The Green Ring of a Restricted Enveloping Algebra in Characteristic 2

Abstract

<jats:p> Let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\Bbbk $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> be an algebraically closed field of characteristic 2 and let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathfrak {fsl}(2)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>fsl</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be the unique, up to isomorphism, 3-dimensional simple Lie algebra over <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\Bbbk $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> . Denote by <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathfrak {m}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> the minimal 2-envelope of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathfrak {fsl}(2)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>fsl</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and by <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathfrak {u}(\mathfrak {m})$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> its corresponding restricted enveloping algebra. The non-isomorphic finite-dimensional indecomposable <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathfrak {u}(\mathfrak {m})$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -modules were classified in [1]. In this paper, the Green ring (or representation ring) for <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathfrak {u}(\mathfrak {m})$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is calculated. Also, the semisimplification of the representation category of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathfrak {u}(\mathfrak {m})$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is determined. </jats:p>