Related papers: Generalized affine buildings for semisimple algebr…

Let $\textbf{U}^+$ be the positive part of the quantum group $\textbf{U}$ associated with a generalized Cartan matrix. In the case of finite type, Lusztig constructed the canonical basis $\textbf{B}$ of $\textbf{U}^+$ via two approaches.…

Semifields are semirings in which every nonzero element has a multiplicative inverse. A rough classification uses the characteristic of the semifield, that is the isomorphism type of the semifield generated by the two neutral elements. For…

We give a Thurston-like definition for laminations on higher Teichmuller spaces associated to a surface $S$ and a semi-simple group $G$ for $G-SL_m$ and $PGL_m$. The case $G=SL_2$ or $PGL_2$ corresponds to the classical theory of…

A constructive method for decomposing finite dimensional representations of semisimple real Lie algebras is developed. The method is illustrated by an example. We also discuss an implementation of the algorithm in the language of the…

We prove a characterization of profinite algebras, i.e., topological algebras that are isomorphic to a projective limit of finite discrete algebras. In general profiniteness concerns both the topological and algebraic characteristics of a…

The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for…

In this paper, we present a simple lattice-theoretic characterization for affine buildings of type A. We introduce a class of modular lattices, called uniform modular lattices, and show that uniform modular lattices and affine buildings of…

The complete algebraic structure of semisimple finite group algebra of a generalized strongly monomial group is provided. This work extends the work of Broche and del R{\'{\i}}o on strongly monomial groups. The theory is complimented by an…

The affine line and the punctured affine line over a finite field F are taken as benchmarks for the problem of describing geometric \'etale fundamental groups. To this end, using a reformulation of Tannaka duality we construct for a…

We propose a general method for constructing boundary integrable Gaudin models associated with (twisted) affine algebras ${\cal G}^{(k)} (k=1, 2)$, where ${\cal G}$ is a simple Lie algebra or superalgebra. Many new integrable Gaudin models…

We prove that an algebraic stack with affine stabilizers over an arbitrary base is \'etale-locally a quotient stack around any point with a linearly reductive stabilizer. This generalizes earlier work by the authors of this article (stacks…

In this article we describe the general and special projectivity groups for all irreducible residues of all thick, irreducible, spherical buildings of type $ \mathsf{B_{n}}$, $ \mathsf{C_{n}}$ and $\mathsf{F_4}$, and rank at least 3. This…

In this paper, we show that the building at infinity of a two-dimensional affine R-building is a generalized polygon endowed with a valuation satisfying some specific axioms. Specializing to the discrete case of affine buildings, this…

This work demonstrates that a complete description of the interaction of matter and all forces, gravitational and non-gravitational, can in fact be realized within a quantum affine algebraic framework. Using the affine group formalism, we…

We study amenability of affine algebras (based on the notion of almost-invariant finite-dimensional subspace), and apply it to algebras associated with finitely generated groups. We show that a group G is amenable if and only if its group…

We consider the problem of classifying gradings by groups on a finite-dimensional algebra $A$ (with any number of multilinear operations) over an algebraically closed field. We introduce a class of gradings, which we call almost fine, such…

We show that for algebraic groups over local fields of characteristic zero, the following are equivalent: Every homomorphism has a closed image, every unitary representation decomposes into a direct sum of finite-dimensional and mixing…

In classical geometric algebra, there have been several treatments of affine and projective planes based on fields. In this thesis we approach affine and projective planes from a constructive point of view and we base our geometry on local…

We study the local analytic classification of affine structures with logarithmic pole on complex surfaces. With this result in hand, we can get the local classification of the logarithmic parallelizable d-webs, d $\ge$ 3.

In this note, we prove that an affine cellular algebra $A$ is semisimple if and only if the scheme associated to $A$ is reduced and 0-dimensional, and the bilinear forms with respect to all layers of $A$ are isomorphisms. Moreover, if the…