A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of Unity

This paper extends an algorithm and canonical embedding in [6] to a large class of quantum algebras. It applies to iterated Ore extensions over a field satisfying some suitable assumptions which cover those of Cauchon’s original setting but also allows for roots of unity. The extended algorithm constructs a quantum affine space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:math> from the original quantum algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> via a series of change of variables within the division ring of fractions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">Frac</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The canonical embedding takes a completely prime ideal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>⊲</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> to a completely prime ideal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>⊲</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:mrow> </mml:math> such that when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> is a PI algebra, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">PI</mml:mi> <mml:mtext>-</mml:mtext> <mml:mi mathvariant="normal">deg</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>/</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">PI</mml:mi> <mml:mtext>-</mml:mtext> <mml:mi mathvariant="normal">deg</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>′</mml:mi> </mml:msup> <mml:mo>/</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . When the quantum parameter is a root of unity, combining our construction with results from [2] allows us to state an explicit formula for the PI degree of completely prime quotient algebras. This paper ends with a method to construct a maximum dimensional irreducible representation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>/</mml:mo> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> given a suitable irreducible representation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>′</mml:mi> </mml:msup> <mml:mo>/</mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> is PI.