Showing papers on "Hopf algebra published in 2014"

Hopf Algebras in Combinatorics

Abstract: These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf algebras in general, we study the Hopf algebra of symmetric functions, including Zelevinsky's axiomatic characterization of it as a "positive self-adjoint Hopf algebra" and its application to the representation theory of symmetric and (briefly) finite general linear groups. The notes then continue with the quasisymmetric and the noncommutative symmetric functions, some Hopf algebras formed from graphs, posets and matroids, and the Malvenuto-Reutenauer Hopf algebra of permutations. Among the results surveyed are the Littlewood-Richardson rule and other symmetric function identities, Zelevinsky's structure theorem for PSHs, the antipode formula for P-partition enumerators, the Aguiar-Bergeron-Sottile universal property of QSym, the theory of Lyndon words, the Gessel-Reutenauer bijection, and Hazewinkel's polynomial freeness of QSym. The notes are written with a graduate student reader in mind, being mostly self-contained but requiring a good familiarity with multilinear algebra and -- for the representation-theory applications -- basic group representation theory.

On finite-dimensional Hopf algebras

Abstract: This is a survey on the state-of-the-art of the classification of finite-dimensional complex Hopf algebras. This general question is addressed through the consideration of different classes of such Hopf algebras. Pointed Hopf algebras constitute the class best understood; the classification of those with abelian group is expected to be completed soon and there is substantial progress in the non-abelian case.

Noncommutative connections on bimodules and Drinfeld twist deformation

Abstract: Given a Hopf algebra H, we study modules and bimodules over an algebra A that carry an H-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For quasitriangular Hopf algebras and bimodules with an extra quasi-commutativity property we induce connections on the tensor product over A of two bimodules from connections on the individual bimodules. This construction applies to arbitrary connections, i.e. not necessarily H-equivariant ones, and further extends to the tensor algebra generated by a bimodule and its dual. Examples of these noncommutative structures arise in deformation quantization via Drinfeld twists of the commutative differential geometry of a smooth manifold, where the Hopf algebra H is the universal enveloping algebra of vector fields (or a finitely generated Hopf subalgebra). We extend the Drinfeld twist deformation theory of modules and algebras to morphisms and connections that are not necessarily H-equivariant. The theory canonically lifts to the tensor product structure.

The Bannai-Ito polynomials as Racah coefficients of the sl_{-1}(2) algebra

Abstract: The Bannai-Ito polynomials are shown to arise as Racah coeffi- cients for sl−1(2). This Hopf algebra has four generators including an involu- tion and is defined with both commutation and anticommutation relations. It is also equivalent to the parabosonic oscillator algebra. The coproduct is used to show that the Bannai-Ito algebra acts as the hidden symmetry algebra of the Racah problem for sl−1(2). The Racah coefficients are recovered from a related Leonard pair.

Radford's biproducts and Yetter-Drinfeld modules for monoidal Hom-Hopf algebras

Abstract: Let (H, α) be a monoidal Hom-bialgebra and (B, β) be a left (H, α)-Hom-module algebra and also a left (H, α)-Hom-comodule coalgebra. Then in this paper, we first introduce the notion of a Hom-smash coproduct, which is a monoidal Hom-coalgebra. Second, we find sufficient and necessary conditions for the Hom-smash product algebra structure and the Hom-smash coproduct coalgebra structure on B ⊗ H to afford B ⊗ H a monoidal Hom-bialgebra structure, generalizing the well-known Radford's biproduct, where the conditions are equivalent to that (B, β) is a bialgebra in the category of Hom-Yetter-Drinfeld modules HYDHH. Finally, we illustrate the category of Hom-Yetter-Drinfeld modules HYDHH and prove that the category HYDHH is a braided monoidal category.

Mathematical aspects of scattering amplitudes

Abstract: In these lectures we discuss some of the mathematical structures that appear when computing multi-loop Feynman integrals. We focus on a specific class of special functions, the so-called multiple polylogarithms, and discuss introduce their Hopf algebra structure. We show how these mathematical concepts are useful in physics by illustrating on several examples how these algebraic structures are useful to perform analytic computations of loop integrals, in particular to derive functional equations among polylogarithms.

κ-Deformed Phase Space, Hopf Algebroid and Twisting

Abstract: Hopf algebroid structures on the Weyl algebra (phase space) are presented. We define the coproduct for the Weyl generators from Leibniz rule. The codomain of the coproduct is modified in order to obtain an algebra structure. We use the dual base to construct the target map and antipode. The notion of twist is analyzed for -deformed phase space in Hopf algebroid setting. It is outlined how the twist in the Hopf algebroid setting reproduces the full Hopf algebra structure of -Poincar e algebra. Several examples of realizations are worked out in details.

Hopf algebras in non-associative Lie theory

Abstract: We review the developments in the Lie theory for non-associative products from 2000 to date and describe the current understanding of the subject in view of the recent works, many of which use non-associative Hopf algebras as the main tool.

On quadri-algebras

Abstract: We describe the Koszul dual of the operad Quad of quadri-algebras, show the koszularity of Quad and give the formal series of Quad and its dual, which proves a conjecture due to Aguiar and Loday. A notion of quadri-bialgebra is also introduced, with applications to the Hopf algebras FQSym and WQSym.

Classifying Bicrossed Products of Hopf Algebras

Abstract: Let A and H be two Hopf algebras. We shall classify up to an isomorphism that stabilizes A all Hopf algebras E that factorize through A and H by a cohomological type object ${\mathcal H}^{2} (A, H)$ . Equivalently, we classify up to a left A-linear Hopf algebra isomorphism, the set of all bicrossed products A ⋈ H associated to all possible matched pairs of Hopf algebras $(A, H, \triangleleft, \triangleright)$ that can be defined between A and H. In the construction of ${\mathcal H}^{2} (A, H)$ the key role is played by special elements of $CoZ^{1} (H, A) \times {\rm Aut}\,_{\rm CoAlg}^1 (H)$ , where CoZ 1 (H, A) is the group of unitary cocentral maps and ${\rm Aut}\,_{\rm CoAlg}^1 (H)$ is the group of unitary automorphisms of the coalgebra H. Among several applications and examples, all bicrossed products H 4 ⋈ k[C n ] are described by generators and relations and classified: they are quantum groups at roots of unity H 4n, ω which are classified by pure arithmetic properties of the ring ℤ n . The Dirichlet’s theorem on primes is used to count the number of types of isomorphisms of this family of 4n-dimensional quantum groups. As a consequence of our approach the group Aut Hopf(H 4n, ω ) of Hopf algebra automorphisms is fully described.

Hopf algebras and Markov chains: two examples and a theory

Abstract: The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural “rock-breaking” process, and Markov chains on simplicial complexes. Many of these chains can be explicitly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.

Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras

Abstract: The question of whether or not a Hopf algebra $H$ is faithfully flat over a Hopf subalgebra $A$ has received positive answers in several particular cases: when $H$ (or more generally, just $A$) is commutative, or cocommutative, or pointed, or when $K$ contains the coradical of $H$. We prove the statement in the title, adding the class of cosemisimple Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting "exact sequence" $A\to H\to C$ is always a cosemisimple coalgebra, and that the expectation $H\to A$ is positive when $H$ is a CQG algebra.

Hopf algebra of permutation pattern functions

Abstract: We study permutation patterns from an algebraic combinatorics point of view. Using analogues of the classical shuffle and infiltration products for word, we define two new Hopf algebras of permutations related to the notion of permutation pattern. We show several remarkable properties of permutation patterns functions, as well their occurrence in other domains.

Cambrian Hopf Algebras

Abstract: Cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the conditions for classical binary search trees. Based on the bijective correspondence between signed permutations and leveled Cambrian trees, we define the Cambrian Hopf algebra generalizing J.-L. Loday and M. Ronco's algebra on binary trees. We describe combinatorially the products and coproducts of both the Cambrian algebra and its dual in terms of operations on Cambrian trees. We also define multiplicative bases of the Cambrian algebra and study structural and combinatorial properties of their indecomposable elements. Finally, we extend to the Cambrian setting different algebras connected to binary trees, in particular S. Law and N. Reading's Baxter Hopf algebra on quadrangulations and S. Giraudo's equivalent Hopf algebra on twin binary trees, and F. Chapoton's Hopf algebra on all faces of the associahedron.

Examples of support varieties for Hopf algebras with noncommutative tensor products

Abstract: The representations of some Hopf algebras have curious behavior: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. We explain a family of examples of such Hopf algebras and their modules, and classify left, right, and two-sided ideals in their stable module categories.

Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras

Abstract: Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Faa di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants. We also discuss Mobius inversion in certain Hopf algebras built from Bell polynomials.

The Green Ring of Drinfeld Double D ( H 4 )

Abstract: In this paper, we study the Green ring (or the representation ring) of Drinfeld quantum double D(H4) of Sweedler’s four-dimensional Hopf algebra H4. We first give the decompositions of the tensor products of finite dimensional indecomposable modules into the direct sum of indecomposable modules over D(H4). Then we describe the structure of the Green ring r(D(H4)) of D(H4) and show that r(D(H4)) is generated, as a ring, by infinitely many elements subject to a family of relations.

Decomposition spaces, incidence algebras and Möbius inversion

Abstract: We introduce the notion of decomposition space as a general framework for incidence algebras and Mobius inversion: it is a simplicial infinity-groupoid satisfying an exactness condition weaker than the Segal condition, which expresses decomposition. We work on the objective level of homotopy linear algebra with coefficients in infinity-groupoids, developed along the way. To any (complete) decomposition space there is associated an incidence (co)algebra (with coefficients in infinity-groupoids), shown to satisfy a sign-free version of the Mobius inversion principle. Examples of decomposition spaces beyond Segal spaces are given by the Waldhausen S-construction and by Schmitt restriction species. Imposing certain homotopy finiteness conditions yields the notion of Mobius decomposition space, an extension of the notion of Mobius category of Leroux. We take a functorial viewpoint throughout, emphasising conservative ULF functors, and show that most reduction procedures in the classical theory are examples of this notion, and in particular that many are examples of decalage of decomposition spaces. Our main theorem concerns the Lawvere-Menni Hopf algebra of Mobius intervals, which contains the universal Mobius function (but does not come from a Mobius category): we establish that Mobius intervals form a decomposition space, which is in some sense universal. NOTE: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov (arXiv:1212.3563) who call it unital 2-Segal space. Our theory is quite orthogonal to theirs.

Quantum groups of GL(2) representation type

Abstract: We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear form. A detailed study of these Hopf algebras gives us an isomorphic classification and the description of their corepresentation categories.

From Hopf algebras to tensor categories

Abstract: This is a survey on spherical Hopf algebras. We give criteria to decide when a Hopf algebra is spherical and collect examples. We discuss tilting modules as a mean to obtain a fusion subcategory of the non-degenerate quotient of the category of representations of a suitable Hopf algebra.

Module categories over affine group schemes

Abstract: Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $G$ be an affine group scheme over $k$. We classify the indecomposable exact module categories over the rigid tensor category $\text{Coh}_f(G)$ of coherent sheaves of finite dimensional $k-$vector spaces on $G$, in terms of $(H,\psi)-$equivariant coherent sheaves on $G$. We deduce from it the classification of indecomposable {\em geometrical} module categories over $\Rep(G)$. When $G$ is finite, this yields the classification of {\em all} indecomposable exact module categories over the finite tensor category $\Rep(G)$. In particular, we obtain a classification of twists for the group algebra $k[G]$ of a finite group scheme $G$. Applying this to $u(\mathfrak {g})$, where $\mathfrak {g}$ is a finite dimensional $p-$Lie algebra over $k$ with positive characteristic, produces (new) finite dimensional noncommutative and noncocommutative triangular Hopf algebras in positive characteristic. We also introduce and study group scheme theoretical categories, and study isocategorical finite group schemes.

Green Rings of Pointed Rank One Hopf algebras of Non-nilpotent Type

Abstract: In this paper, we continue our study of the Green rings of finite dimensional pointed Hopf algebras of rank one initiated in \cite{WLZ}, but focus on those Hopf algebras of non-nilpotent type. Let $H$ be a finite dimensional pointed rank one Hopf algebra of non-nilpotent type. We first determine all non-isomorphic indecomposable $H$-modules and describe the Clebsch-Gordan formulas for them. We then study the structures of both the Green ring $r(H)$ and the Grothendieck ring $G_0(H)$ of $H$ and establish the precise relation between the two rings. We use the Cartan map of $H$ to study the Jacobson radical and the idempotents of $r(H)$. It turns out that the Jacobson radical of $r(H)$ is exactly the kernel of the Cartan map, a principal ideal of $r(H)$, and $r(H)$ has no non-trivial idempotents. Besides, we show that the stable Green ring of $H$ is a transitive fusion ring. This enables us to calculate Frobenius-Perron dimensions of objects of the stable category of $H$. Finally, as an example, we present both the Green ring and the Grothendieck ring of the Radford Hopf algebra.