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Register a new userOn the Vertex Operator Representation of Lie Algebras of Matrices
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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.
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Let $Gamma$ be a generic subgroup of the multiplicative group $mathbb{C}^*$ of nonzero complex numbers. We define a class of Lie algebras associated to $Gamma$, called twisted $Gamma$-Lie algebras, which is a natural generalization of the twisted affine Lie algebras. Starting from an arbitrary even sublattice $Q$ of $mathbb Z^N$ and an arbitrary finite order isometry of $mathbb Z^N$ preserving $Q$, we construct a family of twisted $Gamma$-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted $Gamma$-Lie algebras. As application, this recovers a number of known vertex operator realizations for infinite dimensional Lie algebras, such as twisted affine Lie algebras, extended affine Lie algebras of type $A$, trigonometric Lie algebras of series $A$ and $B$, unitary Lie algebras, and $BC$-graded Lie algebras.
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.
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.
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.
The subjects in the title are interwoven in many different and very deep ways. I recently wrote several expository accounts [64-66] that reflect a certain range of developments, but even in their totality they cannot be taken as a comprehensive survey. In the format of a 30-page contribution aimed at a general mathematical audience, I have decided to illustrate some of the basic ideas in one very interesting example - that of HilbpC2, nq, hoping to spark the curiosity of colleagues in those numerous fields of study where one should expect applications.
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