Supersymmetric W-algebras

We develop a general theory of $W$-algebras in the context of supersymmetric vertex algebras. We describe the structure of $W$-algebras associated with odd nilpotent elements of Lie superalgebras in terms of their free generating sets. As an application, we produce explicit free generators of the $W$-algebra associated with the odd principal nilpotent element of the Lie superalgebra $\mathfrak{gl}(n+1|n).$Comment: 24page

Supersymmetric extension of universal enveloping vertex algebras

In this paper, we study the construction of the supersymmetric extensions of vertex algebras. In particular, for $N = n \in \mathbb{Z}_{+}$, we show the universal enveloping $N = n$ supersymmetric (SUSY) vertex algebra of an $N = n$ SUSY Lie conformal algebra can be extended to an $N = n' > n$ SUSY vertex algebra. We also show the $N = 2$ SUSY affine vertex algebra of level $0$ associated with a Lie superalgebra, which is an $N = 2$ SUSY extension of the affine vertex algebra of level $0$, can be embedded as an $N = 2$ SUSY vertex subalgebra into an $N = 2$ superconformal vertex algebra.Comment: 25 pages; added references and some comments in section

Structure of classical W-algebras

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 171-172).The first part of the thesis provides three equivalent definitions of a classical finite W-algebra Wfin(g, f) and two equivalent definitions of a classical affine W-algebra W(g, f, k) associated to a Lie algebra g, a nilpotent element f and k [epsilon] C. A classical affine W-algebra W(g, f, k) has a Hamiltonian operator H and the H-twisted Zhu algebra of W(g, f, k) is the classical finite W-algebra Wfin (9, f). A classical finite (resp. affine) W-algebra is isomorphic to a polynomial (resp. differential polynomial) ring. I compute Poisson brackets (resp. Poisson [lambda]-brackets) between generating elements of a classical finite (resp. affine) W-algebra when f is a minimal nilpotent. In the second part, I introduce a classical finite (resp. affine) fractional W-algebra Wfin(g, Am, k) (resp. Wfin(g, Am,k)), where Am = -fz- m - pz-m-1 [epsilon] g((z)) for a certain p [epsilon] g and an integer m >/= 0. If m = 0, then the algebra Wfin(g, Am, k) (resp. W(g, Am, k)) is isomorphic to Wfin(g, f) (resp. W(g, f, k)). I show that an affine fractional W-algebra W(g, Am, k) has a Hamiltonian operator H and the H-twisted Zhu-algebra of W(g, Am, k) is Wfin(g, Am, k). As in ordinary W-algebras cases, a classical finite (resp. affine) fractional W-algebra is isomorphic to a polynomial (resp. differential polynomial) ring. In particular, I show explicit forms of generators and compute brackets (resp.[lambda]-brackets) between them when f is a minimal nilpotent. Using generalized Drinfel'd and Sokolov reduction, I find an infinite sequence of integrable systems related to an affine fractional W-algebra when Am is a semisimple element in g((z)). In the last part, I introduce generalized Drinfeld-Sokolov reductions and Hamiltonian ODEs associated to classical finite W-algebras and finite fractional W-algebras. Also, I find integrals of motion of the Hamiltonian ODEs using Drinfel'd-Sokolov reductions. It is an open problem whether these equations are Lenard integrable.by Uhi Rinn Suh.Ph.D

Generators of supersymmetric classical WW-algebras

Let $\mathfrak{g}$ be a Lie superalgebra of type $\mathfrak{sl}$ or$\mathfrak{osp}$ with an odd principal nilpotent element $f$. We consider amatrix $\mathcal{A}_{\mathfrak{g},f}$ determined by $\mathfrak{g}$ and $f$ andfind a generating set of the supersymmetric classical $W$-algebra$\mathcal{W}(\bar{\mathfrak{g}},f)$ using the row determinant of$\mathcal{A}_{\mathfrak{g},f}$