Related papers: Some remarks on periodic gradings

The symmetry algebra $P_\infty = W_\infty \oplus H \oplus I_\infty$ of integrable systems is defined. As an example the classical Sophus Lie point symmetries of all higher KP equations are obtained. It is shown that one (``positive'') half…

We study the computational complexity of deciding whether a given set of term equalities and inequalities has a solution in an $\omega$-categorical algebra $\mathfrak{A}$. There are $\omega$-categorical groups where this problem is…

Fix an integral semisimple element $\lambda$ in the Lie algebra $\mathfrak{g}$ of a complex reductive algebraic group $G$. Let $L$ denote the centralizer of $\lambda$ in $G$ and let $\mathfrak{g}(-1)$ denote the $-1$ eigenspace of…

Many interesting C*-algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution C*-algebra of a symplectic groupoid. Toward this end, I define…

We show that one can construct a classical affine W-algebra via a classical BRST complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a…

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related:…

We study the subalgebra of the lattice vertex operator algebra $V_{\sqrt{2}A_2}$ consisting of the fixed points of an automorphism which is induced from an order 3 isometry of the root lattice $A_2$. We classify the simple modules for the…

Let $S^{\cdot}$ be a noetherian graded algebra over a commutative $k$-algebra $A$, where $k$ is a commutative ring, and assume it is a module over a Lie algebroid ${\mathfrak g}_{A/k}$. If $S^\cdot$ is semi-simple over ${\mathfrak g}_{A/k}$…

We extend known prequantization procedures for Poisson and presymplectic manifolds by defining the prequantization of a Dirac manifold P as a principal U(1)-bundle Q with a compatible Dirac-Jacobi structure. We study the action of Poisson…

Given a finite-dimensional complex Lie algebra g equipped with a nondegenerate, symmetric, invariant bilinear form B, let V_k(g,B) denote the universal affine vertex algebra associated to g and B at level k. For any reductive group G of…

It is a classical fact in Poisson geometry that the cotangent bundle of a Poisson manifold has the structure of a Lie algebroid. Manifestations of this structure are the Lichnerowicz differential on multivector fields (calculating Poisson…

Let $G$ be a connected real reductive group with maximal compact subgroup $K$ of the same rank as $G$. In the recent paper of Huang, Pand\v{z}i\'{c} and Vogan, it was shown that the admissible $\Theta$--stable parabolic subalgebras…

Let $\g$ be a reductive Lie algebra and $V$ a $\g$-semisimple module. In this article, we study the category $\G$ of graded finite-dimensional representations of $\g \ltimes V$. We produce a large class of truncated subcategories, which are…

In this paper we prove that any Poisson structure on a sheaf of Lie algebroids admits a weak deformation quantization, and give a sufficient condition for such a Poisson structure to admit an actual deformation quantization. We also answer…

The infinite affine Lie algebras of type ABCD, also called $\widehat{\mathfrak{gl}}(\infty)$, $\widehat{\mathfrak{o}}(\infty)$, $\widehat{\mathfrak{sp}}(\infty)$, are equivalent to subalgebras of the quantum $W_{1+\infty}$ algebras. They…

A finite-dimensional Lie algebra is called an A-algebra if all of its nilpotent subalgebras are abelian. These arise in the study of constant Yang-Mills potentials and have also been particularly important in relation to the problem of…

It is shown that the elliptic algebra ${\cal A}_{q,p}(\hat{sl}(2)_c)$ at the critical level c=-2 has a multidimensional center containing some trace-like operators t(z). A family of Poisson structures indexed by a non-negative integer and…

We explore the (noncommutative) geometry of locally simple representations of the diagonal locally finite Lie algebras $\mathfrak{sl}(n^\infty)$, $\mathfrak o(n^\infty)$, and $\mathfrak{sp}(n^\infty)$. Let $\mathfrak g_\infty$ be one of…

Recently Kontsevich solved the classification problem for deformation quantizations of all Poisson structures on a manifold. In this paper we study those Poisson structures for which the explicit methods of Fedosov can be applied, namely…

The link between (super)-affine Lie algebras as Poisson brackets structures and integrable hierarchies provides both a classification and a tool for obtaining superintegrable hierarchies. The lack of a fully systematic procedure for…