dorsal/arxiv
View SchemaGroup-like elements in quantum groups, and Feigin's conjecture
| Authors | Arkady Berenstein |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9605016 |
| URL | https://arxiv.org/abs/q-alg/9605016 |
Abstract
Let $A$ be an arbitrary symmetrizable Cartan matrix of rank $r$, and ${\bf n}={\bf n_+}$ be the standard maximal nilpotent subalgebra in the Kac-Moody algebra associated with $A$ (thus, ${\bf n}$ is generated by $E_1,\ldots,E_r$ subject to the Serre relations). Let $\hat U_q({\bf n})$ be the completion (with respect to the natural grading) of the quantized enveloping algebra of ${\bf n}$. For a sequence ${\bf i}=(i_1,\ldots,i_m)$ with $1\le i_k\le r$, let $P_{\bf i}$ be a skew polynomial algebra generated by $t_1,\ldots,t_m$ subject to the relations $t_lt_k=q^{C_{i_k,i_l}}t_kt_l$ ($1\le k<l\le m$) where $C=(C_{ij})=(d_ia_{ij})$ is the symmetric matrix corresponding to $A$. We construct a group-like element ${\bf e}_{\bf i}\in P_{\bf i}\bigotimes \hat U_q({\bf n})$. This element gives rise to the evaluation homomorphism $\psi_{\bf i}:{\bf C}_q[N]\to P_{\bf i}$ given by $\psi_{\bf i}(x)=x({\bf e}_{\bf i})$, where ${\bf C}_q[N]=U_q({\bf n})^0$ is the restricted dual of $U_q({\bf n})$. Under a well-known isomorphism of algebras ${\bf C}_q[N]$ and $U_q({\bf n})$, the map $\psi_{\bf i}$ identifies with Feigin's homomorphism $\Phi({\bf i}): U_q({\bf n})\to P_{\bf i}$. We prove that the image of $\psi_{\bf i}$ generates the skew-field of fractions ${\cal F}(P_{\bf i})$ if and only if ${\bf i}$ is a reduced expression of some element $w$ in the Weyl group $W$; furthermore, in the latter case, ${\rm Ker}~\psi_{\bf i}$ depends only on $w$ (so we denote $I_w:={\rm Ker}~\psi_{\bf i}$). This result generalizes the results in [5], [6] to the case of Kac-Moody algebras. We also construct an element ${\cal R}_w\in \big({\bf C}_q[N]/I_w\big)\bigotimes \hat U_q({\bf n})$ which specializes to ${\bf e}_{\bf i}$ under the embedding ${\bf C}_q[N]/I_w\hookrightarrow P_{\bf i}$. The elements ${\cal R}_w$ are closely
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"date_created": "2026-03-02T18:01:28.128000Z",
"date_modified": "2026-03-02T18:01:28.128000Z",
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"abstract": "Let $A$ be an arbitrary symmetrizable Cartan matrix of rank $r$, and ${\\bf\nn}={\\bf n_+}$ be the standard maximal nilpotent subalgebra in the Kac-Moody\nalgebra associated with $A$ (thus, ${\\bf n}$ is generated by $E_1,\\ldots,E_r$\nsubject to the Serre relations). Let $\\hat U_q({\\bf n})$ be the completion\n(with respect to the natural grading) of the quantized enveloping algebra of\n${\\bf n}$. For a sequence ${\\bf i}=(i_1,\\ldots,i_m)$ with $1\\le i_k\\le r$, let\n$P_{\\bf i}$ be a skew polynomial algebra generated by $t_1,\\ldots,t_m$ subject\nto the relations $t_lt_k=q^{C_{i_k,i_l}}t_kt_l$ ($1\\le k\u003cl\\le m$) where\n$C=(C_{ij})=(d_ia_{ij})$ is the symmetric matrix corresponding to $A$. We\nconstruct a group-like element ${\\bf e}_{\\bf i}\\in P_{\\bf i}\\bigotimes \\hat\nU_q({\\bf n})$. This element gives rise to the evaluation homomorphism\n$\\psi_{\\bf i}:{\\bf C}_q[N]\\to P_{\\bf i}$ given by $\\psi_{\\bf i}(x)=x({\\bf\ne}_{\\bf i})$, where ${\\bf C}_q[N]=U_q({\\bf n})^0$ is the restricted dual of\n$U_q({\\bf n})$. Under a well-known isomorphism of algebras ${\\bf C}_q[N]$ and\n$U_q({\\bf n})$, the map $\\psi_{\\bf i}$ identifies with Feigin\u0027s homomorphism\n$\\Phi({\\bf i}): U_q({\\bf n})\\to P_{\\bf i}$. We prove that the image of\n$\\psi_{\\bf i}$ generates the skew-field of fractions ${\\cal F}(P_{\\bf i})$ if\nand only if ${\\bf i}$ is a reduced expression of some element $w$ in the Weyl\ngroup $W$; furthermore, in the latter case, ${\\rm Ker}~\\psi_{\\bf i}$ depends\nonly on $w$ (so we denote $I_w:={\\rm Ker}~\\psi_{\\bf i}$). This result\ngeneralizes the results in [5], [6] to the case of Kac-Moody algebras. We also\nconstruct an element ${\\cal R}_w\\in \\big({\\bf C}_q[N]/I_w\\big)\\bigotimes \\hat\nU_q({\\bf n})$ which specializes to ${\\bf e}_{\\bf i}$ under the embedding ${\\bf\nC}_q[N]/I_w\\hookrightarrow P_{\\bf i}$. The elements ${\\cal R}_w$ are closely",
"arxiv_id": "q-alg/9605016",
"authors": [
"Arkady Berenstein"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Group-like elements in quantum groups, and Feigin\u0027s conjecture",
"url": "https://arxiv.org/abs/q-alg/9605016"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "c0b93567-764f-444e-8a4c-27ff62923a4f",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
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}