Hopf Subalgebra Research Articles

Flatness of Noetherian Hopf algebras over coideal subalgebras

It is proved in the paper that a Noetherian residually finite-dimensional Hopf algebra H is a flat module over any right Noetherian right coideal subalgebra A. In the case when A is a Hopf subalgebra we get faithful flatness. These results are obtained by verifying the existence of classical quotient rings of A and H. It is also proved that the antipode of either right or left Noetherian residually finite-dimensional Hopf algebra is bijective. As a consequence, such a Hopf algebra is right and left Noetherian simultaneously.

Values of the $$\mathfrak{sl}_2$$ Weight System on Complete Bipartite Graphs

A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassiliev's theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight system corresponding to the colored Jones polynomial. This weight system can be easily defined in terms of the Lie algebra $\mathfrak{sl}_2$, but this definition is too cumbersome from the computational point of view, so that the values of this weight system are known only for some limited classes of chord diagrams. In the present paper we give a formula for the values of the $\mathfrak{sl}_2$ weight system for a class of chord diagrams whose intersection graphs are complete bipartite graphs with no more than three vertices in one of the parts. Our main computational tool is the Chmutov--Varchenko reccurence relation. Furthermore, complete bipartite graphs with no more than three vertices in one of the parts generate Hopf subalgebras of the Hopf algebra of graphs, and we deduce formulas for the projection onto the subspace of primitive elements along the subspace of decomposable elements in these subalgebras. We compute the values of the $\mathfrak{sl}_2$ weight system for the projections of chord diagrams with such intersection graphs. Our results confirm certain conjectures due to S.K.Lando on the values of the weight system $\mathfrak{sl}_2$ at the projections of chord diagrams on the space of primitive elements.

Multigraded Dyson–Schwinger systems

We study systems of combinatorial Dyson–Schwinger equations with an arbitrary number N of coupling constants. The considered Hopf algebra of Feynman graphs is NN-graded, and we wonder if the graded subalgebra generated by the solution is Hopf or not. We first introduce a family of pre-Lie algebras that we classify, dually providing systems generating a Hopf subalgebra; we also describe the associated groups as extensions of groups of formal diffeomorphisms on several variables. We then consider systems coming from Feynman graphs of a quantum field theory. We show that if the number N of independent coupling constants is the number of interactions of the considered quantum field theory, then the generated subalgebra is Hopf. For QED, φ3, and QCD, we also prove that this is the minimal value of N. All these examples are generalizations of the first family of Dyson–Schwinger systems in the one coupling constant case, called fundamental. We also give a generalization of the second family, called cyclic.

A Hopf algebra of subword complexes (Extended abstract)

We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras.

Actions of cocommutative Hopf algebras

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.

Opposite skew left braces and applications

Given a skew left brace B, we introduce the notion of an “opposite” skew left brace B′, which is closely related to the concept of the opposite of a group, and provide several applications. Skew left braces are closely linked with both solutions to the Yang-Baxter Equation and Hopf-Galois structures on Galois field extensions. We show that the set-theoretic solution to the YBE given by B′ is the inverse to the solution given by B. Every Hopf-Galois structure on a Galois field extension L/K gives rise to a skew left brace B; if the underlying Hopf algebra is not commutative, then one can construct an additional “opposite” Hopf-Galois structure (see [1], which relates the Hopf-Galois module structures of each, and refers to the structures as “commuting”); the corresponding skew left brace to this second structure is precisely B′. We show how left ideals (and a newly introduced family of quasi-ideals) of B′ allow us to identify the intermediate fields of L/K which occur as fixed fields of sub-Hopf algebras under this correspondence and to identify which of these are Galois, or Hopf-Galois, over K. Finally, we use the opposite to connect the inverse solution to the YBE and the structure of the Hopf algebra H acting on L/K; this allows us to identify the group-like elements of H.

The unrolled quantum group inside Lusztig’s quantum group of divided powers

In this letter we prove that the unrolled small quantum group, appearing in quantum topology, is a Hopf subalgebra of Lusztig’s quantum group of divided powers. We do so by writing down non-obvious primitive elements with the correct adjoint action. As application, we explain how this gives a realization of the unrolled quantum group as operators on a conformal field theory and match some calculations on this side. In particular, our results explain a prominent weight shift that appears in Feigin and Tipunin (Logarithmic CFTs connected with simple Lie algebras, preprint, 2010. arXiv:1002.5047 ). Our result extends to other Nichols algebras of diagonal type, including super-Lie algebras.

A note on Masuoka’s theorem for semisimple irreducible Hopf algebras

Masuoka proved (Proc Am Math Soc 137(6):1925–1932, 2009) that a finite-dimensional irreducible Hopf algebra H in positive characteristic is semisimple if and only if it is commutative semisimple if and only if the Hopf subalgebra generated by all primitives is semisimple. In this note, we give another proof of this result by using Hochschild cohomology of coalgebras.

Characteristic subgroup lattices and Hopf–Galois structures

The Hopf–Galois structures on normal field extensions [Formula: see text] with [Formula: see text] are in one-to-one correspondence with the set of regular subgroups [Formula: see text] of [Formula: see text], the group of permutations of [Formula: see text] as a set, that are normalized by the left regular representation [Formula: see text]. Each such [Formula: see text] corresponds to a Hopf algebra [Formula: see text] that acts on [Formula: see text]. Such regular subgroups need not be isomorphic to [Formula: see text] but must have the same order. One can divide all such [Formula: see text] into collections [Formula: see text], where [Formula: see text] is the set of those regular [Formula: see text] normalized by [Formula: see text] and isomorphic to a given abstract group [Formula: see text], where [Formula: see text]. There exists an injective correspondence between the characteristic subgroups of a given [Formula: see text] and the set of subgroups of [Formula: see text] stemming from the Galois correspondence between sub-Hopf algebras of [Formula: see text] and intermediate fields [Formula: see text], where [Formula: see text]. We utilize this correspondence to show that for certain pairs [Formula: see text], the collection [Formula: see text] must be empty. This also shows that for these [Formula: see text], there do not exist skew braces with additive group isomorphic to [Formula: see text] and circle group isomorphic to [Formula: see text].

Normality and short exact sequences of Hopf-Galois structures

Every Hopf-Galois structure on a finite Galois extension K/k where corresponds uniquely to a regular subgroup , normalized by , in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra which acts on K/k is . For a given such N we consider the Hopf-Galois structure arising from a subgroup that is also normalized by . This subgroup gives rise to a Hopf sub-algebra with fixed field . By the work of Chase and Sweedler, this yields a Hopf-Galois structure on the extension K/F where the action arises by base changing HP to which is an F-Hopf algebra. We examine this analogy with classical Galois theory, and also examine how the Hopf-Galois structure on K/F relates to that on K/k. We will also pay particular attention to how the Greither-Pareigis enumeration/construction of those HP acting on K/F relates to that of the HN which act on K/k. In the process we also examine short exact sequences of the Hopf algebras which act, whose exactness is directly tied to the descent theoretic description of these algebras.

A semi-abelian extension of a theorem by Takeuchi

We prove that the category of cocommutative Hopf algebras over a field is a semi-abelian category. This result extends a previous special case of it, based on the Milnor–Moore theorem, where the field was assumed to have zero characteristic. Takeuchi's theorem asserting that the category of commutative and cocommutative Hopf algebras over a field is abelian immediately follows from this new observation. We also prove that the category of cocommutative Hopf algebras over a field is action representable. We make some new observations concerning the categorical commutator of normal Hopf subalgebras, and this leads to the proof that two definitions of crossed modules of cocommutative Hopf algebras are equivalent in this context.

Weak quasi-symmetric functions, Rota–Baxter algebras and Hopf algebras

We introduce the Hopf algebra of quasi-symmetric functions with semigroup exponents generalizing the Hopf algebra QSym of quasi-symmetric functions. As a special case we obtain the Hopf algebra QSymN˜ of weak quasi-symmetric functions, which provides a framework for the study of a question proposed by G.-C. Rota relating symmetric type functions and Rota–Baxter algebras. We provide the transformation formulas between the weak monomial and fundamental quasi-symmetric functions, which extends the corresponding results for quasi-symmetric functions. Moreover, we show that QSym is a Hopf subalgebra and a Hopf quotient algebra of QSymN˜. Rota's question is addressed by identifying QSymN˜ with the free commutative unitary Rota–Baxter algebra Ш(x) of weight 1 on one generator x, which also allows us to equip Ш(x) with a Hopf algebra structure.

An In-Depth Look at Quotient Modules

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 its tensor powers up to similarity as modules, or 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 Q and its tensor powers in terms of annihilator ideals, sigma categories, trace ideals, Burnside ring formulas, and if over a semisimple Hopf algebra, the depth of Q in terms of the McKay quiver and the Green ring.

Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups

We define and study dense Frechet subalgebras of compact quantum groups realised as smooth domains associated with a Dirac type operator with compact resolvent. Further, we construct spectral triples on compact matrix quantum groups in terms of Clebsch–Gordon coefficients and the eigenvalues of the Dirac operator {mathcal{D}}. Grotendieck’s theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on them. It is also shown that regular pseudo-differential operators are closed under compositions. As a by-product, we develop elements of the distribution theory and corresponding Fourier analysis. We give applications of our construction to obtain sufficient conditions for Lp − Lq boundedness of coinvariant linear operators. We provide necessary and sufficient conditions for algebraic differential calculi on Hopf subalgebras of compact quantum groups to extend to our proposed smooth subalgebra {{C}^infty_mathcal {D}}. We check explicitly that these conditions hold true on the quantum SU2q for both its 3-dimensional and 4-dimensional calculi.

Indicators of Hopf algebras in positive characteristic

The notion of n-th indicator for a finite-dimensional Hopf algebra was introduced by Kashina, Montgomery, and Ng as invariance of the tensor category of its representations. In this short note, we show that the indicators appearing in positive characteristic all share the same sequence pattern if we assume the coradical of the Hopf algebra is a local Hopf subalgebra.

Skew braces and the Galois correspondence for Hopf Galois structures

Let L/K be a Galois extension of fields with Galois group Γ, and suppose L/K is also an H-Hopf Galois extension. Using the recently uncovered connection between Hopf Galois structures and skew left braces, we introduce a method to quantify the failure of surjectivity of the Galois correspondence from subHopf algebras of H to intermediate subfields of L/K, given by the Fundamental Theorem of Hopf Galois Theory. Suppose L⊗KH=LN where N≅(G,⋆). Then there exists a skew left brace (G,⋆,∘) where (G,∘)≅Γ. We show that there is a bijective correspondence between the set of intermediate fields E between K and L that correspond to K-subHopf algebras of H and a set of sub-skew left braces of G that we call the ∘-stable subgroups of (G,⋆). Counting these subgroups and comparing that number with the number of subgroups of Γ≅(G,∘) describes how far the Galois correspondence for the H-Hopf Galois structure is from being surjective. The method is illustrated by a variety of examples.

On Hopf algebras over quantum subgroups

Using the standard filtration associated with a generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf subalgebra isomorphic to the smallest non-pointed non-cosemisimple Hopf algebra K and the corresponding infinitesimal module is an indecomposable object in YDKK (we assume that the diagrams are Nichols algebras). As a byproduct, we obtain new Nichols algebras of dimension 8 and new Hopf algebras of dimension 64.

On weak peak quasisymmetric functions

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.

Bounds on the number of ideals in finite commutative nilpotent $\mathbb{F}_p$-algebras

Let $A$ be a finite commutative nilpotent $\mathbb{F}_p$-algebra structure on $G$, an elementary abelian group of order $p^n$. If $K/k$ is a Galois extension of fields with Galois group $G$ and $A^p = 0$, then corresponding to $A$ is an $H$-Hopf Galois structure on $K/k$ of type $G$. For that Hopf Galois structure we may study the image of the Galois correspondence from $k$-subHopf algebras of $H$ to subfields of $K$ containing $k$ by utilizing the fact that the intermediate subfields correspond to the $\mathbb{F}_p$-subspaces of $A$, while the subHopf algebras of $H$ correspond to the ideals of $A$. We obtain upper and lower bounds on the proportion of subspaces of $A$ that are ideals of $A$, and test the bounds on some examples.

A propos de l'algèbre de Hopf des mots tassés WMat

In this article we study the packed words Hopf algebra WMat introduced by Duchamp, Hoang-Nghia et Tanasa. We start by explaining that WMat is not cofree, giving its antipode and describing its graded dual. We consider then a Hopf sub-algebra of permutations called SH. Its graded dual SH⊛ has a quadri-algebra structure, so it has a double dendriform algebra structure too. Thereafter, we introduce ISPW, a Hopf algebra of increasing strict packed words. It is graded, connected and cocommutative so is isomorphic to the enveloping algebra of its primitive elements. We describe some families of primitive elements. We prove that ISPW and non commutative symmetric functions are isomorphic. We define then an extended compositions Hopf algebra Ce. It is not cocommutative but its primitive elements and those from ISPW are linked. We give an interpretation of Ce in terms of a semi-direct coproduct Hopf algebra. By using this, we can define two actions groups. We finish by giving an explicit isomorphism between ISPW⊛ and QSym and another one between ISPW and NSym.