arXiv subjects

L. Rybnikov

Publications and source records attributed to L. Rybnikov.

6 recordsLinked to original sources

arXiv2026

$(\mathfrak{gl}_{n},\mathfrak{gl}_{m})$-duality and Olshanski homomorphism

We show that the images of the Bethe subalgebras of the Yangians $Y(\mathfrak{gl}_{n})$ and $Y(\mathfrak{gl}_{m})$ under the homomorphisms to $U(\mathfrak{gl}_{n+m})$ given by the Olshanski centralizer construction coincide. We use this result to obtain the $(\mathfrak{gl}_{n},\mathfrak{gl}_{m})$-duality of the trigonometric Gaudin model and the XXX-spin chain. The duality is obtained in an explicit way relating the generating differential operator on one side and the generating difference operator on the other, thus agreeing with the result of Mukhin, Tarasov and Varchenko arXiv:math/0605172.

math.QA↗

arXiv2023

Gaudin model and Deligne's category

We show that the construction of the higher Gaudin Hamiltonians associated to the Lie algebra $\mathfrak{gl}_{n}$ admits an interpolation to any complex $n$. We do this using the Deligne's category $\mathcal{D}_{t}$, which is a formal way to define the category of finite-dimensional representations of the group $GL_{n}$, when $n$ is not necessarily a natural number. We also obtain interpolations to any complex $n$ of the no-monodromy conditions on a space of differential operators of order $n$, which are considered to be a modern form of the Bethe ansatz equations. We prove that the relations in the algebra of higher Gaudin Hamiltonians for complex $n$ are generated by our interpolations of the no-monodromy conditions. Our constructions allow us to define what it means for a pseudo-deifferential operator to have no monodromy. Motivated by the Bethe ansatz conjecture for the Gaudin model associated with the Lie superalgebra $\mathfrak{gl}_{n\vert n'}$, we show that a ratio of monodromy-free differential operators is a pseudo-differential operator without monodromy.

math.QA↗

arXiv2008

A Lie-theoretic construction of spherical symplectic reflection algebras

We propose a construction of the spherical subalgebra of a symplectic reflection algebra of an arbitrary rank corresponding to a star-shaped affine Dynkin diagram. Namely, it is obtained from the universal enveloping algebra of a certain semi-simple Lie algebra by the process of quantum Hamiltonian reduction. As an application, we propose a construction of finite-dimensional representations of the spherical subalgebra.

math.QA↗

arXiv2007

Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra $U(\g)$ of a semisimple Lie algebra $\g$. This family is parameterized by collections of pairwise distinct complex numbers $z_1,...,z_n$. We obtain some new commutative subalgebras in $U(\g)^{\otimes n}$ as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.

math.QA↗

arXiv2004

Deformation quantization of submanifolds and reductions via Duflo-Kirillov-Kontsevich map

We propose the following receipt to obtain the quantization of the Poisson submanifold $N$ defined by the equations $f_i=0$ (where $f_i$ are Casimirs) from the known quantization of the manifold $M$: one should consider factor algebra of the quantized functions on $M$ by the images of $D(f_i)$, where $D: Fun(M) \to Fun(M)\otimes \CC[\hbar]$ is Duflo-Kirillov-Kontsevich map. We conjecture that this algebra is isomorphic to quantization of $Fun(N)$ with Poisson structure inherited from $M$. Analogous conjecture concerning the Hamiltonian reduction saying that "deformation quantization commutes with reduction" is presented. The conjectures are checked in the case of $S^2$ which can be quantized as a submanifold, as a reduction and using recently found explicit star product. It's shown that all the constructions coincide.

hep-th↗

arXiv2004

Rational Lax operators and their quantization

We investigate the construction of the quantum commuting hamiltonians for the Gaudin integrable model. We prove that [Tr L^k(z), Tr L^m(u) ]=0, for k,m < 4 . However this naive receipt of quantization of classically commuting hamiltonians fails in general, for example we prove that [Tr L^4(z), Tr L^2(u) ] \ne 0. We investigate in details the case of the one spin Gaudin model with the magnetic field also known as the model obtained by the "argument shift method". Mathematically speaking this method gives maximal Poisson commutative subalgebras in the symmetric algebra S(gl(N)). We show that such subalgebras can be lifted to U(gl(N)), simply considering Tr L(z)^k, k\le N for N<5. For N=6 this method fails: [Tr L_{MF}(z)^6, L_{MF}(u)^3]\ne 0 . All the proofs are based on the explicit calculations using r-matrix technique. We also propose the general receipt to find the commutation formula for powers of Lax operator. For small power exponents we find the complete commutation relations between powers of Lax operators.

hep-th↗