dorsal/arxiv
View SchemaCentralizer construction for twisted Yangians
| Authors | Alexander Molev, Grigori Olshanski |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9712050 |
| URL | https://arxiv.org/abs/q-alg/9712050 |
| Journal | Selecta Mathematica 6 (2000), no. 3, 269--317. |
Abstract
For each of the classical Lie algebras $g(n)=o(2n+1), sp(2n), o(2n)$ of type B, C, D we consider the centralizer of the subalgebra $g(n-m)$ in the universal enveloping algebra $U(g(n))$. We show that the $n$th centralizer algebra can be naturally projected onto the $(n-1)$th one, so that one can form the projective limit of the centralizer algebras as $n\to\infty$ with $m$ fixed. The main result of the paper is a precise description of this limit (or stable) centralizer algebra, denoted by $A_m$. We explicitly construct an algebra isomorphism $A_m=Z\otimes Y_m$, where $Z$ is a commutative algebra and $Y_m$ is the so-called twisted Yangian associated to the rank $m$ classical Lie algebra of type B, C, or D. The algebra $Z$ may be viewed as the algebra of virtual Laplace operators; it is isomorphic to the algebra of polynomials with countably many indeterminates. The twisted Yangian $Y_m$ (and hence the algebra $A_m$) can be described in terms of a system of generators with quadratic and linear defining relations which are conveniently presented in R-matrix form involving the so-called reflection equation. This extends the earlier work on the type A case by the second author.
{
"annotation_id": "e76c539d-3eb9-49bb-8351-4b45f5229d2a",
"date_created": "2026-03-02T18:01:28.644000Z",
"date_modified": "2026-03-02T18:01:28.644000Z",
"file_hash": "fde9f2f22c8b26e7568e006112022f44f0afac9caf54a6c0e409c65237d5f5a7",
"private": false,
"record": {
"abstract": "For each of the classical Lie algebras $g(n)=o(2n+1), sp(2n), o(2n)$ of type\nB, C, D we consider the centralizer of the subalgebra $g(n-m)$ in the universal\nenveloping algebra $U(g(n))$. We show that the $n$th centralizer algebra can be\nnaturally projected onto the $(n-1)$th one, so that one can form the projective\nlimit of the centralizer algebras as $n\\to\\infty$ with $m$ fixed. The main\nresult of the paper is a precise description of this limit (or stable)\ncentralizer algebra, denoted by $A_m$. We explicitly construct an algebra\nisomorphism $A_m=Z\\otimes Y_m$, where $Z$ is a commutative algebra and $Y_m$ is\nthe so-called twisted Yangian associated to the rank $m$ classical Lie algebra\nof type B, C, or D. The algebra $Z$ may be viewed as the algebra of virtual\nLaplace operators; it is isomorphic to the algebra of polynomials with\ncountably many indeterminates. The twisted Yangian $Y_m$ (and hence the algebra\n$A_m$) can be described in terms of a system of generators with quadratic and\nlinear defining relations which are conveniently presented in R-matrix form\ninvolving the so-called reflection equation. This extends the earlier work on\nthe type A case by the second author.",
"arxiv_id": "q-alg/9712050",
"authors": [
"Alexander Molev",
"Grigori Olshanski"
],
"categories": [
"q-alg",
"math.QA",
"nlin.SI",
"solv-int"
],
"journal_ref": "Selecta Mathematica 6 (2000), no. 3, 269--317.",
"title": "Centralizer construction for twisted Yangians",
"url": "https://arxiv.org/abs/q-alg/9712050"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "0ab81011-b634-4960-a4b7-8b0b50fb71df",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}