Related papers: On the centralizers in the Weyl algebra

For a finite dimensional representation $V$ of a finite reflection group $W$, we consider the rational Cherednik algebra $\mathsf{H}_{t,c}(V,W)$ associated with $(V,W)$ at the parameters $t\neq 0$ and $c$. The Dunkl total angular momentum…

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $\ell$ over an algebraically closed field $\Bbbk$ of characteristic zero, and let $(e,h,f)$ be an $\mathfrak{sl}_2$-triple of g. Denote by $\mathfrak{g}^{e}$ the…

For any polynomial $P \in \mathbb{C}[X_1,X_2,...,X_n]$, we describe a $\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of $F(P)$…

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…

We show that induction and restriction functors for inclusions of nilCoxeter algebras provide a categorical realization of the algebra of polynomial differential operators in one variable.

We show that if $P$ is a $PD_3$-complex and $g\in\pi_1(P)$ has finite order $>1$ and infinite centraliser then $\pi_1(P)$ retracts onto $Z/2Z\oplus\mathbb{Z}$. If $P$ is an irreducible closed 3-manifold then it follows from the Projective…

We determine criteria for the prime spectrum of an ambiskew polynomial algebra $R$ over an algebraically closed field $K$ to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra…

To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In…

It is shown that the kernel of the Fock representation of a certain Wick algebra with braided operator of coefficients T, || T ||<= 1, coincides with the largest quadratic Wick ideal. Improved conditions on the operator T for the Fock inner…

Let k be an algebraically closed field of zero characteristic. The Lie algebra W_2 of all k-derivations of the polynomial ring k[x, y] naturally acts on the polynomial ring k[x, y] and also on the field of rational functions k(x, y). For a…

Let L be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We provide necessary and also some sufficient conditions in order for its Poisson center and semi-center to be polynomial algebras over…

We introduce the notion of Kravchuk derivations of the polynomial algebra. We prove that any element of the kernel of the derivation gives a polynomial identity satisfied by the Kravchuk polynomials. Also, we prove that any kernel element…

This papers studies centralizers of an element, $a$, in the nucleus of a non-associative algebra with a special type of valuation. We prove that the centralizer of $a$ is a free module of finite rank over the algebra generated by $a$.

In this note we generalize the definition of partial permutations of Ivanov and Kerov and we build a universal algebra which projects onto the m-centraliser algebra defined by Creedon. We use it to present a new proof for the polynomiality…

We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group $W=G(m,p,n)$ and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category…

Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric Witt algebra of all derivations of $P_n$. We describe all…

We study maximal Poisson-commutative subalgebras in the Poisson algebra $S(\mathfrak{g})$ of a semisimple Lie algebra $\mathfrak{g}$ constructed by Mischenko and Fomenko with the help of the argument shift method. We prove that these…

Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)\ge 0, or p_i(x)=0 for a given list of polynomials p_1,...,p_m. We prove several statements that fit into the following…

Let $H(R, \phi, z)$ be a generalized Weyl algebra associated with a ring $R$, its central element $z\in Z(R)$ and an automorphism $\phi,$ such that for some $l \geq 1$, $\phi^l(z)-z$ is nilpotent and $(z,\phi^i(z))=R$ for all $0<i<l$. We…

Fix a Dynkin diagram and let p be a coweight. When does there exist an element w of the corresponding Weyl group such that w is p-minuscule and w(p) is dominant? We answer this question for general Coxeter groups. We express and prove these…