Invariant differential operators for non-compact Lie groups: The main su(n, n) cases

In the present paper we continue the project of systematic explicit construction of invariant differential operators. On the example of the non-compact group SO(p, 9 − p) (for p = 5, 6) we give the classification multiplets of indecomposable elementary representations induced from a new (relative to earlier considerations) choice of parabolic subgroup MAN so that the factor-group G/M is an Einstein manifold. This classification includes the data for the relevant invariant differential operators.

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra so∗(10). We use the maximal Heisenberg parabolic subalgebra P=M⊕A⊕N with M=su(3,1)⊕su(2)≅so∗(6)⊕so(3). We give the main and the reduced multiplets of indecomposable elementary representations. This includes the explicit parametrization of the intertwining differential operators between the ERS. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebras so(p,q) (with p+q=10, p≥q≥2) with maximal Heisenberg parabolic subalgebra: P′=M′⊕A′⊕N′, M′=so(p−2,q−2)⊕sl(2,IR), M′C≅MC.

In the present paper we continue the project of systematic classification and construction of invariant differential operators for non-compact semisimple Lie groups. This time we make the stress on one of the main building blocks, namely the Verma modules and the corresponding parabolic subalgebras. In particular, we start the study of the relation between the parabolic subalgebras of real semisimple Lie algebras and of their complexification. Two cases are given in more detail: the conformal algebra of 4D Minkowski space-time and the minimal parabolics of classical real semisimple Lie algebras.

Over the \((1, n)\)-dimensional real superspace, we consider the Lie superalgebra \(\mathcal {K}(n)\) of contact vector fields on \({\mathbb {R}}^{1|n}\). We classify \(\mathfrak {osp}(n|2)\)-invariant linear differential operators acting on the superspaces of weighted densities. This result allows us to compute the first differential cohomology of \(\mathcal {K}(n)\) with coefficients in the superspace of weighted densities, vanishing on the Lie superalgebra \(\mathfrak {osp}(n|2)\). We explicitly give 1-cocycles spanning these cohomology spaces.

Relative BGG sequences; II. BGG machinery and invariant operators

Using our previous results on the systematic construction of invariant differential operators for non-compact semisimple Lie groups we classify the special reduced multiplets and minimal representations in the case of SO(p,q).KeywordsIntertwining Differential OperatorsMain MultipletDiscrete Series RepresentationsIrrepsDynkin LabelsThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A new systematic method to find the relative invariant differentiation operators is developed. We incorporate this new approach with Lie's infinitesimal method to study the general class y′′′ = f(x, y, y′, y′′) under general point equivalence transformations in the generic case.As a result, all third‐order differential invariants, relative and absolute invariant differentiation operators are determined for third‐order ODEs y′′′ = f(x, y, y′, y′′), which are not quadratic in the second‐order derivative. These relative invariant differentiation operators are used to determine the fourth‐order differential invariants and absolute invariant differentiation operators in a degenerate case of interest. As an application, invariant descriptions of all the canonical forms in the complex planewith four infinitesimal symmetries for third‐order ODEs y′′′ = f(x, y, y′, y′′), which are not quadratic in the second‐order derivative, are provided. Copyright © 2015 John Wiley & Sons, Ltd.

Ricci-corrected derivatives and invariant differential operators

Spectral decomposition of invariant differential operators on certain nilpotent homogeneous spaces

Asymptotic Behavior of Spherical Functions on Semisimple Symmetric Spaces

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.

Let G 0 = K p be the Cartan motion group associated with a noncompact semisimple Riemannian symmetric pair (G, K) .Let a be a maximal abelian subspace of p and let p = a + q be the corresponding orthogonal decomposition.A flat horocycle in p is a G 0 -translate of q .A conical distribution on the space 0 of flat horocycles is an eigendistribution of the algebra D( 0 ) of G 0 -invariant differential operators on 0 which is invariant under the left action of the isotropy subgroup of G 0 fixing q .We prove that the space of conical distributions belonging to each generic eigenspace of D( 0 ) is onedimensional, and we classify the set of all conical distributions on 0 when G/K has rank one.We also consider the question of the irreducibility of the natural representation of G 0 on the eigenspaces of D( 0 ) .

In a perturbative approach Einstein-Hilbert gravity is quantized about a flat background. In order to render the model power counting renormalizable, higher order curvature terms are added to the action. They serve as Pauli-Villars type regulators and require an expansion in the number of fields in addition to the standard expansion in the number of loops. Renormalization is then performed within the BPHZL scheme, which provides the action principle to construct the Slavnov-Taylor identity and invariant differential operators. The final physical state space of the Einstein-Hilbert theory is realized via the quartet mechanism of Kugo and Ojima. Renormalization group and Callan-Symanzik equation are derived for the Green functions and, formally, also for the $S$-matrix.

We introduce and study the concept of a variety of dialgebras which is closely related to the concept of a variety of conformal algebras: Each dialgebra of a given variety embeds into an appropriate conformal algebra of the same variety. In particular, the Leibniz algebras are exactly Lie dialgebras, and each Leibniz algebra embeds into a conformal Lie algebra.

We consider new class of classicalr-matrices forD=4 conformal Lie algebra. These r-matrices do satisfy the classical Yang-Baxter equation and as two-tensors belong to the tensor product of Borel subalgebras. In such a way we generalize the lowest order of known nonstandard quantum deformation ofsl(2) to the Lie algebrasl(4)≅so(6). As an exercise we interpret nonstandard deformation ofsl(2) as describing quantumD=1 conformal algebra with fundamental mass parameter. Further we describeD=4 conformal bialgebras with deformation parameters equal to the inverse of fundamental masses. It appears that forD=4 the deformation of the Poincare algebra sector coincides with “null plane” quantum Poincare algebra.