相关论文: A new complex reflection group in $PU(9,1)$ and th…

We compute the fundamental group of a toroidal compactification of a Hermitian locally symmetric space $D/\Gamma$, without assuming either that $\Gamma$is neat or that it is arithmetic. We also give bounds for the first Betti number.

We study the mirrors of five-parameter Calabi-Yau threefolds first studied by Hulek and Verrill in the context of observed modular behaviour of the zeta functions for Calabi-Yau manifolds. Toric geometry allows for a simple explicit…

Let $\Gamma$ be a dual polar graph with diameter $D \geqslant 3$, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field $\mathbb{F}_q$ equipped with a non-degenerate form (alternating,…

We study relations between reflections in (positive or negative) points in the complex hyperbolic plane. It is easy to see that the reflections in the points q_1,q_2 obtained from p_1,p_2 by moving p_1,p_2 along the geodesic generated by…

Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (=a finite union of hyperplanes) whose Levi-Civita connection is of Dunkl…

Let $\Gamma$ be a finite group acting on a simple Lie algebra $\mathfrak{g}$ and acting on a $s$-pointed projective curve $(\Sigma, \vec{p}=\{p_1, \dots, p_s\})$ faithfully (for $s\geq 1$). Also, let an integrable highest weight module…

We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma$ be a finite index subgroup of $\mathrm{SL}(2,\mathbb{Z})$ with an action on a complex simple Lie algebra $\mathfrak…

Let $\Gamma$ be a graph product of finite groups, with finite underlying graph, and let $\Delta$ be the associated right-angled building. We prove that a uniform lattice $\Lambda$ in the cubical automorphism group Aut$(\Delta)$ is weakly…

This paper extends the foundational reflection theory of Nichols algebras to the setting of some certain coquasi-Hopf algebras. Our primary motivation arises from the classification of pointed finite-dimensional coquasi-Hopf algebras. We…

As is known, the irreducible projective representations (Reps) of anti-unitary groups contain three different situations, namely, the real, the complex and quaternion types with torsion number 1,2,4 respectively. This subtlety increases the…

Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices…

We discuss an interesting duality known to occur for certain complex reflection groups, namely the duality groups. Our main construction yields a concrete, representation theoretic realisation of this duality. This allows us to naturally…

We describe the mirror of the Z orbifold as a representation of a class of generalized Calabi-Yau manifolds that can be realized as manifolds of dimension five and seven. Despite their dimension these correspond to superconformal theories…

In a discrete group generated by hyperplane reflections in the $n$-dimensional hyperbolic space, the reflection length of an element is the minimal number of hyperplane reflections in the group that suffices to factor the element. For a…

If $\Delta$ and $\Gamma$ are two derivations of a commutative algebra $A$ such that $\Delta\Gamma-\Gamma\Delta=\Delta$ is locally nilpotent, one can endow $A$ with a new product $\ast$ whose filtered semiclassical limit is the Poisson…

Let $(G,G^\Gamma)$ be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs $(G,G^\Gamma)$ such that there exists at least one nontrivial unitarizable simple $(\mathfrak{g},K)$-module $\pi_K$ that is…

Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary $C^{\infty}$-smooth hypersurface $\gamma\subset\mathbb R^{n+1}$ that is either a…

A discrete subgroup $\Gamma$ of a locally compact group $H$ is called a uniform lattice if the quotient $H/\Gamma$ is compact. Such an $H$ is called an envelope of $\Gamma$. In this paper we study the problem of classifying envelopes of…

We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we…

The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, the symmetric homology is related to…