Maximal Parabolic Subalgebras Research Articles

Heisenberg Parabolic Subgroup of SO∗(10) and 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.

Shuffling functors and spherical twists on [formula omitted

For a semisimple complex Lie algebra g, the BGG category O is of particular interest in representation theory. It is known that Irving's shuffling functors Shw, indexed by elements w∈W of the Weyl group, induce an action of the braid group BW associated to W on the derived categories Db(Oλ) of blocks of O.We show that for maximal parabolic subalgebras p of sln corresponding to the parabolic subgroup Wp=Sn−1×S1 of Sn, the derived shuffling functors LShsi are instances of Seidel and Thomas' spherical twist functors. Namely, we show that certain parabolic indecomposable projectives Pp(w) are spherical objects, and the associated twist functors are naturally isomorphic to LShw[1] as auto-equivalences of Db(Op).We give an overview of the main properties of the BGG category O, the construction of shuffling and spherical twist functors, and give some examples how to determine images of both. To this end, we employ the equivalence of blocks of O and the module categories of certain path algebras.

Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems

The notion of hidden symmetry algebra used in the context of exactly solvable systems (typically a non semisimple Lie algebra) is re-examined from the purely algebraic way, analyzing subspaces of commuting polynomials that generate finite-dimensional quadratic algebras. By construction, these algebras do not depend on the choice of realizations by vector fields of the underlying Lie algebra, allowing to propose a new approach to analyze polynomial algebras as those subspaces in an enveloping algebra that commute with a given algebraic Hamiltonian. These polynomial algebras play an important role in context of superintegrability, but are still poorly understood from an algebraic point of view. Among the main results, we present finitely generated quadratic algebras of dimensions 4, 5 and 6, as well as cubic algebras of dimensions 3 and 5, and various Abelian algebras, all of dimension 3. Based on the observation how superintegrability is associated with exact solvability, we propose a procedure that connects the underlying Lie algebra with algebraic integrals of motion. As the integrals constructed in such way are now independent on the realization, alternative choices of realizations can provide new explicit models with the same symmetry algebra. In this paper, we consider examples of such equivalent Hamiltonians in terms of differential operators for the three cases and connected to the underlying Lie algebra gl(2,R)⋉R2⊕T1 as well as to the maximal parabolic subalgebra of gl(3,R). We also point out differences between the enveloping algebra of Lie algebras and the enveloping algebra of the related differential operators realization.

Homogeneous Symplectic 4-Manifolds and Finite Dimensional Lie Algebras of Symplectic Vector Fields on the Symplectic 4-Space

We classify the finite type (in the sense of E. Cartan theory of prolongations) subalgebras $\mathfrak{h}\subset\mathfrak{sp}(V)$, where $V$ is the symplectic 4-dimensional space, and show that they satisfy $\mathfrak{h}^{(k)}=0$ for all $k>0$. Using this result, we reduce the problem of classification of graded transitive finite-dimensional Lie algebras $\mathfrak{g}$ of symplectic vector fields on $V$ to the description of graded transitive finite-dimensional subalgebras of the full prolongations $\mathfrak{p}_1^{(\infty)}$ and $\mathfrak{p}_2^{(\infty)}$, where $\mathfrak{p}_1$ and $\mathfrak{p}_2$ are the maximal parabolic subalgebras of $\mathfrak{sp}(V)$. We then classify all such $\mathfrak{g}\subset\mathfrak{p}_i^{(\infty)}$, $i=1,2$, under some assumptions and describe the associated homogeneous symplectic 4-manifolds $(M=G/K,\omega)$. We prove that any reductive homogeneous symplectic manifold (of any dimension) admits an invariant torsion free symplectic connection, i.e., it is a homogeneous Fedosov manifold, and give conditions for uniqueness of the Fedosov structure. Finally, we show that any nilpotent symplectic Lie group (of any dimension) admits a natural invariant Fedosov structure which is Ricci-flat.

Blow-ups and infinitesimal automorphisms of CR-manifolds

For a real-analytic connected CR-hypersurface M of CR-dimension ngeqslant 1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra mathfrak {s}={mathfrak {s}}(M): (i) either dim {mathfrak {s}}=n^2+4n+3 and M is spherical everywhere; (ii) or dim {mathfrak {s}}leqslant n^2+2n+2+delta _{2,n} and in the case of equality M is spherical and has fixed signature of the Levi form in the complement to its Levi-degeneracy locus. A version of this result is proved for the Lie group of global automorphisms of M. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry.

Jordan algebraic interpretation of maximal parabolic subalgebras: exceptional Lie algebras

With this paper we start a programme aiming at connecting two vast scientific areas: Jordan algebras and representation theory. Within representation theory, we focus on non-compact, real forms of semisimple Lie algebras and groups as well as on the modern theory of their induced representations, in which a central role is played by the parabolic subalgebras and subgroups. The aim of the present paper and its sequels is to present a Jordan algebraic interpretations of maximal parabolic subalgebras; in this first paper, we confine ourselves to maximal parabolic subalgebras of the non-compact real forms of finite-dimensional exceptional Lie algebras, in particular focussing on Jordan algebras of rank 2 and 3.

On the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras

In this article, we explicitly list all reducible scalar-type generalized Verma modules for all maximal parabolic subalgebras of the simple complex Lie algebras.

Gelfand-Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules

The Gelfand-Kirillov dimension is an invariant which can measure the size of infinite-dimensional algebraic structures. In this article, we show that it can also measure the reducibility of scalar generalized Verma modules. In particular, we use it to determine the reducibility of scalar generalized Verma modules associated with maximal parabolic subalgebras in the Hermitian symmetric case.

Subprime solutions of the classical Yang–Baxter equation

We introduce a new family of classical r-matrices for the Lie algebra sln that lies in the Zariski boundary of the Belavin–Drinfeld space M of quasi-triangular solutions to the classical Yang–Baxter equation. In this setting M is a finite disjoint union of components; exactly ϕ(n) of these components are SLn-orbits of single points. These points are the generalized Cremmer–Gervais r-matrices ri,n which are naturally indexed by pairs of positive coprime integers, i and n, with i<n. A conjecture of Gerstenhaber and Giaquinto states that the boundaries of the Cremmer–Gervais components contain r-matrices having maximal parabolic subalgebras pi,n⊆sln as carriers. We prove this conjecture in the cases when n≡±1 (mod i). The subprime linear functionals f∈pi,n⁎ and the corresponding principal elements H∈pi,n play important roles in our proof. Since the subprime functionals are Frobenius precisely in the cases when n≡±1 (mod i), this partly explains our need to require these conditions on i and n. We conclude with a proof of the GG boundary conjecture in an unrelated case, namely when (i,n)=(5,12), where the subprime functional is no longer a Frobenius functional.

Local Type I Metrics with Holonomy in G2*

By [arXiv:1604.00528], a list of possible holonomy algebras for pseudo-Riemannian manifolds with an indecomposable torsion free ${\rm G}_{2}^*$-structure is known. Here indecomposability means that the standard representation of the algebra on ${\mathbb R}^{4,3}$ does not leave invariant any proper non-degenerate subspace. The dimension of the socle of this representation is called the type of the Lie algebra. It is equal to one, two or three. In the present paper, we use Cartan's theory of exterior differential systems to show that all Lie algebras of Type I from the list in [arXiv:1604.00528] can indeed be realised as the holonomy of a local metric. All these Lie algebras are contained in the maximal parabolic subalgebra $\mathfrak p_1$ that stabilises one isotropic line of ${\mathbb R}^{4,3}$. In particular, we realise $\mathfrak p_1$ by a local metric.

Borel–de Siebenthal pairs, global Weyl modules and Stanley–Reisner rings

We develop the theory of integrable representations for an arbitrary standard maximal parabolic subalgebra of an affine Lie algebra. We see that such subalgebras can be thought of as arising in a natural way from a Borel–de Siebenthal pair of semisimple Lie algebras. We see that although there are similarities with the representation theory of the standard maximal parabolic subalgebra there are also very interesting and non-trivial differences; including the fact that there are examples of non-trivial global Weyl modules which are irreducible and finite-dimensional. We also give a presentation of the endomorphism ring of the global Weyl module; although these are no longer polynomial algebras we see that for certain parabolics these algebras are Stanley–Reisner rings which are both Koszul and Cohen–Macaulay.

Quantization and injective submodules of differential operator modules

The Lie algebra of vector fields on Rm acts naturally on the spaces of differential operators between tensor field modules. Its projective subalgebra is isomorphic to slm+1, and its affine subalgebra is a maximal parabolic subalgebra of the projective subalgebra with Levi factor glm. We prove two results. First, we realize explicitly all injective objects of the parabolic category Oglm(slm+1) of glm-finite slm+1-modules, as submodules of differential operator modules. Second, we study projective quantizations of differential operator modules, i.e.,slm+1-invariant splittings of their order filtrations. In the case of modules of differential operators from a tensor density module to an arbitrary tensor field module, we determine when there exists a unique projective quantization, when there exists no projective quantization, and when there exist multiple projective quantizations.

Generalised Jantzen filtration of exceptional Lie superalgebras

Let g be an exceptional Lie superalgebra, and let p be the maximal parabolic subalgebra which contains the distinguished Borel subalgebra and has a purely even Levi subalgebra. For any parabolic Verma module in the parabolic category Op, it is shown that the Jantzen filtration is the unique Loewy filtration, and the decomposition numbers of the layers of the filtration are determined by the coefficients of inverse Kazhdan–Lusztig polynomials. An explicit description of the submodule lattices of the parabolic Verma modules is given, and formulae for characters and dimensions of the finite-dimensional simple modules are obtained.

SLICES FOR MAXIMAL PARABOLIC SUBALGEBRAS OF A SEMISIMPLE LIE ALGEBRA

Let p denote a maximal (truncated) parabolic subalgebra of a simple Lie algebra $$ \mathfrak{g} $$ . It is known that the Poisson centre Y ( $$ \mathfrak{p} $$ ) is a polynomial algebra in many cases. We construct a slice for the coadjoint action of p, thus extending a theorem of Kostant. The role of the principal $$ \mathfrak{s}\mathfrak{l} $$ 2-triple is played by an adapted pair.

A Steinberg type decomposition theorem for higher level Demazure modules

We study Demazure modules which occur in a level ℓ irreducible integrable representation of an affine Lie algebra. We also assume that they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie algebra. We prove that such a module is isomorphic to the fusion product of ‘‘prime’’ Demazure modules, where the prime factors are indexed by dominant integral weights which are either a multiple of ℓ or take value less than ℓ on all simple coroots. Our proof depends on a technical result which we prove in all the classical cases and G2. Calculations with mathematica show that this result is correct for small values of the level. Using our result, we show that there exist generalizations of Q-systems to pairs of weights where one of the weights is not necessarily rectangular and is of a different level. Our results also allow us to compare the multiplicities of an irreducible representation occurring in the tensor product of certain pairs of irreducible representations, i.e., we establish a version of Schur positivity for such pairs of irreducible modules for a simple Lie algebra.

DEMAZURE MODULES OF LEVEL TWO AND PRIME REPRESENTATIONS OF QUANTUM AFFINE

We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in Hernandez and Leclerc (Duke Math. J.154 (2010), 265–341; Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics &amp; Statitics, Volume 40, pp. 175–193 (2013)), in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting) to $\mathfrak{sl}_{n+1}[t]$-stable prime Demazure modules in level-two integrable highest-weight representations of the classical affine Lie algebra. It was proved in Chari et al. (arXiv:1408.4090) that a stable Demazure module is isomorphic to the fusion product of stable prime Demazure modules. Our next result proves that such a fusion product is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine $\mathfrak{sl}_{n+1}$.

Invariant differential operators for non-compact Lie groups: the SO* (12) case

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra so* (12). We give the main multiplets of indecomposable elementary representations. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebra so(6, 6) with suitably chosen maximal parabolic subalgebra.

Invariant differential operators for non-compact Lie groups: the reduced SU(3,3) multiplets

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras $su(n,n)$. Earlier were given the main multiplets of indecomposable elementary representations for $n\leq 4$, and the reduced ones for $n=2$. Here we give all reduced multiplets containing physically relevant representations including the minimal ones for the algebra $su(3,3)$. Due to the recently established parabolic relations the results are valid also for the algebra $sl(6,\mathbb{R})$ with suitably chosen maximal parabolic subalgebra.

Multiplet classification for SU(n,n)

In the present paper we review our project of systematic construction of invariant differential operators on the example of the non-compact algebras su(n,n) for n = 2, 3,4. We give explicitly the main multiplets of indecomposable elementary representations and some reduced multiplets. We give explicitly the minimal representations. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebras sl(2n, R) and when n = 2k for the algebras su* (4k) with suitably chosen maximal parabolic subalgebras.

Weyl Modules for the Hyperspecial Current Algebra

We develop the theory of global and local Weyl modules for the hyperspecial maximal parabolic subalgebra of type $A_{2n}^{(2)}$. We prove that the dimension of a local Weyl module depends only on its highest weight, thus establishing a freeness result for global Weyl modules. Furthermore, we show that the graded local Weyl modules are level one Demazure modules for the corresponding affine Lie algebra. In the last section we derive the same results for the special maximal parabolic subalgebras of the twisted affine Lie algebras not of type $A_{2n}^{(2)}$.