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We study the spectral properties of Schr\"odinger operators on a compact connected Riemannian manifold $M$ without boundary in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, if $M$ carries an…
On any Reflection Equation algebra corresponding to a skew-invertible Hecke symmetry (i.e. a special type solution of the Quantum Yang-Baxter Equation) we define analogs of the partial derivatives. Together with elements of the initial…
We present a simple, systematic and practical method to construct conformally invariant equations in arbitrary Riemann spaces. This method that we call "Weyl-to-Riemann" is based on two features of Weyl geometry. i) A Weyl space is defined…
We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine…
We obtain a gauge-invariant relativistic quantum geometry by using a Weylian-like manifold with a geometric scalar field which provides a gauge-invariant relativistic quantum theory in which the algebra of the Weylian-like field depends on…
This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The…
We study isomorphisms between generalized Weyl algebras, giving a complete answer to the quantum case of this problem for R=k[h].
We introduce and study action of quantum groups on skew polynomial rings and related rings of quotients. This leads to a ``q-deformation'' of the Gel'fand-Kirillov conjecture which we partially prove. We propose a construction of…
The coinvariant algebra of a Weyl group plays a fundamental role in several areas of mathematics. The fake degrees are the graded multiplicities of the irreducible modules of a Weyl group in its coinvariant algebra, and they were computed…
These notes describe some links between the group $\mathrm{SL}_2(\mathbb{R})$, the Heisenberg group and hypercomplex numbers---complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this…
Bell and Zhang have shown that if $A$ and $B$ are two connected graded algebras finitely generated in degree one that are isomorphic as ungraded algebras, then they are isomorphic as graded algebras. We exploit this result to solve the…
Gelfand-Naimark-Stone duality provides an algebraic counterpart of compact Hausdorff spaces in the form of uniformly complete bounded archimedean $\ell$-algebras. In [4] we extended this duality to completely regular spaces. In this article…
Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning…
This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The…
Let $k$ be an algebraically closed field and $A$ a $\mathbb{Z}$-graded finitely generated simple $k$-algebra which is a domain of Gelfand-Kirillov dimension 2. We show that the category of $\mathbb{Z}$-graded right $A$-modules is equivalent…
We review and expand upon recent work demonstrating that Weyl invariant theories can be broken "inertially," which does not depend upon a potential. This can be understood in a general way by the "current algebra" of these theories,…
We present the general theory of relativity in the language of a non-Riemannian geometry, namely, Weyl geometry. We show that the new mathematical formalism may lead to different pictures of the same gravitational phenomena, by making use…
In this survey article we discuss the question: to what extent is an algebraic variety determined by its ring of differential operators? In the case of affine curves, this question leads to a variety of mathematical notions such as the Weyl…
For an arbitrary simple Lie algebra $\g$ and an arbitrary root of unity $q,$ the closed subsets of the Weyl alcove of the quantum group $U_q(\g)$ are classified. Here a closed subset is a set such that if any two weights in the Weyl alcove…
We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF.…