Title: Quadratic relations of the deformed W-superalgebra Wq,tA(M,N)
Abstract: Abstract We find the free field construction of the basic W -current and screening currents for the deformed W -superalgebra <?CDATA ${\mathcal{W}}_{q,t}\left(A(M,N)\right)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">W</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:mfenced> </mml:math> associated with Lie superalgebra of type A ( M , N ). Using this free field construction, we introduce the higher W -currents and obtain a closed set of quadratic relations among them. These relations are independent of the choice of Dynkin diagrams for the Lie superalgebra A ( M , N ), though the screening currents are not. This allows us to define <?CDATA ${\mathcal{W}}_{q,t}\left(A(M,N)\right)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">W</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:mfenced> </mml:math> by generators and relations.