Connected Hopf Algebras Research Articles

Coideal subalgebras of pointed and connected Hopf algebras

Let H H be a pointed Hopf algebra with abelian coradical. Let A ⊇ B A\supseteq B be left (or right) coideal subalgebras of H H that contain the coradical of H H . We show that A A has a PBW basis over B B , provided that H H satisfies certain mild conditions. In the case that H H is a connected graded Hopf algebra of characteristic zero and A A and B B are both homogeneous of finite Gelfand-Kirillov dimension, we show that A A is a graded iterated Ore extension of B B . These results turn out to be conceptual consequences of a structure theorem for each pair S ⊇ T S\supseteq T of homogeneous coideal subalgebras of a connected graded braided bialgebra R R with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well-behaved PBW basis of S S over T T . The approach to the structure theorem is constructive by means of a combinatorial method based on Lyndon words and braided commutators, which is originally developed by V. K. Kharchenko [Algebra Log. 38 (1999), pp. 476–507, 509] for primitively generated braided Hopf algebras of diagonal type. Since in our context we don’t priorilly assume R R to be primitively generated, new methods and ideas are introduced to handle the corresponding difficulties, among others.

The Eigenvalues of Hyperoctahedral Descent Operators and Applications to Card-Shuffling


 We extend an algebra of Mantaci and Reutenauer, acting on the free associative algebra, to a vector space of operators acting on all graded connected Hopf algebras. These operators are convolution products of certain involutions, which we view as hyperoctahedral variants of Patras's descent operators. We obtain the eigenvalues and multiplicities of all our new operators, as well as a basis of eigenvectors for a subclass akin to Adams operations. We outline how to apply this eigendata to study Markov chains, and examine in detail the case of card-shuffles with flips or rotations.

On actions of connected Hopf algebras

Let H H be a connected Hopf algebra acting on an algebra A A . Working over a base field having characteristic 0 0 , we show that for a given prime (semi-prime, completely prime) ideal I I of A A , the largest H H -stable ideal of A contained in I I is also prime (semi-prime, completely prime). We also prove a similar result for certain subrings of convolution algebras.

Lie groups of controlled characters of combinatorial Hopf algebras

In this article groups of “controlled” characters of a combinatorial Hopf algebra are considered from the perspective of infinite-dimensional Lie theory. A character is controlled in our sense if it satisfies certain growth bounds, e.g. exponential growth. We study these characters for combinatorial Hopf algebras. Following Loday and Ronco, a combinatorial Hopf algebra is a graded and connected Hopf algebra which is a polynomial algebra with an explicit choice of basis (usually identified with combinatorial objects such as trees, graphs, etc.). If the growth bounds and the Hopf algebra are compatible we prove that the controlled characters forminfinite-dimensional Lie groups. Further, we identify the Lie algebra and establish regularity results (in the sense of Milnor) for these Lie groups. The general construction principle exhibited here enables to treat a broad class of examples from physics, numerical analysis and control theory. Groups of controlled characters appear in renormalisation of quantum field theories, numerical analysis and control theory in the guise of groups of locally convergent power series. The results presented here generalise the construction of the (tame) Butcher group, also known as the controlled character group of the Butcher–Connes–Kreimer Hopf algebra.

The structure of connected (graded) Hopf algebras

In this paper, we establish a structure theorem for connected graded Hopf algebras over a field of characteristic 0 by claiming the existence of a family of homogeneous generators and a total order on the index set that satisfy some desirable conditions. The approach to the structure theorem is constructive, based on the combinatorial properties of Lyndon words and the standard bracketing on words. As a surprising consequence of the structure theorem, we show that connected graded Hopf algebras of finite Gelfand-Kirillov dimension over a field of characteristic 0 are all iterated Hopf Ore extensions of the base field. In addition, some keystone facts of connected Hopf algebras over a field of characteristic 0 are observed as corollaries of the structure theorem, without the assumptions of having finite Gelfand-Kirillov dimension (or affineness) on Hopf algebras or of that the base field is algebraically closed.

On the structure and cohomology ring of connected Hopf algebras

Let p be a prime, and k be a field of characteristic p. We investigate the algebra structure and the structure of the cohomology ring for the connected Hopf algebras of dimension p3, which appear in the classification obtained in [20]. The list consists of 23 algebras together with two infinite families. We identify the Morita type of the algebra, and in almost all cases this is sufficient to clarify the structure of the cohomology ring.

GK dimension of some connected Hopf algebras

While it was identified that the growth of any connected Hopf algebras is either a positive integer or infinite, we have yet to determine the Gelfand–Kirillov (GK) dimension of a given connected Hopf algebra. We use the notion of anti-cocommutative elements introduced in Wang, D. G., Zhang, J. J., Zhuang, G. (2013). Coassociative lie algebras. Glasgow Math. J. 55(A):195–215 to analyze the structure of connected Hopf algebras generated by anti-cocommutative elements and compute the GK dimension of said algebras. Additionally, we apply these results to compare global dimension of connected Hopf algebras and the dimension of their corresponding Lie algebras of primitive elements.

Primitive Deformations of Quantum p-Groups

In this paper, working over an algebraically closed field k of prime characteristic p, we introduce a concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional Hopf algebras which are Hopf deformations of restricted universal enveloping algebras. We illustrate this technique for the case when the restricted Lie algebra has dimension 3. Together with our previous classification results, we provide a complete classification of p3-dimensional connected Hopf algebras over k of characteristic p > 2.

The Heisenberg product: from Hopf algebras and species to symmetric functions

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 product the Heisenberg product, that is not graded and its highest and lowest degree-terms are the classical external and internal products (and their namesakes in different contexts). In order to define it, we start from the two opposite more general extremes: species in the “commutative context”, and endomorphisms of Hopf algebras in the “non-commutative” environment. Both specialize to the space of commutative symmetric functions where the definitions coincide. We also deal with the different coproducts that these objects carry, to which we add the Heisenberg coproduct for quasi-symmetric functions, and study their Hopf algebra compatibility particularly for symmetric and non commutative symmetric functions. We obtain combinatorial formulas for the structure constants of the new product that extend, generalize and unify results due to Garsia, Remmel, Reutenauer and Solomon. In the space of quasi-symmetric functions, we describe explicitly the new operations in terms of alphabets.

Quantum homogeneous spaces of connected Hopf algebras

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.

Character groups of Hopf algebras as infinite-dimensional Lie groups

In this article character groups of Hopf algebras are studied from the perspective of infinite-dimensional Lie theory. For a graded and connected Hopf algebra we obtain an infinite-dimensional Lie group structure on the character group with values in a locally convex algebra. This structure turns the character group into a Baker-Campbell-Hausdorff-Lie group which is regular in the sense of Milnor. Furthermore, we show that certain subgroups associated to Hopf ideals become closed Lie subgroups of the character group. If the Hopf algebra is not graded, its character group will in general not be a Lie group. However, we show that for any Hopf algebra the character group with values in a weakly complete algebra is a pro-Lie group in the sense of Hofmann and Morris.

Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic

We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair associated with these extensions, we construct a cohomology group, which classifies all the extensions up to equivalence. Moreover, we present a 1–1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points, where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted Lie algebra extensions.

The characteristic polynomial of the Adams operators on graded connected Hopf algebras

The Adams operators $\Psi_n$ on a Hopf algebra $H$ are the convolution powers of the identity of $H$. We study the Adams operators when $H$ is graded connected. They are also called Hopf powers or Sweedler powers. The main result is a complete description of the characteristic polynomial (both eigenvalues and their multiplicities) for the action of the operator $\Psi_n$ on each homogeneous component of $H$. The eigenvalues are powers of $n$. The multiplicities are independent of $n$, and in fact only depend on the dimension sequence of $H$. These results apply in particular to the antipode of $H$ (the case $n=-1$). We obtain closed forms for the generating function of the sequence of traces of the Adams operators. In the case of the antipode, the generating function bears a particularly simple relationship to the one for the dimension sequence. In case H is cofree, we give an alternative description for the characteristic polynomial and the trace of the antipode in terms of certain palindromic words. We discuss parallel results that hold for Hopf monoids in species and $q$-Hopf algebras.

Connected Hopf algebras of Gelfand-Kirillov dimension four

We classify connected Hopf algebras of Gelfand-Kirillov dimension 4 over an algebraically closed field of characteristic zero.

PROPERTIES OF DUAL MODULES OVER STEENROD ALGEBRA

Properties of annulators and modules generated by annulators, including dual modules over Steenrod algebra are studied. Properties of Kroneker pairing are proved using general properties of Steenrod algebra and dual algebra as graded connected Hopf algebras. Isomorphisms between modules generated by annulators and dual modules over dual Stennrod algebra are proved. It is shown that these modules are Hopf comodules induced by coproduct in dual Steenrod algebra. All generators of these modules are found. The method of finding basis of module of indecomposable elements, viewed as vector space over cyclic field for some of the studied modules

Classification of connected Hopf algebras of dimension [formula omitted] I

Let p be a prime, and k be an algebraically closed field of characteristic p. In this paper, we provide the classification of connected Hopf algebras of dimension p3, except for the case when the primitive space of the Hopf algebra is a two-dimensional abelian restricted Lie algebra. Each isomorphism class is presented by generators x, y, z with relations and Hopf algebra structures. Let μ be the multiplicative group of (p2+p−1)-th roots of unity. When the primitive space is one-dimensional and p is odd, there is an infinite family of isomorphism classes, which is naturally parameterized by Ak1/μ.

Connected Hopf algebras and iterated Ore extensions

We investigate when a skew polynomial extension T=R[x;σ,δ] of a Hopf algebra R admits a Hopf algebra structure, substantially generalising a theorem of Panov. When this construction is applied iteratively in characteristic 0 one obtains a large family of connected noetherian Hopf algebras of finite Gelfand–Kirillov dimension, including for example all enveloping algebras of finite dimensional solvable Lie algebras and all coordinate rings of unipotent groups. The properties of these Hopf algebras are investigated.

Hopf algebras, distributive (Laplace) pairings and hash products: a unified approach to tensor product decompositions of group characters

We show for bicommutative graded connected Hopf algebras that a certain distributive (Laplace) subgroup of the convolution monoid of 2-cochains parameterizes certain well behaved Hopf algebra deformations. Using the Laplace group, or its Frobenius subgroup, we define higher derived hash products, and develop a general theory to study their main properties. Applying our results to the (universal) bicommutative graded connected Hopf algebra of symmetric functions, we show that classical tensor product and character decompositions, such as those for the general linear group, mixed co- and contravariant or rational characters, orthogonal and symplectic group characters, Thibon and reduced symmetric group characters, are special cases of higher derived hash products. In the appendix we discuss a relation to formal group laws.

COASSOCIATIVE LIE ALGEBRAS

AbstractA coassociative Lie algebra is a Lie algebra equipped with a coassociative coalgebra structure satisfying a compatibility condition. The enveloping algebra of a coassociative Lie algebra can be viewed as a coalgebraic deformation of the usual universal enveloping algebra of a Lie algebra. This new enveloping algebra provides interesting examples of non-commutative and non-cocommutative Hopf algebras and leads to the classification of connected Hopf algebras of Gelfand–Kirillov dimension four in Wang et al. (Trans. Amer. Math. Soc., to appear).

Connected Hopf algebras of dimension [formula omitted

Let H be a finite-dimensional connected Hopf algebra over an algebraically closed field k of characteristic p>0. We provide the algebra structure of the associated graded Hopf algebra grH. Then, we study the case when H is generated by a Hopf subalgebra K and another element and the case when H is cocommutative. When H is a restricted universal enveloping algebra, we give a specific basis for the second term of the Hochschild cohomology of the coalgebra H with coefficients in the trivial H-bicomodule k. Finally, we classify all connected Hopf algebras of dimension p2 over k.