our approach is to present such algebras by quivers with relations. If $R$ is the localisation of $\mathbb Z[q]$ at the…
Borchers and Wiesbrock have demonstrated certain results concerning the one-parameter semigroups of endomorphisms of von Neumann algebras that appear as lightlike translations in the theory of algebras of local observables. These results are abstracted and analyzed as essentially operator-theoretic. Criteria are then demonstrated for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms. All this is combined to establish a von Neumann-algebraic converse to the Borchers-Wiesbrock results. This analysis is then applied to questions of isotony and co…
