Related papers: A new class of supermatrix algebras defined by tra…
We introduce a construction that, given a pair (u,v) of complex Hadamard matrices of the same order, generates infinitely many biunitary matrices of varying (and distinct) orders. As a key application, this framework yields nested sequences…
Suppose that a Lie type algebra L over a field K admits a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H such that the fixed-point subalgebra of F is trivial and the fixed-point subalgebra of H is…
We take a unifying and new approach toward polynomial and trigonometric approximation in an arbitrary number of variables, resulting in a precise and general ready-to-use tool that anyone can easily apply in new situations of interest. The…
The Neumann--Zagier matrices of an ideal triangulation are integer matrices with symplectic properties whose entries encode the number of tetrahedra that wind around each edge of the triangulation. They can be used as input data for the…
Newton's inequalities $c_n^2 \ge c_{n-1}c_{n+1}$ are shown to hold for the normalized coefficients $c_n$ of the characteristic polynomial of any $M$- or inverse $M$-matrix. They are derived by establishing first an auxiliary set of…
In this book i treat linear algebra over division ring. A system of linear equations over a division ring has properties similar to properties of a system of linear equations over a field. However, noncommutativity of a product creates a…
We introduce a general class of automorphisms of rotation algebras, the noncommutative Furstenberg transformations. We prove that fully irrational noncommutative Furstenberg transformations have the tracial Rokhlin property, which is a…
Given a finite collection of $C^1$ complex vector fields on a $C^2$ manifold $M$ such that they and their complex conjugates span the complexified tangent space at every point, the classical Newlander-Nirenberg theorem gives conditions on…
We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element g of degree 1, T/gT is a twisted homogeneous coordinate ring of an elliptic…
In this document we study the uniform local path connectivity of sets of $m$-tuples of pairwise commuting normal matrices with some additional constraints. More specifically, given given $\varepsilon>0$, a fixed metric $\eth$ in…
Suppose a group $\Gamma$ acts on a scheme $X$ and a Lie superalgebra $\mathfrak{g}$. The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from $X$ to $\mathfrak{g}$. We classify the irreducible…
In this set of lectures we give a pedagogical introduction to the way in which the nilpotency of a super-de Rham operator can be exploited for the construction of gauge theories in superspace. We begin with a discussion of how the…
The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a transitive group of isometries are obtained. These conditions are Intrinsic, Deductive, Explicit and ALgorithmic, and they offer an IDEAL labeling…
This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate…
As an analogue of the topological boundary of discrete groups $\Gamma$, we define the noncommutative topological boundary of tracial von Neumann algebras $(M, \tau)$ and apply it to generalize the main results of [AHO23], showing that for a…
We systematically analyze the large-$N$ limit of the superconformal index of $\mathcal{N}=1$ superconformal theories having a quiver description. The index of these theories is known in terms of unitary matrix integrals, which we calculate…
Trigonometric invariants are defined for each Weyl group orbit on the root lattice. They are real and periodic on the coroot lattice. Their polynomial algebra is spanned by a basis which is calculated by means of an algorithm. The…
We give a new definition for the crossed-product of a C*-algebra A by a *-endomorphism \alpha, which depends not only on the pair (A,\alpha) but also on the choice of a transfer operator (defined in the paper). With this we generalize some…
We develop a first and second fundamental theorem for $n$--tuples of bosonic and fermionic matrices, by developing graded analogues of the classical case.
Gabriel's Theorem, and the work of Bernstein, Gelfand and Ponomarev established a connection between the theory of quiver representations and the theory of simple Lie algebras. Lie superalgebras have been studied from many perspectives, and…