The Experts below are selected from a list of 321 Experts worldwide ranked by ideXlab platform
Leonid Rybnikov - One of the best experts on this subject based on the ideXlab platform.
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ON CLASSICAL LIMITS OF BETHE SubalgebraS IN YANGIANS
Transformation Groups, 2021Co-Authors: Aleksei Ilin, Leonid RybnikovAbstract:The Yangian Y (
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On classical limits of Bethe Subalgebras in Yangians
arXiv: Quantum Algebra, 2020Co-Authors: Aleksei Ilin, Leonid RybnikovAbstract:The Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra $U(\mathfrak{g}[t])$ and the coordinate ring of the first congruence subgroup $\mathcal{O}(G_1[[t^{-1}]])$. Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Yangian. Bethe Subalgebras form a natural family of commutative Subalgebras depending on a group element $C$ of the adjoint group $G$. The images of these algebras in tensor products of fundamental representations give all integrals of the quantum XXX Heisenberg magnet chain. We describe the associated graded of Bethe Subalgebras as Subalgebras in $U(\mathfrak{g}[t])$ and in $\mathcal{O}(G_1[[t^{-1}]])$ for all semisimple $C\in G$. We show that associated graded in $U(\mathfrak{g}[t])$ of the Bethe Subalgebra assigned to the identity of $G$ is the universal Gaudin Subalgebra of $U(\mathfrak{g}[t])$ obtained from the center of the corresponding affine Kac-Moody algebra at the critical level. This generalizes Talalaev's formula for generators of the universal Gaudin Subalgebra to $\mathfrak{g}$ of any type. In particular, this shows that higher Hamiltonians of the Gaudin magnet chain can be quantized without referring to the Feigin-Frenkel center at the critical level. Using our general result on associated graded of Bethe Subalgebras, we compute some limits of Bethe Subalgebras corresponding to regular semisimple $C\in G$ as $C$ goes to an irregular semisimple group element $C_0$. We show that this limit is the product of the smaller Bethe Subalgebra and a quantum shift of argument Subalgebra in the universal enveloping algebra of the centralizer of $C_0$ in $\mathfrak{g}$. This generalizes the Nazarov-Olshansky solution of Vinberg's problem on quantization of shift of argument Subalgebras.
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Bethe Subalgebras in Yangians and the Wonderful Compactification
Communications in Mathematical Physics, 2019Co-Authors: Aleksei Ilin, Leonid RybnikovAbstract:Let $${\mathfrak {g}}$$ g be a complex simple Lie algebra. We study the family of Bethe Subalgebras in the Yangian $$Y({\mathfrak {g}})$$ Y ( g ) parameterized by the corresponding adjoint Lie group G . We describe their classical limits as Subalgebras in the algebra of polynomial functions on the formal Lie group $$G_1[[t^{-1}]]$$ G 1 [ [ t - 1 ] ] . In particular we show that, for regular values of the parameter, these Subalgebras are free polynomial algebras with the same Poincaré series as the Cartan Subalgebra of the Yangian. Next, we extend the family of Bethe Subalgebras to the De Concini–Procesi wonderful compactification $$\overline{G}\supset G$$ G ¯ ⊃ G and describe the Subalgebras corresponding to generic points of any stratum in $$\overline{G}$$ G ¯ as Bethe Subalgebras in the Yangian of the corresponding Levi Subalgebra in $${\mathfrak {g}}$$ g . In particular, we describe explicitly all Bethe Subalgebras corresponding to the closure of the maximal torus in the wonderful compactification.
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Gelfand–Tsetlin Degeneration of Shift of Argument Subalgebras in Types B, C and D
Arnold Mathematical Journal, 2019Co-Authors: Leonid Rybnikov, Mikhail ZavalinAbstract:The universal enveloping algebra of any semisimple Lie algebra $$\mathfrak {g}$$ g contains a family of maximal commutative Subalgebras, called shift of argument Subalgebras, parametrized by regular Cartan elements of $$\mathfrak {g}$$ g . For $$\mathfrak {g}=\mathfrak {gl}_n$$ g = gl n the Gelfand–Tsetlin commutative Subalgebra in $$U(\mathfrak {g})$$ U ( g ) arises as some limit of Subalgebras from this family. We study the analogous limit of shift of argument Subalgebras for classical Lie algebras ( $$\mathfrak {g}=\mathfrak {sp}_{2n}$$ g = sp 2 n or $$\mathfrak {so}_{n}$$ so n ). The limit Subalgebra is described explicitly in terms of Bethe Subalgebras in twisted Yangians $$Y^-(2)$$ Y - ( 2 ) and $$Y^+(2)$$ Y + ( 2 ) , respectively. We index the eigenbasis of such limit Subalgebra in any irreducible finite-dimensional representation of $$\mathfrak {g}$$ g by Gelfand–Tsetlin patterns of the corresponding type, and conjecture that this indexing is, in appropriate sense, natural. According to Halacheva et al. (Crystals and monodromy of Bethe vectors. arXiv:1708.05105 , 2017) such eigenbasis has a natural $$\mathfrak {g}$$ g -crystal structure. We conjecture that this crystal structure coincides with that on Gelfand–Tsetlin patterns defined by Littelmann in Cones, crystals, and patterns (Transform Groups 3(2):145–179, 1998 ).
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Bethe Subalgebras in Yangians and the Wonderful Compactification
Communications in Mathematical Physics, 2019Co-Authors: Aleksei Ilin, Leonid RybnikovAbstract:Let \({\mathfrak {g}}\) be a complex simple Lie algebra. We study the family of Bethe Subalgebras in the Yangian \(Y({\mathfrak {g}})\) parameterized by the corresponding adjoint Lie group G. We describe their classical limits as Subalgebras in the algebra of polynomial functions on the formal Lie group \(G_1[[t^{-1}]]\). In particular we show that, for regular values of the parameter, these Subalgebras are free polynomial algebras with the same Poincare series as the Cartan Subalgebra of the Yangian. Next, we extend the family of Bethe Subalgebras to the De Concini–Procesi wonderful compactification \(\overline{G}\supset G\) and describe the Subalgebras corresponding to generic points of any stratum in \(\overline{G}\) as Bethe Subalgebras in the Yangian of the corresponding Levi Subalgebra in \({\mathfrak {g}}\). In particular, we describe explicitly all Bethe Subalgebras corresponding to the closure of the maximal torus in the wonderful compactification.
Aleksei Ilin - One of the best experts on this subject based on the ideXlab platform.
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ON CLASSICAL LIMITS OF BETHE SubalgebraS IN YANGIANS
Transformation Groups, 2021Co-Authors: Aleksei Ilin, Leonid RybnikovAbstract:The Yangian Y (
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On classical limits of Bethe Subalgebras in Yangians
arXiv: Quantum Algebra, 2020Co-Authors: Aleksei Ilin, Leonid RybnikovAbstract:The Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra $U(\mathfrak{g}[t])$ and the coordinate ring of the first congruence subgroup $\mathcal{O}(G_1[[t^{-1}]])$. Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Yangian. Bethe Subalgebras form a natural family of commutative Subalgebras depending on a group element $C$ of the adjoint group $G$. The images of these algebras in tensor products of fundamental representations give all integrals of the quantum XXX Heisenberg magnet chain. We describe the associated graded of Bethe Subalgebras as Subalgebras in $U(\mathfrak{g}[t])$ and in $\mathcal{O}(G_1[[t^{-1}]])$ for all semisimple $C\in G$. We show that associated graded in $U(\mathfrak{g}[t])$ of the Bethe Subalgebra assigned to the identity of $G$ is the universal Gaudin Subalgebra of $U(\mathfrak{g}[t])$ obtained from the center of the corresponding affine Kac-Moody algebra at the critical level. This generalizes Talalaev's formula for generators of the universal Gaudin Subalgebra to $\mathfrak{g}$ of any type. In particular, this shows that higher Hamiltonians of the Gaudin magnet chain can be quantized without referring to the Feigin-Frenkel center at the critical level. Using our general result on associated graded of Bethe Subalgebras, we compute some limits of Bethe Subalgebras corresponding to regular semisimple $C\in G$ as $C$ goes to an irregular semisimple group element $C_0$. We show that this limit is the product of the smaller Bethe Subalgebra and a quantum shift of argument Subalgebra in the universal enveloping algebra of the centralizer of $C_0$ in $\mathfrak{g}$. This generalizes the Nazarov-Olshansky solution of Vinberg's problem on quantization of shift of argument Subalgebras.
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Bethe Subalgebras in Yangians and the Wonderful Compactification
Communications in Mathematical Physics, 2019Co-Authors: Aleksei Ilin, Leonid RybnikovAbstract:Let $${\mathfrak {g}}$$ g be a complex simple Lie algebra. We study the family of Bethe Subalgebras in the Yangian $$Y({\mathfrak {g}})$$ Y ( g ) parameterized by the corresponding adjoint Lie group G . We describe their classical limits as Subalgebras in the algebra of polynomial functions on the formal Lie group $$G_1[[t^{-1}]]$$ G 1 [ [ t - 1 ] ] . In particular we show that, for regular values of the parameter, these Subalgebras are free polynomial algebras with the same Poincaré series as the Cartan Subalgebra of the Yangian. Next, we extend the family of Bethe Subalgebras to the De Concini–Procesi wonderful compactification $$\overline{G}\supset G$$ G ¯ ⊃ G and describe the Subalgebras corresponding to generic points of any stratum in $$\overline{G}$$ G ¯ as Bethe Subalgebras in the Yangian of the corresponding Levi Subalgebra in $${\mathfrak {g}}$$ g . In particular, we describe explicitly all Bethe Subalgebras corresponding to the closure of the maximal torus in the wonderful compactification.
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Bethe Subalgebras in Yangians and the Wonderful Compactification
Communications in Mathematical Physics, 2019Co-Authors: Aleksei Ilin, Leonid RybnikovAbstract:Let \({\mathfrak {g}}\) be a complex simple Lie algebra. We study the family of Bethe Subalgebras in the Yangian \(Y({\mathfrak {g}})\) parameterized by the corresponding adjoint Lie group G. We describe their classical limits as Subalgebras in the algebra of polynomial functions on the formal Lie group \(G_1[[t^{-1}]]\). In particular we show that, for regular values of the parameter, these Subalgebras are free polynomial algebras with the same Poincare series as the Cartan Subalgebra of the Yangian. Next, we extend the family of Bethe Subalgebras to the De Concini–Procesi wonderful compactification \(\overline{G}\supset G\) and describe the Subalgebras corresponding to generic points of any stratum in \(\overline{G}\) as Bethe Subalgebras in the Yangian of the corresponding Levi Subalgebra in \({\mathfrak {g}}\). In particular, we describe explicitly all Bethe Subalgebras corresponding to the closure of the maximal torus in the wonderful compactification.
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Bethe Subalgebras in Yangians and the wonderful compactification.
Communications in Mathematical Physics, 2019Co-Authors: Aleksei Ilin, Leonid RybnikovAbstract:We study the family of Bethe Subalgebras in the Yangian $Y(\mathfrak{g})$ parameterized by the corresponding adjoint Lie group $G$. We describe their classical limits as Subalgebras in the algebra of polynomial functions on the formal Lie group $G_1[[t^{-1}]]$. In particular we show that, for regular values of the parameter, these Subalgebras are free polynomial algebras with the same Poincare series as the Cartan Subalgebra of the Yangian. Next, we extend the family of Bethe Subalgebras to the De Concini--Procesi wonderful compactification $\overline{G}\supset G$ and describe the Subalgebras corresponding to generic points of any stratum in $\overline{G}$ as Bethe Subalgebras in the Yangian of the corresponding Levi Subalgebra in $\mathfrak{g}$.
Konrad Pióro - One of the best experts on this subject based on the ideXlab platform.
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Subalgebra lattices of a partial unary algebra
Demonstratio Mathematica, 2012Co-Authors: Konrad PióroAbstract:Necessary and sufficient conditions will be found for quadruples of lattices to be isomorphic to lattices of weak, relative, strong Subalgebras and initial segments, respectively, of one partial unary algebra. To this purpose we will start with a characterization of pairs of lattices that are weak and strong Subalgebra lattices of one partial unary algebra, respectively. Next, we will describe the initial segment lattice of a partial unary algebra. Applying this result, pairs of lattices of strong Subalgebras and initial segments will be characterized. Further, we will characterize pairs of lattices of relative and strong Subalgebras and also other pairs of Subalgebra lattices of one partial unary algebra.
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On the Subalgebra Lattice of Unary Algebras
Acta Mathematica Hungarica, 1999Co-Authors: Konrad PióroAbstract:We characterize pairs 〈L, A〉, where Lis a lattice and Ais a unary partial algebra, such that the strong Subalgebra lattice Ss(A) is isomorphic to L. Moreover, we find necessary and sufficient conditions for arbitrary unary partial algebras to have isomorphic strong Subalgebra lattices. Observe, that for a total algebra A, the lattice Ss(A) is the usual well-known Subalgebra lattice. Thus in particular we solve these two problems for total unary algebras and their lattices of (also total) Subalgebras.
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On a strong property of the weak Subalgebra lattice
Algebra Universalis, 1998Co-Authors: Konrad PióroAbstract:In the present paper, we apply results from [Pio1] to prove that for an arbitrary total and locally finite unary algebra A of finite unary type K, its weak Subalgebra lattice uniquely determines its strong Subalgebra lattice (recall that in the case of total algebras the strong Subalgebra lattice is the well-known lattice of all (total) Subalgebras). More precisely, we prove that for every unary partial algebra B of the same unary type K, if weak Subalgebra lattices of A and B are isomorphic (with A as above), then the strong Subalgebra lattices of A and B are isomorphic, and moreover B is also total and locally finite. At the end of this paper we also show the necessity of all the three conditions for A.
David A. Towers - One of the best experts on this subject based on the ideXlab platform.
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On the Subalgebra lattice of a Leibniz algebra
arXiv: Rings and Algebras, 2020Co-Authors: Salvatore Siciliano, David A. TowersAbstract:In this paper we begin to study the Subalgebra lattice of a Leibniz algebra. In particular, we deal with Leibniz algebras whose Subalgebra lattice is modular, upper semi-modular, lower semi-modular, distributive, or dually atomistic. The fact that a non-Lie Leibniz algebra has fewer one-dimensional Subalgebras in general results in a number of lattice conditions being weaker than in the Lie case.
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On abelian Subalgebras and ideals of maximal dimension in supersolvable lie algebras
arXiv: Rings and Algebras, 2011Co-Authors: Manuel Ceballos, David A. TowersAbstract:In this paper, the main objective is to compare the abelian Subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian Subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian Subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian Subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two. Throughout the paper, we also give several examples to clarify some results.
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the index complex of a maximal Subalgebra of a lie algebra
Proceedings of the Edinburgh Mathematical Society, 2011Co-Authors: David A. TowersAbstract:Let M be a maximal Subalgebra of the Lie algebra L. A Subalgebra C of L is said to be a completion for M if C is not contained in M but every proper Subalgebra of C that is an ideal of L is contained in M. The set I(M) of all completions of M is called the index complex of M in L. We use this concept to investigate the influence of the maximal Subalgebras on the structure of a Lie algebra, in particular finding new characterisations of solvable and supersolvable Lie algebras.
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Supplements to maximal Subalgebras of Lie algebras
arXiv: Rings and Algebras, 2010Co-Authors: David A. TowersAbstract:For a Lie algebra $L$ and a Subalgebra $M$ of $L$ we say that a Subalgebra $U$ of $L$ is a {\em supplement} to $M$ in $L$ if $L = M + U$. We investigate those Lie algebras all of whose maximal Subalgebras have abelian supplements, those that have nilpotent supplements, those that have nil supplements, and those that have supplements with the property that their derived algebra is inside the maximal Subalgebra being supplemented. For the algebras over an algebraically closed field of characteristic zero in the last three of these classes we find complete descriptions; for those in the first class partial results are obtained.
Branimir Šešelja - One of the best experts on this subject based on the ideXlab platform.
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Lattice of partially ordered fuzzy Subalgebras
Fuzzy Sets and Systems, 1996Co-Authors: Branimir ŠešeljaAbstract:Abstract Fuzzy Subalgebras are defined to be mappings from an algebra to a poset (which can be a lattice, and also the interval [0,1]), so that every level subset is an ordinary Subalgebra. Every fuzzy Subalgebra has a canonical representation, a mapping from the algebra to the poset of levels. Necessary and sufficient conditions under which a fuzzy Subalgebra can be explicitly described by a suitable formula in lattice theoretic terms are given. Classes of algebras for which the poset of fuzzy Subalgebras is a lattice are characterized.
