Complex Lie Algebra Research Articles

Unipotent ideals for spin and exceptional groups

In the monograph [27], we define the notion of a unipotent representation of a complex reductive group. The representations we define include, as a proper subset, all special unipotent representations in the sense of [4] and form the (conjectural) building blocks of the unitary dual. In [27] we provide combinatorial formulas for the infinitesimal characters of all unipotent representations of linear classical groups. In this paper, we establish analogous formulas for spin and exceptional groups, thus completing the determination of the infinitesimal characters of all unipotent ideals. Using these formulas, we prove an old conjecture of Vogan: all unipotent ideals are maximal. For G a real reductive Lie group (not necessarily complex), we introduce the notion of a unipotent representation attached to a rigid nilpotent orbit (in the complexified Lie algebra of G). Like their complex group counterparts, these representations form the (conjectural) building blocks of the unitary dual. Using the atlas software (and the work of [2]) we show that if G is a real form of a simple group of exceptional type, all such representations are unitary.

WHEN IS A COMPLETION OF THE UNIVERSAL ENVELOPING ALGEBRA A BANACH PI-ALGEBRA?

AbstractWe prove that a Banach algebra B that is a completion of the universal enveloping algebra of a finite-dimensional complex Lie algebra $\mathfrak {g}$ satisfies a polynomial identity if and only if the nilpotent radical $\mathfrak {n}$ of $\mathfrak {g}$ is associatively nilpotent in B. Furthermore, this holds if and only if a certain polynomial growth condition is satisfied on $\mathfrak {n}$ .

Twisting functors and Gelfand–Tsetlin modules over semisimple Lie algebras

We associate to an arbitrary positive root [Formula: see text] of a complex semisimple finite-dimensional Lie algebra [Formula: see text] a twisting endofunctor [Formula: see text] of the category of [Formula: see text]-modules. We apply this functor to generalized Verma modules in the category [Formula: see text] and construct a family of [Formula: see text]-Gelfand–Tsetlin modules with finite [Formula: see text]-multiplicities, where [Formula: see text] is a commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] generated by a Cartan subalgebra of [Formula: see text] and by the Casimir element of the [Formula: see text]-subalgebra corresponding to the root [Formula: see text]. This covers classical results of Andersen and Stroppel when [Formula: see text] is a simple root and previous results of the authors in the case when [Formula: see text] is a complex simple Lie algebra and [Formula: see text] is the maximal root of [Formula: see text]. The significance of constructed modules is that they are Gelfand–Tsetlin modules with respect to any commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] containing [Formula: see text]. Using the Beilinson–Bernstein correspondence we give a geometric realization of these modules together with their explicit description. We also identify a tensor subcategory of the category of [Formula: see text]-Gelfand–Tsetlin modules which contains constructed modules as well as the category [Formula: see text].

Birational geometry for the covering of a nilpotent orbit closure

Let \({\mathfrak {g}}\) be a complex classical simple Lie algebra and let O be a nilpotent orbit of \({\mathfrak {g}}\). The fundamental group \(\pi _1(O)\) is finite. Take the universal covering \(\pi ^0: X^0 \rightarrow O\). Then \(\pi ^0\) extends to a finite cover \(\pi : X \rightarrow {\bar{O}}\). By using the Kirillov–Kostant form \(\omega _{KK}\) on O, the normal affine variety X becomes a conical symplectic variety. In this article we give an explicit construction of a Q-factorial terminalization of X.

The Representation Theory of Brauer Categories I: Triangular Categories

This is the first in a series of papers in which we study representations of the Brauer category and its allies. We define a general notion of triangular category that abstracts key properties of the triangular decomposition of a semisimple complex Lie algebra, and develop a highest weight theory for them. We show that the Brauer category, the partition category, and a number of related diagram categories admit this structure.

Fully commutative elements and spherical nilpotent orbits

Let g be a complex simple Lie algebra, with fixed Borel subalgebra b⊂g 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 connections with the spherical nilpotent orbits in g. If g is not of type G2, it is shown that an element w∈W is fully commutative if and only if the subalgebra of b determined by the inversions of w lies in the closure of a spherical nilpotent orbit. A similar characterization is also given for the ad-nilpotent ideals of b, which are parametrized by suitable elements in the affine Weyl group of g thanks to the work of Cellini and Papi.

Contravariant pairings between standard Whittaker modules and Verma modules

We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category O to the Miličić–Soergel category N. We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings. We show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard Whittaker modules. We show that costandard modules give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the associated flag variety, which are defined using holonomic duality of D-modules. We prove that with these costandard modules, blocks of category N have the structure of highest weight categories and we establish a BGG reciprocity theorem for N.

Commutative Polarisations and the Kostant Cascade

Let \(\mathfrak {g}\) be a complex simple Lie algebra. We classify the parabolic subalgebras \(\mathfrak {p}\) of \(\mathfrak {g}\) such that the nilradical of \(\mathfrak {p}\) has a commutative polarisation. The answer is given in terms of the Kostant cascade. It requires also the notion of an optimal nilradical and some properties of abelian ideals in a Borel subalgebra of \(\mathfrak {g}\).

Combinatorics of the q-characters of Hernandez-Leclerc modules

Let g be a complex simple Lie algebra and Uq(gˆ) be the corresponding untwisted quantum affine algebra. In this paper, we give a path description of the q-characters of Hernandez-Leclerc modules, show that up to spectral parameter shift, the equivalent classes of Hernandez-Leclerc modules in the Grothendieck ring of the category of finite-dimensional Uq(sln+1ˆ)-modules are cluster variables in the cluster algebra introduced by Hernandez and Leclerc, and finally prove that the geometric q-character formula conjecture is true for Hernandez-Leclerc modules.

Kazhdan–Lusztig conjecture via zastava spaces

Abstract We deduce the Kazhdan–Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category 𝒪 {\mathcal{O}} from the equivariant geometric Satake correspondence and the analysis of torus fixed points in zastava spaces. We make similar speculations for the affine Lie algebras and 𝒲 {\mathscr{W}} -algebras.

Classification of real trivectors in dimension nine

In this paper we classify real trivectors in dimension 9. The corresponding classification over the field C of complex numbers was obtained by Vinberg and Elashvili in 1978. One of the main tools used for their classification was the construction of the representation of SL(9,C) on the space of complex trivectors of C9 as a theta-representation corresponding to a Z/3Z-grading of the simple complex Lie algebra of type E8. This divides the trivectors into three groups: nilpotent, semisimple, and mixed trivectors. Our classification follows the same pattern. We use Galois cohomology, first and second, to obtain the classification over R.

A commutative algebra approach to multiplicative Hom-Lie algebras

Let be a finite-dimensional complex Lie algebra and be the affine variety of all multiplicative Hom-Lie algebras on . We use a method of computational ideal theory to describe , showing that consists of two 1-dimensional and one 3-dimensional irreducible components and for . We construct a new family of multiplicative Hom-Lie algebras on the Heisenberg Lie algebra and characterize the affine varieties and . We also study the derivation algebra of a multiplicative Hom-Lie algebra D on and, under some hypotheses on D, we prove that the Hilbert series is a rational function.

Non-semisimple Tensor Categories and Their Semisimplification

Finite tensor categories are, despite their many applications and great interest, notoriously hard to classify. Among them, the semisimple ones (called fusion categories) have been intensively studied. Those with non-integral dimensions form a remarkable class. Already more than 20 years ago, tilting modules have been proposed as a source of such fusion categories. In this way, the Verlinde categories associated to the pair of a simple complex Lie algebra \mathfrak{g} and an integer level k have been recovered in a purely algebraic framework–called semisimplification of tensor categories. Recently efforts to understand how to go beyond these examples emerged. This mini-workshop aims at bringing together experts from various branches of representation theory and topological field theory to deepen our understanding of finite tensor categories and to compare new ways to understand semisimplification.

Birational geometry for the covering of a nilpotent orbit closure II

Let O be a nilpotent orbit of a complex semisimple Lie algebra g and let π:X→O¯ be the finite covering associated with the universal covering of O. In the previous article [14] we have explicitly constructed a Q-factorial terminalization X˜ of X when g is classical. In this article we count how many non-isomorphic Q-factorial terminalizations X has. We construct the universal Poisson deformation of X˜ over H2(X˜,C) and look at the action of the Weyl group W(X) on H2(X˜,C). The main result is an explicit geometric description of W(X).

The Lie algebra $\it{\mathcal{K}_5}$: three variable models and their connection with special functions

We construct the relation between the 5-dimensional complex Lie algebra $\it{\mathcal{K}_5}$ and certain special functions. In this paper, we discuss the new three variable models of the irreducible representations of the Lie algebra $\it{\mathcal{K}_5}$. We use a two-fold Euler type integral transformation to obtain the new transformed models and hence various new recurrence relations in terms of Kampe de Feriet function are derived.

Crystal pop-stack sorting and type [formula omitted] crystal lattices

Given a complex simple Lie algebra g and a dominant weight λ, let Bλ be the crystal poset associated to the irreducible representation of g with highest weight λ. In the first part of the article, we introduce the crystal pop-stack sorting operatorPop◊:Bλ→Bλ, a noninvertible operator whose definition extends that of the pop-stack sorting map and the recently-introduced Coxeter pop-stack sorting operators. Every forward orbit of Pop◊ contains the minimal element of Bλ, which is fixed by Pop◊. We prove that the maximum size of a forward orbit of Pop◊ is the Coxeter number of the Weyl group of g. In the second part of the article, we characterize exactly when a type A crystal is a lattice.

Automorphic Lie Algebras and Modular Forms

AbstractWe introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma $ be a finite index subgroup of $\textrm {SL}(2,\mathbb Z)$ with an action on a complex simple Lie algebra $\mathfrak g$, which can be extended to $\textrm {SL}(2,{\mathbb {C}})$. We show that the Lie algebra of the corresponding $\mathfrak {g}$-valued modular forms is isomorphic to the extension of $\mathfrak {g}$ over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups $\Gamma (N), \, N\leq 6$, is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras.

BGG categories in prime characteristics

Let \({\mathfrak {g}}\) be a simple complex Lie algebra. In this paper we study the BGG category \({\mathcal {O}}_q\) for the quantum group \(U_q({\mathfrak {g}})\) with q being a root of unity in a field K of characteristic \(p >0\). We first consider the simple modules in \({\mathcal {O}}_q\) and prove a Steinberg tensor product theorem for them. This result reduces the problem of determining the corresponding irreducible characters to the same problem for a finite subset of finite dimensional simple modules. Then we investigate more closely the Verma modules in \({\mathcal {O}}_q\). Except for the special Verma module, which has highest weight \(-\rho \), they all have infinite length. Nevertheless, we show that each Verma module has a certain finite filtration with an associated strong linkage principle. The special Verma module turns out to be both simple and projective/injective. This leads to a family of projective modules in \({\mathcal {O}}_q\), which are also tilting modules. We prove a reciprocity law, which gives a precise relation between the corresponding family of characters for indecomposable tilting modules and the family of characters of simple modules with antidominant highest weights. All these results are of particular interest when \(q = 1\), and we have paid special attention to this case.

Generalized punctual Hilbert schemes and 𝔤-complex structures

We define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple properties with Hitchin components, and which are conjecturally homeomorphic to them. For simple complex Lie algebras, this generalizes the higher complex structure. For real Lie algebras, this should give an alternative description of the Hitchin–Kostant–Rallis section.

A note on branching of V(ρ)

Let g be a complex simple Lie algebra and let g0 be the sub-algebra fixed by a diagram automorphism of g. Let G be the complex, simply-connected, simple algebraic group with Lie algebra g, and let G0 be the connected subgroup of G with Lie algebra g0. Let ρ be the half sum of positive roots of g. In this article, we give a necessary and sufficient condition for a highest weight g0-representation V0(dμ) to occur in the representation resg0V(dρ), for any saturation factor d of the pair (G0,G).