Scholarlink

<p>To determine if the endomorphism ring of a projective-injective module is symmetric is of wide interest in representation theory, ring theory, algebraic Lie theory and even in mathematical physics. In this paper, we consider the question for a large class of algebras which includes all quasi-hereditary truncations and quotients of (quantized, cyclotomic) Schur algebras and certain quotients of Hecke algebras of type <i>A</i>. We prove that the endomorphism algebra of any projective-injective module over algebras in this class is symmetric. As a consequence, endomorphism algebras of projective-injective polynomial modules over quantum general linear groups are symmetric. For a quantum group <i>U</i><sub><i>q</i></sub> over the field of complex numbers with <i>q</i> a primitive <i>l</i>-th root of unity where <i>l</i> is odd, we show that the endomorphism algebras of finite-dimensional projective-injective modules are symmetric. We present an explicit example showing that endomorphism algebras of projective-injective modules in the BGG category <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\cal{O}}_{q}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mi>q</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> for <i>U</i><sub><i>q</i></sub> need not be symmetric, thereby disproving a conjecture by Andersen and Mazorchuk.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Endomorphism algebras of projective-injective modules

  • 期刊论文

  • 发表日期：

    2025年6月29日

查看全文
  • Ming Fang,
  • Jun Hu

摘要

To determine if the endomorphism ring of a projective-injective module is symmetric is of wide interest in representation theory, ring theory, algebraic Lie theory and even in mathematical physics. In this paper, we consider the question for a large class of algebras which includes all quasi-hereditary truncations and quotients of (quantized, cyclotomic) Schur algebras and certain quotients of Hecke algebras of type A. We prove that the endomorphism algebra of any projective-injective module over algebras in this class is symmetric. As a consequence, endomorphism algebras of projective-injective polynomial modules over quantum general linear groups are symmetric. For a quantum group Uq over the field of complex numbers with q a primitive l-th root of unity where l is odd, we show that the endomorphism algebras of finite-dimensional projective-injective modules are symmetric. We present an explicit example showing that endomorphism algebras of projective-injective modules in the BGG category \({\cal{O}}_{q}\) O q for Uq need not be symmetric, thereby disproving a conjecture by Andersen and Mazorchuk.