Related papers: Quantum Dynamical R-matrices and Quantum Frobenius…

Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category O of the Borel subalgebra of an arbitrary untwisted quantum affine…

For the integrable $N$-particle Calogero-Moser system with elliptic potential it is shown that the Lax operator found by Krichever possesses a classical $r$-matrix structure. The $r$-matrix is a natural generalisation of the matrix found…

In this paper we introduce a new model for the quantum-mechanical system of the hydrogen atom. We start with a four-dimensional Lorentzian quadratic space $(V,q)$ and let $C \subset V$ be the corresponding cone. The Hilbert space of our…

Coalgebras generalize various kinds of dynamical systems occuring in mathematics and computer science. Examples of systems that can be modeled as coalgebras include automata and Markov chains. We will present a coalgebraic representation of…

We construct a rational integrable system (the rational top) on a coadjoint orbit of ${\rm SL}_N$ Lie group. It is described by the Lax operator with spectral parameter and classical non-dynamical skew-symmetric $r$-matrix. In the case of…

We develop a wave mechanics formalism for qubit geometry using holomorphic functions and Mobius transformations, providing a geometric perspective on quantum computation. This framework extends the standard Hilbert space description,…

Fix a smooth, complete algebraic curve $X$ over an algebraically closed field $k$ of characteristic zero. To a reductive group $G$ over $k$, we associate an algebraic stack $\operatorname{Par}_G$ of quantum parameters for the geometric…

We construct the classical r-matrix structure for the Lax formulation of BC_N Ruijsenaars-Schneider systems proposed in hep-th 0006004. The r-matrix structure takes a quadratic form similar to the A_N Ruijsenaars-Schneider Poisson bracket…

It is proposed the scheme of quantum mechanics, in which a Hilbert space and the linear operators are not primary elements of the theory. Instead of it certain variant of the algebraic approach is considered. The elements of noncommutative…

Deterministic dynamical models are discussed which can be described in quantum mechanical terms. -- In particular, a local quantum field theory is presented which is a supersymmetric classical model. The Hilbert space approach of Koopman…

We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical $R$-matrix even at the classical level,…

It is shown that the elliptic Ruijsenaars-Schneider model can be obtained from the cotangent bundle over the two-dimensional current group by means of the Hamiltonian reduction procedure.

Deterministic dynamical models are discussed which can be described in quantum mechanical terms. In particular, a local quantum field theory is presented which is a supersymmetric classical model. -- The Hilbert space approach of Koopman…

This thesis presents our results on Liouville integrable systems of Calogero-Ruijsenaars type: 1. We prove an explicit formula providing canonical spectral coordinates for the rational Calogero-Moser system. 2. We explore action-angle…

n-symplectic geometry, a generalization of symplectic geometry on the cotangent bundle of a manifold M, is formulated on the bundle of linear frames LM using the Rn-valued soldering 1-form as the generalized n-symplectic potential. In this…

Canonical coordinates for the Schr\"odinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schr\"odinger equation, considered as a classical field theory, shares with Liouville completely…

Schwinger's quantization scheme is extended in order to solve the problem of the formulation of quantum mechanics on a space with a group structure. The importance of Killing vectors in a quantization scheme is showed. Usage of these…

In this paper we explicitly prove that Integrable System solved by Quantum Inverse Scattering Method can be described with the pure algebraic object (Universal R-matrix) and proper algebraic representations. Namely, on the example of the…

We reconsider the quantization of symbols defined on the product between a nilpotent Lie algebra and its dual. To keep track of the non-commutative group background, the Lie algebra is endowed with the Baker-Campbell-Hausdorff product,…

Herein, we introduce a strategy to decompose an arbitrary square matrix into a linear combination of non-unitaries (LCNU) where each non-unitary term is embedded into a unitary matrix. The result is a linear combination of unitaries (LCU)…