Related papers: The Krein transform and semi-bounded extensions of…

We construct quasi-Hopf algebras quantizing double extensions of the Manin pairs of Drinfeld, associated to a curve with a meromorphic differential, and the Lie algebra sl(2). This construction makes use of an analysis of the vertex…

We construct analogs of the Gelfand-Zetlin algebras in the Reflection Equation algebras, corresponding to Hecke symmetries, mainly to those coming from the quantum groups U_q(sl(N)). Corresponding semiclassical (i.e. Poisson) counterparts…

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…

Inspired by earlier works on representations of the Temperley-Lieb algebra we introduce a novel family of representations of the algebra. This may be seen as a generalization of the so called asymmetric twin representation. The underlying…

A modification of Kaluza-Klein theory is proposed which is general enough to admit an arbitrary finite noncommutative internal geometry. It is shown that the existence of a non-trival extension to the total geometry of a linear connection…

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on $\mathbb{R}^{n}$ with linear boundary conditions on (a relatively open part of) a…

In a recent series of papers we have analyzed a certain deformation of the canonical commutation relations producing an interesting functional structure which has been proved to have some connections with physics, and in particular with…

We introduce a new paradigm for proving the Schur $P$-positivity. Generalizing dual equivalence, we give an axiomatic definition for a family of involutions on a set of objects to be a queer dual equivalence, and we prove whenever such a…

We explain the concept of Krein signature in Hamiltonian and $\mathcal{PT}$-symmetric systems on the case study of the one-dimensional Gross-Pitaevskii equation with a real harmonic potential and an imaginary linear potential. These…

It is considered a differentiable manifold equipped with a pseudo-Riemannian metric and an almost contact 3-struc\-ture so that an almost contact metric structure and two almost contact B-metric structures are generated. There are…

A new multiplication-free transform derived from DHT is introduced: the RHT. Investigations on the properties of the RHT led us to the concept of weak-inversion. Using new constructs, we show that RHT is not involutional like the DHT, but…

The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally…

For a 3-manifold $M$ with boundary, we study the Kauffman module with indeterminate equal to $-1+\epsilon$ where $\epsilon^2=0$. We conjecture an explicit relation between this module and the Reidemeister torsion of $M$ which we prove in…

The Krein--Tannaka duality for compact groups was a generalization the Pontryagin--Van Kampen duality for locally compact abelian groups and a remote predecessor of the theory of tensor categories. It is less known that it found…

Multidimensional contractions of irreducible representations of Cayley--Klein orthogonal algebras in Gel'fand--Zetlin basis are considered. Contracted over different parameters, algebras can turn out to be isomorphic. In this case method of…

We reframe a collection of well-known comparison results in genus zero Gromov-Witten theory in order to relate these to integral transforms between derived categories. This implies that various comparisons among Gromov-Witten theories and…

We introduce a notion of "quasi-right-veering" for closed braids, which plays an analogous role to "right-veering" for open books. We show that a transverse link $K$ in a contact 3-manifold $(M,\xi)$ is non-loose if and only if every braid…

We survey various properties of Krein--von Neumann extensions $S_K$ and the reduced Krein--von Neumann operator $\hatt S_K$ in connection with a strictly positive (symmetric) operator $S$ with nonzero deficiency indices. In particular, we…

Given a smooth curve $C$, we define and study analogues of KLR algebras and quiver Schur algebras, where quiver representations are replaced by torsion sheaves on $C$. In particular, they provide a geometric realization for certain…

Extensions of dual definite subspaces to dual maximal definite ones are described. The concepts of dual quasi maximal subspaces and quasi basis are introduced and studied. The obtained results are applied to the classification of…