Quasi-derivations of Witt and related algebras

Abstract In the present work, we compute quasi-derivations of the Witt algebra and some algebras well related to the Witt algebra. Namely, we prove that each quasi-derivation of the Witt algebra is a sum of a derivation and a $$\frac{1}{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> -derivation; a similar result is obtained for the Virasoro algebra. A different situation appears for Lie algebras $$\mathscr {W}(a,b):$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> <mml:mo>:</mml:mo> </mml:mrow> </mml:math> In the case of $$b=-1,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> they do not have interesting examples of quasi-derivations, but the case of $$b\ne -1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>≠</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> provides some new non-trivial examples of quasi-derivations. We also completely describe all quasi-derivations of $$\mathscr {W}(a,b).$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> As a corollary, we describe the derivations and quasi-derivations of the Novikov–Witt and admissible Novikov–Witt algebras previously constructed by Bai and his co-authors; and $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> -derivations and transposed $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> -Poisson structures on cited Lie algebras. In particular, we proved that each $$\mathscr {W}(a,b)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> admits a non-trivial transposed $$\frac{1}{1-b}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> -Poisson structure.

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