Group orbits and regular partitions of Poisson manifolds

We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties ${\mathcal L}$ of Lagrangian subalgebras of reductive quadratic Lie algebras \d with Poisson structures defined by Lagrangian splittings of \d. In the special case of \g \oplus \g, where \g is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on ${\mathcal L}$ defined by arbitrary Lagrangian splittings of ${\mathfrak g} \oplus {\mathfrak g}$. Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin--Drinfeld splittings as special cases.Comment: 23 pages, AMS Latex, minor changes in v.

Fan-Linear Maps and Fan Algebras

Fan algebras arise from fan-linear maps, a special class of functions defined on partitions of the nonnegative integer lattice in the plane. These algebras are natural objects of study in commutative algebra as they include many classical examples of commutative rings. Additionally, the ubiquity of this structure has only recently been identified, therefore little is known regarding the properties of these algebras. We begin our study by classifying all fan-linear maps via the conditions imposed on them by their domains. This classification includes a general result regarding the universal group of cones in lattices of arbitrary dimension. We then go on to show that the set of all fan-linear maps on any fixed partition is necessarily a finitely-generated affine semigroup. Finally, this leads to the conclusion that the set of fan algebras corresponding to a fixed partition and a fixed set of ideals forms a finitely generated semigroup. This is accomplished through the identification of generating maps in the semigroup of all fan-linear maps with generating algebras and the description of a natural additive operation

Fixed Point Algebras for Easy Quantum Groups

Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their $K$-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the $K$-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group $S_n^+$, the free orthogonal quantum group $O_n^+$ and the quantum reflection groups $H_n^{s+}$. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions

A general framework for the polynomiality property of the structure coefficients of double-class algebras

Take a sequence of couples $(G_n,K_n)_n$, where $G_n$ is a group and $K_n$ is a sub-group of $G_n.$ Under some conditions, we are able to give a formula that shows the form of the structure coefficients that appear in the product of double-classes of $K_n$ in $G_n.$ We show how this can give us a similar result for the structure coefficients of the centers of group algebras. These formulas allow us to re-obtain the polynomiality property of the structure coefficients in the cases of the center of the symmetric group algebra and the Hecke algebra of the pair $(\mathcal{S}_{2n},\mathcal{B}_{n}).$ We also give a new polynomiality property for the structure coefficients of the center of the hyperoctahedral group algebra and the double-class algebra $\mathbb{C}[diag(\mathcal{S}_{n-1})\setminus \mathcal{S}_n\times \mathcal{S}^{opp}_{n-1}/ diag(\mathcal{S}_{n-1})].

Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras

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 Fa\`a 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 M\"obius inversion in certain Hopf algebras built from Bell polynomials.Comment: 37 pages, final version, to appear in IJA

Mullineux involution and twisted affine Lie algebras

We use Naito-Sagaki's work [S. Naito & D. Sagaki, J. Algebra 245 (2001) 395--412, J. Algebra 251 (2002) 461--474] on Lakshmibai-Seshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We characterize the set of Mullineux-fixed partitions in terms of crystal graphs of basic representations of twisted affine Lie algebras of type $A_{2\ell}^{(2)}$ and of type $D_{\ell+1}^{(2)}$. We set up bijections between the set of symmetric partitions and the set of partitions into distinct parts. We propose a notion of double restricted strict partitions. Bijections between the set of restricted strict partitions (resp., the set of double restricted strict partitions) and the set of Mullineux-fixed partitions in the odd case (resp., in the even case) are obtained

JMMO Fock space and Geck-Rouquier classification of simple modules for Hecke algebras

This paper is a survey on the representation theory of Hecke algebras, Ariki-Koike algebras and connections with quantum group