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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…

We construct uncountably many mutually nonisomorphic simple separable stably finite unital exact C$^\ast$-algebras which are not isomorphic to their opposite algebras. In particular, we prove that there are uncountably many possibilities…

In this paper, we study partitions of totally positive integral elements $\alpha$ in a real quadratic field $K$. We prove that for a fixed integer $m \geq 1$, an element with $m$ partition exists in almost all $K$. We also obtain an upper…

I. Penkov and V. Serganova have recently introduced, for any non-degenerate pairing $W\otimes V\to\mathbb C$ of vector spaces, the Lie algebra $\mathfrak{gl}^M=\mathfrak{gl}^M(V,W)$ consisting of endomorphisms of $V$ whose duals preserve…

Let $G$ be a connected complex semisimple Lie group, $\Gamma$ be a cocompact, irreducible and torsionless lattice in $G$ and $K$ be a maximal compact subgroup of $G$. Assume $\Gamma$ acts by left multiplication and $K$ acts by right…

Let $A$ be a matrix with nonnegative real entries. A nonnegative factorization of size $k$ is a representation of $A$ as a sum of $k$ nonnegative rank-one matrices. The space of all such factorizations is a bounded semialgebraic set, and we…

For a field $R$ of characteristic $p\ge 0$ and a matrix $c$ in the full $n\times n$ matrix algebra $M_n(R)$ over $R$, let $S_n(c,R)$ be the centralizer algebra of $c$ in $M_n(R)$. We show that $S_n(c,R)$ is a Frobenius-finite,…

Let $M$ be a II$_1$ factor with a von Neumann subalgebra $Q\subset M$ that has infinite index under any projection in $Q'\cap M$ (e.g., $Q$ abelian; or $Q$ an irreducible subfactor with infinite Jones index). We prove that given any…

Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $\tau,$ we consider the non commutative Arens algebra $L^{\omega}(M, \tau)=\bigcap\limits_{p\geq1}L^{p}(M, \tau)$ and the related algebras $L^{\omega}_2(M,…

The Abelian semigroup expansion is a powerful and simple method to derive new Lie algebras from a given one. Recently it was shown that the $S$-expansion of $\mathfrak{so}\left( 3,2\right) $ leads us to the Maxwell algebra $\mathcal{M}$. In…

Following arguments that the (hidden) M-algebra serves as the maximal super-exceptional tangent space for 11D supergravity, we make explicit here its integration to a (super-Lie) group. This is equipped with a left-invariant extension of…

In this paper, a semigroup algebra consisting of polynomial expressions with coefficients in a field $F$ and exponents in an additive submonoid $M$ of $\mathbb{Q}_{\ge 0}$ is called a Puiseux algebra and denoted by $F[M]$. Here we study the…

Suppose that $f$ is a homomorphism from the mapping class group $\mathcal{M}(N_{g,n})$ of a nonorientable surface of genus $g$ with $n$ boundary components, to $\mathrm{GL}(m,\mathbb{C})$. We prove that if $g\ge 5$, $n\le 1$ and $m\le g-2$,…

Let $k$ be a field of characteristic not two or three. We classify up to isomorphism all finite-dimensional Lie superalgebras $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ over $k$, where $\mathfrak{g}_0$ is a three-dimensional simple…

Let ${\rm M}(V)={\rm M}(n,\mathbb{F}_q)$ denote the algebra of $n\times n$ matrices over $\mathbb{F}_q$, and let ${\rm M}(V)_U$ denote the (maximal reducible) subalgebra that normalizes a given $r$-dimensional subspace $U$ of…

Let $\mathfrak g$ be a Kac-Moody algebra. We show that every homogeneous right coideal subalgebra $U$ of the multiparameter version of the quantized universal enveloping algebra $U_q(\mathfrak{g}),$ $q^m\neq 1$ containing all group-like…

For a right-angled Coxeter system $(W,S)$ and $q>0$, let $\mathcal{M}_q$ be the associated Hecke von Neumann algebra, which is generated by self-adjoint operators $T_s, s \in S$ satisfying the Hecke relation $(\sqrt{q}\: T_s - q) (\sqrt{q}…

We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain properties, which ensure that a II$_1$ factor $M$ is prime or has at most one Cartan subalgebra. For instance, we prove that if there…

For a finite dimensional Lie algebra $\mathfrak{g}$, the Duflo map $S\mathfrak{g}\rightarrow U\mathfrak{g}$ defines an isomorphism of $\mathfrak{g}$-modules. On $\mathfrak{g}$-invariant elements it gives an isomorphism of algebras.…

Let $M$ be a manifold and $\Lambda$ a compact exact connected Lagrangian submanifold of $T^*M$. We can associate with $\Lambda$ a conic Lagrangian submanifold $\Lambda'$ of $T^*(M\times R)$. We prove that there exists a canonical sheaf $F$…