Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of <i>U</i><sub>q</sub>(<i>sl</i><sub>2</sub>)

Let <inline-formula><math display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> denote an algebraically closed field; let <i>q</i> be a nonzero scalar in <inline-formula>&l...

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Main Authors: Hasan Alnajjar, Brian Curtin
Format: Article
Language:English
Published: MDPI AG 2020-09-01
Series:Mathematics
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Online Access:https://www.mdpi.com/2227-7390/8/9/1546
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author Hasan Alnajjar
Brian Curtin
author_facet Hasan Alnajjar
Brian Curtin
author_sort Hasan Alnajjar
collection DOAJ
description Let <inline-formula><math display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> denote an algebraically closed field; let <i>q</i> be a nonzero scalar in <inline-formula><math display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> such that <i>q</i> is not a root of unity; let <i>d</i> be a nonnegative integer; and let <i>X</i>, <i>Y</i>, <i>Z</i> be the equitable generators of <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="italic">U</mi><mi mathvariant="italic">q</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="italic">sl</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> over <inline-formula><math display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>. Let <i>V</i> denote a finite-dimensional irreducible <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="italic">U</mi><mi mathvariant="italic">q</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="italic">sl</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-module with dimension <inline-formula><math display="inline"><semantics><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and let <i>R</i> denote the set of all linear maps from <i>V</i> to itself that act tridiagonally on the standard ordering of the eigenbases for each of <i>X</i>, <i>Y</i>, and <i>Z</i>. We show that <i>R</i> has dimension at most seven. Indeed, we show that the actions of 1, <i>X</i>, <i>Y</i>, <i>Z</i>, <inline-formula><math display="inline"><semantics><mrow><mi>X</mi><mi>Y</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>Y</mi><mi>Z</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math display="inline"><semantics><mrow><mi>Z</mi><mi>X</mi></mrow></semantics></math></inline-formula> on <i>V</i> give a basis for <i>R</i> when <inline-formula><math display="inline"><semantics><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>.
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spelling doaj.art-738608c9557a4c058937ceb9f4aff96d2023-11-20T13:09:58ZengMDPI AGMathematics2227-73902020-09-0189154610.3390/math8091546Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of <i>U</i><sub>q</sub>(<i>sl</i><sub>2</sub>)Hasan Alnajjar0Brian Curtin1Department of Mathematics, The University of Jordan, Amman 11942, JordanDepartment of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave. CMC342, Tampa, FL 33620, USALet <inline-formula><math display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> denote an algebraically closed field; let <i>q</i> be a nonzero scalar in <inline-formula><math display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> such that <i>q</i> is not a root of unity; let <i>d</i> be a nonnegative integer; and let <i>X</i>, <i>Y</i>, <i>Z</i> be the equitable generators of <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="italic">U</mi><mi mathvariant="italic">q</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="italic">sl</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> over <inline-formula><math display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>. Let <i>V</i> denote a finite-dimensional irreducible <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="italic">U</mi><mi mathvariant="italic">q</mi></msub><mrow><mo>(</mo><msub><mi mathvariant="italic">sl</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-module with dimension <inline-formula><math display="inline"><semantics><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and let <i>R</i> denote the set of all linear maps from <i>V</i> to itself that act tridiagonally on the standard ordering of the eigenbases for each of <i>X</i>, <i>Y</i>, and <i>Z</i>. We show that <i>R</i> has dimension at most seven. Indeed, we show that the actions of 1, <i>X</i>, <i>Y</i>, <i>Z</i>, <inline-formula><math display="inline"><semantics><mrow><mi>X</mi><mi>Y</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>Y</mi><mi>Z</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math display="inline"><semantics><mrow><mi>Z</mi><mi>X</mi></mrow></semantics></math></inline-formula> on <i>V</i> give a basis for <i>R</i> when <inline-formula><math display="inline"><semantics><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/8/9/1546finite-dimensional <i>U<sub>q</sub></i>(<i>sl</i><sub>2</sub>)-modulesstandard eigenbasisLeonard pairs
spellingShingle Hasan Alnajjar
Brian Curtin
Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of <i>U</i><sub>q</sub>(<i>sl</i><sub>2</sub>)
Mathematics
finite-dimensional <i>U<sub>q</sub></i>(<i>sl</i><sub>2</sub>)-modules
standard eigenbasis
Leonard pairs
title Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of <i>U</i><sub>q</sub>(<i>sl</i><sub>2</sub>)
title_full Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of <i>U</i><sub>q</sub>(<i>sl</i><sub>2</sub>)
title_fullStr Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of <i>U</i><sub>q</sub>(<i>sl</i><sub>2</sub>)
title_full_unstemmed Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of <i>U</i><sub>q</sub>(<i>sl</i><sub>2</sub>)
title_short Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of <i>U</i><sub>q</sub>(<i>sl</i><sub>2</sub>)
title_sort linear maps that act tridiagonally with respect to eigenbases of the equitable generators of i u i sub q sub i sl i sub 2 sub
topic finite-dimensional <i>U<sub>q</sub></i>(<i>sl</i><sub>2</sub>)-modules
standard eigenbasis
Leonard pairs
url https://www.mdpi.com/2227-7390/8/9/1546
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