Related papers: Graded Algebras, Algebraic Functions, Planar Trees…

The polynomial method has been used recently to obtain many striking results in combinatorial geometry. In this paper, we use affine Hilbert functions to obtain an estimation theorem in finite field geometry. The most natural way to state…

The Z-grading determined by a long simple root of an affine or finite type Lie algebra arises from an adjoint or cominuscule representation of a lower rank semi-simple complex Lie algebra. Analysis of the relationship between the grading…

A simply connected topological space is called \emph{rationally elliptic} if the rank of its total homotopy group and its total (co)homology group are both finite. A well-known Hilali conjecture claims that for a rationally elliptic space…

This review paper contains a concise introduction to highest weight representations of infinite dimensional Lie algebras, vertex operator algebras and Hilbert schemes of points, together with their physical applications to elliptic genera…

We introduce the class of graded Lie-Rinehart algebras as a natural generalization of the one of graded Lie algebras. For $G$ an abelian group, we show that if $L$ is a tight $G$-graded Lie-Rinehart algebra over an associative and…

Let $H$ be a pointed Hopf algebra. We show that under some mild assumptions $H$ and its associated graded Hopf algebra $\gr H$ have the same Gelfand-Kirillov dimension. As an application, we prove that the Gelfand-Kirillov dimension of a…

Multiple elliptic polylogarithms can be written as (multiple) integrals of products of basic hypergeometric functions. The latter are computable, to arbitrary precision, using a q-difference equation and q-contiguous relations.

The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every integer $n$ the number $\varphi_R(n)$, possibly infinite, of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures…

The affine Hilbert function is a classical algebraic object that has been central, among other tools, to the development of the polynomial method in combinatorics. Owing to its concrete connections with Gr\"obner basis theory, as well as…

We study topological Hopf algebras that are holomorphically finitely generated (HFG) as Fr\'echet Arens--Micheal algebras in the sense of Pirkovskii. Some of them, but not all, can be obtained from affine Hopf algebras by applying the…

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically,…

We consider quantum symmetric algebras, FRT bialgebras and, more generally, intertwining algebras for pairs of Hecke symmetries which represent quantum hom-spaces. The paper makes an attempt to investigate Koszulness and Gorensteinness of…

We present a closed form for a multi-variate generating function for the dimensions of the irreducible representations of a semisimple, simply connected linear algebraic group over $\mathbb{C}$ whose highest weights lie in a finitely…

The universal R-matrices and, dually, the coquasitriangular structures of the group Hopf algebra of a finite Abelian group (resp. of an arbitrary Abelian group) are determined. This is used to formulate graded multilinear algebra in terms…

If $R=k[x_1,\ldots,x_n]/I$ is a graded artinian algebra, then the length of $k[x_1,\ldots,x_n]/I^s$ becomes a polynomial in $s$ of degree $n$ for large $s$. If we write this polynomial as $\sum_{i=0}^n(-1)^ie_i{s+n-i-1\choose n-i}$, then…

Let $H$ be a connected graded Hopf algebra over a field of characteristic zero and $K$ an arbitrary graded Hopf subalgebra of $H$. We show that there is a family of homogeneous elements of $H$ and a total order on the index set that satisfy…

Let $A$ be a graded C*-algebra. We characterize Kasparov's K-theory group $\hat{K}_0(A)$ in terms of graded *-homomorphisms by proving a general converse to the functional calculus theorem for self-adjoint regular operators on graded…

Suppose that $h$ and $g$ belong to the algebra $\B$ generated by the rational functions and an entire function $f$ of finite order on ${\Bbb C}^n$ and that $h/g$ has algebraic polar variety. We show that either $h/g\in\B$ or $f=q_1e^p+q_2$,…

We study serial coalgebras by means of their valued Gabriel quivers. In particular, Hom-computable and representation-directed coalgebras are characterized. The Auslander-Reiten quiver of a serial coalgebra is described. Finally, a version…

We classify $n$-representation infinite algebras $\Lambda$ of type \~A. This type is defined by requiring that $\Lambda$ has higher preprojective algebra $\Pi_{n+1}(\Lambda) \simeq k[x_1, \ldots, x_{n+1}] \ast G$, where $G \leq…