Algebra Of Differential Operators Research Articles

Lie algebras of differential operators for matrix valued Laguerre type polynomials

Lie algebras of differential operators for matrix valued Laguerre type polynomials

Quantized Weyl algebras, the double centralizer property, and a new first fundamental theorem for Uq(gln)

Let P:=Pm×n denote the quantized coordinate ring of the space of m × n matrices, equipped with natural actions of the quantized enveloping algebras Uq(glm) and Uq(gln) . Let L and R denote the images of Uq(glm) and Uq(gln) in End(P) , respectively. We define a q-analogue of the algebra of polynomial-coefficient differential operators inside End(P) , henceforth denoted by PD , and we prove that L∩PD and R∩PD are mutual centralizers inside PD . Using this, we establish a new First Fundamental theorem of invariant theory for Uq(gln) . We also compute explicit formulas in terms of q-determinants for generators of the algebras Lh∩PD and Rh∩PD , where Lh and Rh denote the images of the Cartan subalgebras of Uq(glm) and Uq(gln) in End(P) , respectively. Our algebra PD and the algebra Pol(Matm,n)q that is defined in (Shklyarov et al 2004 Int. J. Math. 15 855–94) are related by extension of scalars, but we give a new construction of PD using deformed twisted tensor products.

Courant-Dorfman algebras of differential operators and Dorfman connections of Courant algebroids

We construct an algebra and a complex of multidifferential operators on tensor products of a Courant algebroid E with values in the endomorphism bundle of a smooth vector bundle B, predual of E, extending the standard complex of the Courant-Dorfman algebra of E. Also, we study Dorfman connections of E on B, and show that the Cartan calculus, curvatures of induced connections and basic differential geometric identities of them make sense in this algebra.

On the Bezrukavnikov–Kaledin quantization of symplectic varieties in characteristic p

We prove that after inverting the Planck constant $h$ , the Bezrukavnikov–Kaledin quantization $(X, {\mathcal {O}}_h)$ of symplectic variety $X$ in characteristic $p$ with $H^2(X, {\mathcal {O}}_X) =0$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$ .

Realizations and invariants of Lie algebras

Lie algebras of first-order differential operators with smooth coefficients are considered and their construction methods are reviewed. Obstacles preventing contractions of such operators are emphasized and a constructive procedure for parameterizations is proposed. A number of illustrative examples are considered.

Cosets from equivariant 𝒲-algebras

The equivariant W \mathcal W -algebra of a simple Lie algebra g \mathfrak {g} is a BRST reduction of the algebra of chiral differential operators on the Lie group of g \mathfrak {g} . We construct a family of vertex algebras A [ g , κ , n ] A[\mathfrak {g}, \kappa , n] as subalgebras of the equivariant W \mathcal W -algebra of g \mathfrak {g} tensored with the integrable affine vertex algebra L n ( g ˇ ) L_n(\check {\mathfrak {g}}) of the Langlands dual Lie algebra g ˇ \check {\mathfrak {g}} at level n ∈ Z > 0 n\in \mathbb {Z}_{>0} . They are conformal extensions of the tensor product of an affine vertex algebra and the principal W \mathcal {W} -algebra whose levels satisfy a specific relation. When g \mathfrak {g} is of type A D E ADE , we identify A [ g , κ , 1 ] A[\mathfrak {g}, \kappa , 1] with the affine vertex algebra V κ − 1 ( g ) ⊗ L 1 ( g ) V^{\kappa -1}(\mathfrak {g}) \otimes L_1(\mathfrak {g}) , giving a new and efficient proof of the coset realization of the principal W \mathcal W -algebras of type A D E ADE .

The Lie algebra of the lowest transitively differential group of degree three *

We investigate the real Lie algebra of first-order differential operators with polynomial coefficients, which is subject to the following requirements. (1) The Lie algebra should admit a basis of differential operators with homogeneous polynomial coefficients of degree up to and including three. (2) The generator of the algebra must include the translation operators for all the variables . (3) The Lie algebra is the smallest indecomposable Lie algebra satisfying (1) and (2). It turns out to be a 39-dimensional Lie algebra in six variables (k = 6) and the construction of this algebra is also the simplest possible case in the general construction of the Lie algebras of the transitively differential groups introduced by Guillemin and Sternberg in 1964 involving the coefficients of degree 3. Those algebras and various subalgebras have similarities with algebras related to different applications in physics such as the Schrödinger, Conformal and Galilei algebras with and without central extension. The paper is devoted to the presentation of the structure and different decompositions of the Lie algebra under investigation. It is also devoted to the presentation of relevant Lie subalgebras and the construction of their Casimir invariants using different methods. We rely, in particular, on differential operator realizations, symbolic computation packages, the Berezin bracket and virtual copies of the Lie algebras.

A proof of the polynomial conjecture for nilpotent Lie groups monomial representations

Let G = exp ⁡ ( g ) G = \operatorname {exp}(\mathfrak {g}) be a connected and simply connected real nilpotent Lie group with Lie algebra g \mathfrak g , H = exp ⁡ ( h ) H = \operatorname {exp}(\mathfrak {h}) an analytic subgroup of G G with Lie algebra h \mathfrak h , χ \chi a unitary character of H H and τ = ind H G χ \tau = \text {ind}_H^G \chi the monomial representation of G G induced by χ \chi . Let D τ ( G / H ) D_{\tau }(G/H) be the algebra of the G G -invariant differential operators on the fiber bundle over the base space G / H G/H associated to the data ( H , χ ) (H,\chi ) . We prove the polynomial conjecture due to Corwin-Greenleaf stating that if τ \tau is of finite multiplicities, the algebra D τ ( G / H ) D_{\tau }(G/H) is isomorphic to the algebra C [ Γ τ ] H {{\mathbb C}[{\Gamma }_{\tau }]}^H of the H H -invariant polynomial functions on the affine subspace Γ τ = { ℓ ∈ g ∗ ; ℓ | h = − − 1 d χ } {\Gamma }_{\tau } = \{\ell \in {\mathfrak g}^*; {\ell }|_{\mathfrak h} = -\sqrt {-1}d\chi \} of g ∗ {\mathfrak g}^* . In this case, we show that any non-zero element of D τ ( G / H ) D_{\tau }(G/H) admits a fundamental tempered solution.

Geometric tool kit for higher-spin gravity (Part I): An introduction to the geometry of differential operators

This paper provides an introduction to the algebra and geometry of differential operators and jet bundles. Its point of view is guided by the leitmotiv that higher-spin gravity theories call for higher-order generalizations of Lie derivatives and diffeomorphisms. Nevertheless, the material covered here may be of general interest to anyone working on topics where geometrical (coordinate-free, global, generic) and mathematically rigorous definitions of differential operators are required.

Formula omitted] modules of finite type on affine space

We study the category of modules admitting compatible actions of the Lie algebra V of vector fields on an affine space and the algebra A of polynomial functions. We show that modules in this category which are finitely generated over A, are free. We also show that this pair of compatible actions is equivalent to commuting actions of the algebra of differential operators and the Lie algebra of vector fields vanishing at the origin. This allows us to construct explicit realizations of such modules as gauge modules.

The step-wise construction of solitary solutions to Riccati equations with diffusive coupling

<abstract><p>A novel scheme based on the generalized differential operator and computer algebra was used to construct solitary solutions to a system of Riccati differential equations with diffusive coupling. The presented approach yields necessary and sufficient existence conditions of solitary solutions with respect to the system parameters. The proposed stepwise approach enabled the derivation of the explicit analytic solution, which could not be derived using direct balancing techniques due to the complexity of algebraic relationships. Computational experiments were used to demonstrate the efficacy of proposed scheme.</p></abstract>

Algebras of variable coefficient quantized differential operators

In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, “quantized Hermitian symmetric spaces,” an obvious problem is to describe (quantum) holomorphically induced representations in terms of some manageable structures. An intimately related problem is to decide what the algebras of quantized differential operators with variable coefficients should be. It is an immediate point that even zeroth order operators, given as multiplications by polynomials, have to be specified as, e.g., left or right multiplication operators since the polynomial algebras are replaced by quadratic, non-commutative algebras. In the settings we are interested in, there are bilinear pairings that allow us to define differential operators as duals of multiplication operators. Indeed, there are different choices of pairings which lead to quite different results. We consider three different pairings while specializing to Uq(su(n,n)). The pairings are between quantized generalized Verma modules and quantized holomorphically induced modules. It is a natural demand that the corresponding representations can be expressed by (matrix valued) differential operators. We show that a quantum Weyl algebra Weylq(n,n) introduced by Hayashi [Commun. Math. Phys. 127(1), 129–144 (1990)] plays a fundamental role. In fact, for one pairing, the algebra of differential operators, though inherently depending on a choice of basis, is precisely matrices over Weylq(n,n). We determine explicitly the form of the (quantum) holomorphically induced representations and determine, for the different pairings, if they can be expressed by differential operators.

U(h)-Free Modules over the Lie Algebras of Differential Operators

This paper mainly considers a class of non-weight modules over the Lie algebra of the Weyl type. First, we construct the U(h)-free modules of rank one over the differential operator algebra. Then, we characterize the tensor products of these kind of modules and the quasi-finite highest weight modules. Finally, we undertake such research for the differential operator algebra of multi-variables.

A bispinor formalism for spinning Witten diagrams

We develop a new embedding-space formalism for AdS4 and CFT3 that is useful for evaluating Witten diagrams for operators with spin. The basic variables are Killing spinors for the bulk AdS4 and conformal Killing spinors for the boundary CFT3. The more conventional embedding space coordinates XI for the bulk and PI for the boundary are bilinears in these new variables. We write a simple compact form for the general bulk-boundary propagator, and, for boundary operators of spin ℓ ≥ 1, we determine its conservation properties at the unitarity bound. In our CFT3 formalism, we identify an mathfrak{so} (5, 5) Lie algebra of differential operators that includes the basic weight-shifting operators. These operators, together with a set of differential operators in AdS4, can be used to relate Witten diagrams with spinning external legs to Witten diagrams with only scalar external legs. We provide several applications that include Compton scattering and the evaluation of an R4 contact interaction in AdS4. Finally, we derive bispinor formulas for the bulk-to-bulk propagators of massive spinor and vector gauge fields and evaluate a diagram with spinor exchange.

The Hochschild cohomology of the algebra of differential operators tangent to a central arrangement of lines

The Hochschild cohomology of the algebra of differential operators tangent to a central arrangement of lines

Notes on Higher-Spin Diffeomorphisms

Higher-spin diffeomorphisms are to higher-order differential operators what diffeomorphisms are to vector fields. Their rigorous definition is a challenging mathematical problem which might predate a better understanding of higher-spin symmetries and interactions. Several yes-go and no-go results on higher-spin diffeomorphisms are collected from the mathematical literature in order to propose a generalisation of the algebra of differential operators on which higher-spin diffeomorphisms are well-defined. This work is dedicated to the memory of Christiane Schomblond, who taught several generations of Belgian physicists the formative rigor and delicate beauty of theoretical physics.

SIMPLICITY CRITERIA FOR RINGS OF DIFFERENTIAL OPERATORS

Abstract Let K be a field of arbitrary characteristic, $${\cal A}$$ be a commutative K-algebra which is a domain of essentially finite type (e.g., the algebra of functions on an irreducible affine algebraic variety), $${a_r}$$ be its Jacobian ideal, and $${\cal D}\left( {\cal A} \right)$$ be the algebra of differential operators on the algebra $${\cal A}$$ . The aim of the paper is to give a simplicity criterion for the algebra $${\cal D}\left( {\cal A} \right)$$ : the algebra $${\cal D}\left( {\cal A} \right)$$ is simple iff $${\cal D}\left( {\cal A} \right)a_r^i{\cal D}\left( {\cal A} \right) = {\cal D}\left( {\cal A} \right)$$ for all i ≥ 1 provided the field K is a perfect field. Furthermore, a simplicity criterion is given for the algebra $${\cal D}\left( R \right)$$ of differential operators on an arbitrary commutative algebra R over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators.

The Dunkl kernel and intertwining operator for dihedral groups

Dunkl operators associated with finite reflection groups generate a commutative algebra of differential-difference operators. There exists a unique linear operator called intertwining operator which intertwines between this algebra and the algebra of standard differential operators. There also exists a generalization of the Fourier transform in this context called Dunkl transform.In this paper, we determine an integral expression for the Dunkl kernel, which is the integral kernel of the Dunkl transform, for all dihedral groups. We also determine an integral expression for the intertwining operator in the case of dihedral groups, based on observations valid for all reflection groups. As a special case, we recover the result of Xu (2019) [36]. Crucial in our approach is a systematic use of the link between both integral kernels and the simplex in a suitable high dimensional space.

Ladder relations for a class of matrix valued orthogonal polynomials

AbstractUsing the theory introduced by Casper and Yakimov, we investigate the structure of algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) on, and we derive algebraic and differential relations for these MVOPs. A particular case of importance is that of MVOPs with respect to a matrix weight of the formon the real line, whereis a scalar polynomial of even degree with positive leading coefficient andis a constant matrix.

Reductions of the strict KP hierarchy

Let $$R$$ be a commutative complex algebra and $$ \partial $$ be a $$ \mathbb{C} $$ -linear derivation of $$R$$ such that all powers of $$ \partial $$ are $$R$$ -linearly independent. Let $$R[ \partial ]$$ be the algebra of differential operators in $$ \partial $$ with coefficients in $$R$$ and $$ P{\kern-1.5pt}sd $$ be its extension by the pseudodifferential operators in $$ \partial $$ with coefficients in $$R$$ . In the algebra $$R[ \partial ]$$ , we seek monic differential operators $$ \mathbf{M} _n$$ of order $$n\ge2$$ without a constant term satisfying a system of Lax equations determined by the decomposition of $$ P{\kern-1.5pt}sd $$ into a direct sum of two Lie algebras that lies at the basis of the strict KP hierarchy. Because this set of Lax equations is an analogue for this decomposition of the $$n$$ -KdV hierarchy, we call it the strict $$n$$ -KdV hierarchy. The system has a minimal realization, which allows showing that it has homogeneity properties. Moreover, we show that the system is compatible, i.e., the strict differential parts of the powers of $$M=( \mathbf{M} _n)^{1/n}$$ satisfy zero-curvature conditions, which suffice for obtaining the Lax equations for $$ \mathbf{M} _n$$ and, in particular, for proving that the $$n$$ th root $$M$$ of $$ \mathbf{M} _n$$ is a solution of the strict KP theory if and only if $$ \mathbf{M} _n$$ is a solution of the strict $$n$$ -KdV hierarchy. We characterize the place of solutions of the strict $$n$$ -KdV hierarchy among previously known solutions of the strict KP hierarchy.