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• W2962783104 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an algebraically closed field and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite-dimensional associative basic <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra of the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A equals k upper Q slash upper I> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mi>Q</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>A=kQ/I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Q> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding=application/x-tex>Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a quiver without oriented cycles or double arrows and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an admissible ideal of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k upper Q> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mi>Q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>kQ</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We consider roots of the Tits form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q Subscript upper A> <mml:semantics> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>q_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in particular in the case where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q Subscript upper A> <mml:semantics> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>q_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is weakly non-negative. We prove that for any maximal omnipresent root <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=v> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding=application/x-tex>v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q Subscript upper A> <mml:semantics> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>q_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists an indecomposable <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-module <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that v=<bold>dim</bold> X. Moreover, if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is strongly simply connected, the existence of a maximal omnipresent root of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q Subscript upper A> <mml:semantics> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>q_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is tame of tilted type." @default.
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• W2962783104 date "2013-09-16" @default.
• W2962783104 modified "2023-10-16" @default.
• W2962783104 title "Algebras whose Tits form accepts a maximal omnipresent root" @default.
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