Related papers: The Weyl algebra as a fixed ring

We study holonomic modules for the rings of invariant differential operators on affine commutative domains with finite Krull dimension with respect to arbitrary actions of finite groups. We prove the Bernstein inequality for these rings.…

If $k$ is a field and $R$ is a commutative $k$-algebra, we explore the question of when the ring $D_{R|k}$ of $k$-linear differential operators on $R$ is isomorphic to its opposite ring. Under mild hypotheses, we prove this is the case…

We develop algebraic geometry for general Segal's Gamma-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under Spec Z). The starting observation is that the category obtained by gluing…

The impetus for this study is the work of Dumas and Rigal on the Jordanian deformation of the ring of coordinate functions on $2\times 2$ matrices. We are also motivated by current interest in birational equivalence of noncommutative rings.…

We construct a family of non-invertible topological defects in two-dimensional theories of $n$ Weyl fermions. The construction relies on the existence of $G$-symmetric conformal boundary conditions for $n$ Dirac fermions. Upon unfolding,…

Let G be a complex semisimple group and U its maximal unipotent subgroup. We study the algebra D(G/U) of algebraic differential operators on G/U and also its quasi-classical counterpart: the algebra of regular functions on the cotangent…

Let X be a complex smooth affine irreducible curve, and let D = D(X) be the ring of global differential operators on X. In this paper, we give a geometric classification of left ideals in $ D $ and study the natural action of the Picard…

Building on the Atiyah--Singer holomorphic Lefschetz fixed-point theorem, we define ramification modules associated to the fixed loci of a finite group acting on a compact complex manifold. This allows us to generalize the Chevalley--Weil…

Let X be a smooth or proper variety defined over a finite field. The geometric etale fundamental group of X is a normal subgroup of the Weil group, so conjugation gives it a Weil action. We consider the pro-Q_l-algebraic completion of the…

Let g be a simple Lie algebra, with fixed Borel subalgebra b and with Weyl group W. Expanding on previous work of Fan and Stembridge in the simply laced case, this note aims to study the fully commutative elements of W, and their…

We define a multiple Dirichlet series whose group of functional equations is the Weyl group of the affine Kac-Moody root system $\tilde{A}_n$, generalizing the theory of multiple Dirichlet series for finite Weyl groups. The construction is…

Chevalley's theorem states that for any simple finite dimensional Lie algebra G (1) the restriction homomorphism of the algebra of polynomials on G onto the Cartan subalgebra H induces an isomorphism between the algebra of G-invariant…

Using the language of operated algebras, we construct and investigate a class of operator rings and enriched modules induced by a derivation or Rota-Baxter operator. In applying the general framework to univariate polynomials, one is led to…

Border bases are a generalization of Gr\"obner bases for zero-dimensional ideals in polynomial rings. In this article, we introduce border bases for a non-commutative ring of linear differential operators, namely the rational Weyl algebra.…

The symmetric coinvariant algebra $C[x_1, dots, x_n]_{S_n}$ is the quotient algebra of the polynomial ring by the ideal generated by symmetric polynomials vanishing at the origin. It is known that the algebra is isomorphic to the regular…

Let $A_1(K)=K \langle x,y | yx-xy= 1 \rangle$ be the first Weyl algebra over a characteristic zero field $K$ and let $\alpha$ be the exchange involution on $A_1(K)$ given by $\alpha(x)= y$ and $\alpha(y)= x$. The Dixmier conjecture of…

First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field.…

The first Weyl algebra, $A_1 = k \langle x, y\rangle/(xy-yx - 1)$ is naturally $\mathbb{Z}$-graded by letting $\operatorname{deg} x = 1$ and $\operatorname{deg} y = -1$. Sue Sierra studied $\operatorname{gr}- A_1$, category of graded right…

The Lie algebra $\mathcal{D}$ of regular differential operators on the circle has a universal central extension $\hat{\mathcal{D}}$. The invariant subalgebra $\hat{\mathcal{D}}^+$ under an involution preserving the principal gradation was…

Let $V$ be a symplectic space over $\mathbb{C}$, $dim_\mathbb{C} V=2l$, and let $G$ be a finite subgroup of $Sp(V)$. The invariant regular functions $\mathbb{C}[V]^G$ inherit a Poisson algebra structure and so the quotient variety ${\cal…