International Journal of Mathematics

The skein algebra of an oriented three-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the ${SL}_2(ℂ)$-character variety. In this paper, we determine the quotient of a polynomial ring which is isomorphic to the skein algebra of a group with three generators and two relators. As an application, we give an explicit formula for the skein algebra of the Borromean rings complement in $S^3$.

Let ${\mathbb{Q}}_3$ be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in ${\mathbb{Q}}_3$. By an isotropic curve, we mean a nonconstant holomorphic map from a Riemann surface into ${\mathbb{Q}}_3$, null with respect to the conformal structure of ${\mathbb{Q}}_3$. The relations between isotropic curves and a number of relevant classes of surfaces in Riemannian and Lorentzian spaceforms are discussed.

In this paper, we give an upper bound of the orders of Jacobi forms. For each fixed weight $k$, the growth rate of our upper bound divided by the index $m$ as $m$ goes to infinity is zero. Applying this result to Siegel paramodular forms, we know that the space of all formal series of Jacobi forms is finite dimensional.

An almost Abelian Lie algebra is a non-Abelian Lie algebra with a codimension 1 Abelian ideal. Most 3-dimensional real Lie algebras are almost Abelian, and they appear in every branch of physics that deals with anisotropic media — cosmology, crystallography, etc. In differential geometry and theoretical physics, almost Abelian Lie groups have given rise to some of the simplest solvmanifolds on which various geometric structures such as symplectic, Kähler, spin, etc., are currently studied in explicit terms. However, a systematic study of almost Abelian Lie groups and algebras from mathematics perspective has not been carried out yet, and this paper is the first step in addressing this wide and diverse class of groups and algebras. This paper studies the structure and important algebraic properties of almost Abelian Lie algebras of arbitrary dimension over any field of scalars. A classification of almost Abelian Lie algebras is given. All Lie subalgebras and ideals, automorphisms and derivations, Lie orthogonal operators and quadratic Casimir elements are described exactly.

Let $n>1$. Let ${\{U_{ij}\}}_{1\le i<j\le n}$ be $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$ commuting unitaries on some Hilbert space ${\mathscr{H}}$, and suppose $U_{ji}:=U_{ij}^{\ast }$, $1\le i<j\le n$. An $n$-tuple of isometries $V=(V_1,\dots ,V_n)$ on ${\mathscr{H}}$ is called ${{𝒰}}_n$-twisted isometry with respect to ${\{U_{ij}\}}_{i<j}$ (or simply ${{𝒰}}_n$-twisted isometry if ${\{U_{ij}\}}_{i<j}$ is clear from the context) if $V_i$’s are in the commutator ${\{U_{st}:s\ne t\}}^{\prime }$, and $V_i^{\ast }{V}_j=U_{ij}^{\ast }{V}_j{V}_i^{\ast }$, $i\ne j$

We prove that each ${{𝒰}}_n$-twisted isometry admits a von Neumann–Wold type orthogonal decomposition, and prove that the universal $C^{\ast }$-algebra generated by ${{𝒰}}_n$-twisted isometries is nuclear. We exhibit concrete analytic models of ${{𝒰}}_n$-twisted isometries, and establish connections between unitary equivalence classes of the irreducible representations of the $C^{\ast }$-algebras generated by ${{𝒰}}_n$-twisted isometries and the unitary equivalence classes of the nonzero irreducible representations of twisted noncommutative tori. Our motivation of ${{𝒰}}_n$-twisted isometries stems from the classical rotation $C^{\ast }$-algebras and Heisenberg group $C^{\ast }$-algebras.

In this paper, we generalize some results shown by Das and Peterson. Let $M$ be a ${\mathrm{\text{II}}}_1$-factor acting on $L^2(M)$. We consider certain unital normal completely positive maps on $B(L^2(M))$ which are identity on $M$. We investigate their fixed point spaces and obtain a rigidity result. As an application, we show some results of subfactors.

In this paper, we discuss various results about variation of the local fundamental group of normal complex spaces. It is proved that the finite Galois descent of upper semicontinuity of the local fundamental group holds at a factorial complex analytic germ. We also show by an example that finite Galois descent of upper semicontinuity of the local first homology group at a smooth germ is not true in general. We prove that the local fundamental groups of a normal complex algebraic variety are finite in number upto isomorphism.