Restriction and induction of indecomposable modules over the Temperley–Lieb algebras
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Abstract
Abstract Both the original Temperley–Lieb algebras <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="sans-serif">T</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> and their dilute counterparts <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="sans-serif">d</mml:mi> <mml:mi mathvariant="sans-serif">T</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> form families of filtered algebras: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="sans-serif">T</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⊂</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="sans-serif">T</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="sans-serif">d</mml:mi> <mml:mi mathvariant="sans-serif">T</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⊂</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="sans-serif">d</mml:mi> <mml:mi mathvariant="sans-serif">T</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> , for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>n</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> </mml:mstyle> </mml:math> . For each such inclusion, the restriction and induction of every finite-dimensional indecomposable module over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="sans-serif">T</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="sans-serif">d</mml:mi> <mml:mi mathvariant="sans-serif">T</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mstyle> </mml:math> ) is computed. To accomplish this, a thorough description of each indecomposable is given, including its projective cover and injective hull, some short exact sequences in which it appears, its socle and head, and its extension groups with irreducible modules. These data are also used to prove the completeness of the list of indecomposable modules, up to isomorphism. In fact, two completeness proofs are given—the first is based on elementary homological methods and the second uses Auslander–Reiten theory. The latter proof offers a detailed example of this algebraic tool that may be of independent interest.
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