Related papers: $W_{1+\infty}$ constraints for the hermitian one-m…
By using the free field realizations, we analyze the representation theory of the W_{1+infinity} algebra with c=1. The eigenvectors for the Cartan subalgebra of W_{1+infinity} are parametrized by the Young diagrams, and explicitly written…
We review the recent development in the representation theory of the $W_{1+\infty}$ algebra. The topics that we concern are, Quasifinite representation, Free field realizations, (Super) Matrix Generalization, Structure of subalgebras such…
In the recent study of Virasoro action on characters, we discovered that it gets especially simple for peculiar linear combinations of the Virasoro operators: particular harmonics of $\hat w$-operators. In this letter, we demonstrate that…
We construct the ($\beta$-deformed) higher order total derivative operators and analyze their remarkable properties. In terms of these operators, we derive the higher order constraints for the ($\beta$-deformed) Hermitian matrix models. We…
We provide a generators and relation description of the deformed W_{1+\infty}-algebra introduced in previous joint work of E. Vasserot and the second author. This gives a presentation of the (spherical) cohomological Hall algebra of the…
We describe the general framework for constructing collective--theory Hamiltonians whose hermicity requirements imply a Kac--Moody algebra of constraints on the associated Jacobian. We give explicit examples for the algebras $sl(2)_k$ and…
We introduce a cohomology theory of grading-restricted vertex algebras. To construct the {\it correct} cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to "rational functions valued in the…
We discuss a spectrum generating algebra in the supersymmetric quantum mechanical system which is defined as a series of solutions to a specific differential equation. All Hamiltonians have equally spaced eigenvalues, and we realize both…
Very recently, Bai [Linear Algebra Appl., 681:150-186, 2024 \& Appl. Math. Lett., 166:109510, 2025] studied some concrete structures, and obtained essential algebraic and computational properties of the one-dimensional, two-dimensional and…
The vertex algebra W_{1+\infty,c} with central charge c may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer n\geq 1, it was conjectured in the physics…
We demonstrate that commutativity of numerous one-dimensional subalgebras in $W_{1+\infty}$ algebra, i.e. the existence of many non-trivial integrable systems described in recent arXiv:2303.05273 follows from the subset of relations in…
We classify the quasifinite highest weight modules over a family of subalgebras W_{\infty}^{n} of the central extension W_{1+\infty} of the Lie algebra of differential operators on the circle consisting of operators of order \geq n. We…
We give a spectral theorem for unital representations of Hermitian commutative unital *-algebras by possibly unbounded operators in a pre-Hilbert space. A better result is known for the case in which the *-algebra is countably generated.
We study the even spin $\mathcal{W}_\infty$ which is a universal $\mathcal{W}$-algebra for orthosymplectic series of $\mathcal{W}$-algebras. We use the results of Fateev and Lukyanov to embed the algebra into $\mathcal{W}_{1+\infty}$.…
We derive the loop equations for the one Hermitian matrix model in any dimension. These are a consequence of the Schwinger-Dyson equations of the model. Moreover we show that in leading order of large $N$ the loop equations form a closed…
W_{1+infty} is defined as an infinite dimensional Lie algebra spanned by the unit operator and the Laurent modes of a series of local quasiprimary chiral fields V^l(z) of dimension l+1 (l=0,1,2,...). These fields are neutral with respect to…
We develop methods for systematic construction of superintegrable polynomials in matrix/eigenvalue models. Our consideration is based on a tight connection of superintegrable property of Gaussian Hermitian model and $W_{1 + \infty}$ algebra…
We present a classification of $W$ algebras and superalgebras arising in Abelian as well as non Abelian Toda theories. Each model, obtained from a constrained WZW action, is related with an $Sl(2)$ subalgebra (resp. $OSp(1|2)$ superalgebra)…
We consider complex, weakly non-Hermitian matrices $A = W_1 +i\sqrt{{\tau}_N}W_2$ , where $W_1$ and $W_2$ are Hermitian matrices and $\tau_N = O(N^{-1})$. We first show that for pairs of Hermitian matrices $(W_1 , W_2)$ such that $W_1$…
We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary $W$-algebra. We show that for a minimal simple $W$-algebra…