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Register a new userOn the irreducibility of associated varieties of W-algebras
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We investigate the irreducibility of the nilpotent Slodowy slices that appear as the associated variety of W-algebras. Furthermore, we provide new examples of vertex algebras whose associated variety has finitely many symplectic leaves.
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We show that sheet closures appear as associated varieties of affine vertex algebras. Further, we give new examples of non-admissible affine vertex algebras whose associated variety is contained in the nilpotent cone. We also prove some conjectures from our previous paper and give new examples of lisse affine W-algebras.
We introduce a new family of affine $W$-algebras associated with the centralizers of arbitrary nilpotent elements in $mathfrak{gl}_N$. We define them by using a version of the BRST complex of the quantum Drinfeld--Sokolov reduction. A family of free generators of the new algebras is produced in an explicit form. We also give an analogue of the Fateev--Lukyanov realization for these algebras by applying a Miura-type map.
The polynomial ring $B_r:=mathbb{Q}[e_1,ldots,e_r]$ in $r$ indeterminates is a representation of the Lie algebra of all the endomorphism of $mathbb{Q}[X]$ vanishing at powers $X^j$ for all but finitely many $j$. We determine a $B_r$-valued formal power series in $r+2$ indeterminates which encode the images of all the basis elements of $B_r$ under the action of the generating function of elementary endomorphisms of $mathbb{Q}[X]$, which we call the structural series of the representation. The obtained expression implies (and improves) a formula by Gatto & Salehyan, which only computes, for one chosen basis element, the generating function of its images. For sake of completeness we construct in the last section the $B=B_infty$-valued structural formal power series which consists in the evaluation of the vertex operator describing the bosonic representation of $gl_{infty}(mathbb{Q})$ against the generating function of the standard Schur basis of $B$. This provide an alternative description of the bosonic representation of $gl_{infty}$ due to Date, Jimbo, Kashiwara and Miwa which does not involve explicitly exponential of differential operators.
We introduce a new family of Poisson vertex algebras $mathcal{W}(mathfrak{a})$ analogous to the classical $mathcal{W}$-algebras. The algebra $mathcal{W}(mathfrak{a})$ is associated with the centralizer $mathfrak{a}$ of an arbitrary nilpotent element in $mathfrak{gl}_N$. We show that $mathcal{W}(mathfrak{a})$ is an algebra of polynomials in infinitely many variables and produce its free generators in an explicit form. This implies that $mathcal{W}(mathfrak{a})$ is isomorphic to the center at the critical level of the affine vertex algebra associated with $mathfrak{a}$.
Let L be the space of spinors on the 3-sphere that are the restrictions of the Laurent polynomial type harmonic spinors on C^2. L becomes an associative algebra. For a simple Lie algebra g, the real Lie algebra Lg generated by the tensor product of L and g is called the g-current algebra. The real part K of L becomes a commutative subalgebra of L. For a Cartan subalgebra h of g, h tensored by K becomes a Cartan subalgebra Kh of Lg. The set of non-zero weights of the adjoint representation of Kh corresponds bijectively to the root space of g. Let g=h+e+ f be the standard triangular decomposition of g, and let Lh, Le and Lf respectively be the Lie subalgebras of Lg generated by the tensor products of L with h, e and f respectively . Then we have the triangular decomposition: Lg=Lh+Le+Lf, that is also associated with the weight space decomposition of Lg. With the aid of the basic vector fields on the 3-shpere that arise from the infinitesimal representation of SO(3) we introduce a triple of 2-cocycles {c_k; k=0,1,2} on Lg. Then we have the central extension: Lg+ sum Ca_k associated to the 2-cocycles {c_k; k=0,1,2}. Adjoining a derivation coming from the radial vector field on S^3 we obtain the second central extension g^=Lg+ sum Ca_k + Cn. The root space decomposition of g^ as welll as the Chevalley generators of g^ will be given.
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