Classical affine W-algebras associated to Lie superalgebras (Q2795526)

In this paper the author studies the structure of classical affine W-algebras associated to Lie superalgebras (W-superalgebras). Recall that there are four types of W-algebras, classical affine, classical finite, quantum affine, and quantum finite W-algebras. These types of algebras are endowed with Poission vertex algebras(PVAs), Poisson algebras(PAs), vertex algebras (VAs) and associative algebras (AAs) structures, respectively. PVAs (respectively, PAs) are quasi-classical limits of VAs (respectively) (AAs) and Poisson algebras (respectively, AAs) are finalizations of PVAs (respectively, VAs). The main ingredient of this paper is a classical affine W-algebra, which is endowed with PVA structures. Hence, we expect classical affine W-algebras are obtained by quasi-classical limits of quantum affine W-algebras and chiralizations of classical finite W-algebras. A classical finite W-algebra associated to a Lie (super) algebra \(g\) and its nilpotent \(f\) is defined by the Hamiltonian reduction. A natural way to get a quantum finite W-algebra is by BRST quantization of the Lie (super) algebra complex, called a finite BRST complex. It is proved (Ref. 5,11) that the quantum finite W-algebra associated to a Lie (super) algebra \(g\) and its nilpotent \(f\) can be obtained by a quantum Hamiltonian reduction associated to \(U(g)\). A natural question is that if we can develop a similiar theory for a classical affine W-algebra associated to a Lie superalgebra (classical affine W-superalgebra). The author proves that a classical affine W-superalgebra can be defined via classical BRST complex and via Hamilton reduction. Also he shows that the same argument works for classical and quantum finite W-superalgebra. The last part of this paper is about fractional W-superalgebras. The author defines classical affine fractional W-algebras associated to Lie superalgebras (fractional W-superalgebras) and finds free generators of a classical affine fractional W-superalgebras associated a minimal nilpotent. I see that its a rich study of classical finite W-(super) algebras, classical affine W-(super) algebras, quantum affine W-(super) algebra, quantum finite W-(super) algebras and their relations.