In this paper, we present a candidate for $$\mathcal {N}=(1,1)$$ N = ( 1 , 1 ) extended higher-spin $$AdS_3$$ A d S 3 supergravity with the most general boundary conditions discussed by Grumiller and Riegler recently. We show that the asymptotic symmetry algebra consists of two copies of the $$\mathfrak {osp}(3|2)_k$$ osp ( 3 | 2 ) k affine algebra in the presence of the most general boundary conditions. Furthermore, we impose some certain restrictions on gauge fields on the most general boundary conditions and that leads us to the supersymmetric extension of the Brown–Henneaux boundary conditions. We eventually see that the asymptotic symmetry algebra reduces to two copies of the $$\mathcal {SW}(\frac{3}{2},2)$$ SW ( 3 2 , 2 ) algebra for $$\mathcal {N}=(1,1)$$ N = ( 1 , 1 ) extended higher-spin supergravity.

Casimir W-algebras are shown to exist in such a way that the conformal spins of primary (generating) fields coincide with the orders of independent Casimir operators. We show here that this coincidence can be extended further to the case that these generating fields have the same eigenvalues with the Casimir operators.

We obtain new coupled super Nonlinear Schrödinger equations by using AKNS scheme and soliton connection taking values in N=2 superconformal algebra .

Starting from the well-known quantum Miura-like transformation for the non-simplylaced Lie algebras B N , we give an explicit construction of the Casimir WB 3 algebras. We reserve the notation WB N for the Casimir W algebras of type W(2, 4, 6, . . . , 2N, 2N + 1/2) which contains one fermionic field. It is seen that WB 3 algebra is closed and associative for all values of the central charge c. * This letter is dedicated to the 225th anniversary of the Istanbul Technical University. † 469 Mod. Phys. Lett. A 1999.14:469-477. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SAN DIEGO on 04/12/15. For personal use only.

In this paper, we explicitly construct an asymptotic W5 symmetry algebra of the three-dimensional anti-de Sitter (AdS3) higher spin gravity. We use an sl(5,R)⊕sl(5,R) Lie algebra valued Chern-Simons gauge theory with a negative cosmological constant, and its asymptotic symmetry algebra is explicitly calculated as two copies of the classical W5 algebra with central charge c. Our results can be interpreted as a spin 5 extension of AdS3 gravity and a proof of how the higher spin Ward identities and the asymptotic W5 symmetry algebra is derived from the higher spin bulk field equations of motion. This higher spin asymptotic W5 symmetry algebra contains a finite number of conformal primary spin s: s = 2, 3, 4, 5. We also indicated how to introduce chemical potentials and holonomy conditions associated with these higher spin charges in AdS3 higher spin gravity in a manner that it preserves the asymptotic symmetry algebra.