The structure of the $$\mathcal{N}=4$$ N = 4 supersymmetric linear $$W_{\infty }[\lambda ]$$ W ∞ [ λ ] algebra

Abstract

Abstract For the vanishing deformation parameter $$\lambda $$ λ , the full structure of the (anti)commutator relations in the $$\mathcal{N}=4$$ N = 4 supersymmetric linear $$W_{\infty }[\lambda =0]$$ W ∞ [ λ = 0 ] algebra is obtained for arbitrary weights $$h_1$$ h 1 and $$h_2$$ h 2 of the currents appearing on the left hand sides in these (anti)commutators. The $$w_{1+\infty }$$ w 1 + ∞ algebra can be seen from this by taking the vanishing limit of other deformation parameter q with the proper contractions of the currents. For the nonzero $$\lambda $$ λ , the complete structure of the $$\mathcal{N}=4$$ N = 4 supersymmetric linear $$W_{\infty }[\lambda ]$$ W ∞ [ λ ] algebra is determined for the arbitrary weight $$h_1$$ h 1 together with the constraint $$h_1-3 \le h_2 \le h_1+1$$ h 1 - 3 ≤ h 2 ≤ h 1 + 1 . The additional structures on the right hand sides in the (anti)commutators, compared to the above $$\lambda =0$$ λ = 0 case, arise for the arbitrary weights $$h_1$$ h 1 and $$h_2$$ h 2 where the weight $$h_2$$ h 2 is outside of above region.