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Register a new userOn Segal--Sugawara vectors and Casimir elements for classical Lie algebras
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Authors
A. I. Molev
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We consider the centers of the affine vertex algebras at the critical level associated with simple Lie algebras. We derive new formulas for generators of the centers in the classical types. We also give a new formula for the Capelli-type determinant for the symplectic Lie algebras and calculate the Harish-Chandra images of the Casimir elements arising from the characteristic polynomial of the matrix of generators of each classical Lie algebra.
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For every simple Lie algebra $mathfrak{g}$ we consider the associated Takiff algebra $mathfrak{g}^{}_{ell}$ defined as the truncated polynomial current Lie algebra with coefficients in $mathfrak{g}$. We use a matrix presentation of $mathfrak{g}^{}_{ell}$ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra ${rm U}(mathfrak{g}^{}_{ell})$. A similar matrix presentation for the affine Kac--Moody algebra $widehat{mathfrak{g}}^{}_{ell}$ is then used to prove an analogue of the Feigin--Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. The proof relies on an explicit construction of a complete set of Segal--Sugawara vectors for the Lie algebra $mathfrak{g}^{}_{ell}$.
The conjectures of Alday, Gaiotto and Tachikawa and its generalizations have been mathematically formulated as the existence of an action of a $W$-algebra on the cohomology or $K$-theory of the instanton moduli space, together with a Whitakker vector. However, the original conjectures also predict intertwining properties with the natural higher rank version of the $Ext^1$ operator which was previously studied by Okounkov and the author in [CO], a result which is now sometimes referred to as AGT in rank one [Alb,PSS]. Physically, this corresponds to incorporating matter in the Nekrasov partition functions, an obviously important feature in the physical theory. It is therefore of interest to study how the $Ext^1$ operator relates to the aforementioned structures on cohomology in higher rank, and if possible to find a formulation from which the AGT conjectures follow as a corollary. In this paper, we carry out something analogous using a modified Segal-Sugawara construction for the $hat{mathfrak{sl}}_2mathbb{C}$ structure that appears in Okounkov and Nekrasovs proof of Nekrasovs conjecture [NO] for rank two. This immediately implies the AGT identities when the central charge is one, a case which is of particular interest for string theorists, and because of the natural appearance of the Seiberg-Witten curve in this setup, see for instance Dijkgraaf and Vafa [DV], as well as [IKV].
The symplectic structures on $3$-Lie algebras and metric symplectic $3$-Lie algebras are studied. For arbitrary $3$-Lie algebra $L$, infinite many metric symplectic $3$-Lie algebras are constructed. It is proved that a metric $3$-Lie algebra $(A, B)$ is a metric symplectic $3$-Lie algebra if and only if there exists an invertible derivation $D$ such that $Din Der_B(A)$, and is also proved that every metric symplectic $3$-Lie algebra $(tilde{A}, tilde{B}, tilde{omega})$ is a $T^*_{theta}$-extension of a metric symplectic $3$-Lie algebra $(A, B, omega)$. Finally, we construct a metric symplectic double extension of a metric symplectic $3$-Lie algebra by means of a special derivation.
We give a definition of quaternion Lie algebra and of the quaternification of a complex Lie algebra. By our definition gl(n,H), sl(n,H), so*(2n) ans sp(n) are quaternifications of gl(n,C), sl(n,C), so(n,C) and u(n) respectively. Then we shall prove that a simple Lie algebra admits the quaternification. For the proof we follow the well known argument due to Harich-Chandra, Chevalley and Serre to construct the simple Lie algebra from its corresponding root system. The root space decomposition of this quaternion Lie algebra will be given. Each root sapce of a fundamental root is complex 2-dimensional.
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 $mathfrak{a}$.
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