Complex Semisimple Lie Algebra Research Articles

Unrestricted Quantum Moduli Algebras, II: Noetherianity and Simple Fraction Rings at Roots of 1

We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra $\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mathbb{C}(q)$. Moreover, we show that these two properties still hold on $\mathbb{C}\big[q,q^{-1}\big]$ for the integral version of the quantum graph algebra. We also study the specializations $\mathcal{L}_{0,n}^\epsilon$ of the quantum graph algebra at a root of unity $\epsilon$ of odd order, and show that $\mathcal{L}_{0,n}^\epsilon$ and its invariant algebra under the quantum group $U_\epsilon(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute.

Whittaker vectors in singular Whittaker modules

Let g be a complex semisimple Lie algebra with Borel subalgebra b and corresponding nilradical n . We show that singular Whittaker modules M are simple if and only if the space Wh M of Whittaker vectors is 1-dimensional. For arbitrary locally n -finite g -modules V, an immediate corollary is that the dimension of Wh V is bounded by the composition length of V.

Character formulas in category 𝒪_{𝓅}

Let O p \mathcal {O}_p denote the characteristic p > 0 p>0 version of the ordinary category O \mathcal {O} for a semisimple complex Lie algebra. In this paper we give some (formal) character formulas in O p \mathcal {O}_p . First we concentrate on the irreducible characters. Here we give explicit formulas for how to obtain all irreducible characters from the characters of the finitely many restricted simple modules as well as the characters of a small number of infinite dimensional simple modules in O p \mathcal {O}_p with specified highest weights. We next prove a strong linkage principle for Verma modules which allow us to split O p \mathcal {O}_p into a finite direct sum of linkage classes. There are corresponding translation functors and we use these to further cut down the set of irreducible characters needed for determining all others. Then we show that the twisting functors on O \mathcal {O} carry over to twisting functors on O p \mathcal {O}_p , and as an application we prove a character sum formula for Jantzen-type filtrations of Verma modules with antidominant highest weights. Finally, we record formulas relating the characters of the two kinds of tilting modules in O p \mathcal {O}_p .

Dirac cohomology for the BGG category [formula omitted

We study Dirac cohomology HDg,h(M) for modules belonging to category O of a finite dimensional complex semisimple Lie algebra. We start by studying the generalized infinitesimal character decomposition of M⊗S, with S being a spin module of h⊥. As a consequence, “Vogan’s conjecture” holds, and we prove a nonvanishing result for HDg,h(M) while we show that in the case of a Hermitian symmetric pair (g,k) and an irreducible unitary module M∈O, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in M. In the last part, we show that the higher Dirac cohomology and index introduced by Pandžić and Somberg satisfy nice homological properties for M∈O.

Stokes phenomena, Poisson–Lie groups and quantum groups

Let g be a complex semisimple Lie algebra, b±⊂g a pair of opposite Borel subalgebras, and r∈b−⊗b+ the corresponding solution of the classical Yang–Baxter equations. Let G be the simply–connected Poisson–Lie group corresponding to (g,r), H⊂B±⊂G the subgroups with Lie algebras h=b−∩b+ and b±, and G⁎=B+×HB− the Poisson–Lie group dual of G. G–valued Stokes phenomena were used by Boalch [3,4] to give a canonical, analytic linearisation of the Poisson–Lie group structure on G⁎. Ug–valued Stokes phenomena were used by the first author to construct a twist killing the KZ associator, and therefore give a transcendental construction of the Drinfeld–Jimbo quantum group Uħg[23]. In the present paper, we show that the former construction can be obtained as semiclassical limit of the latter. Along the way, we also show that the R–matrix of Uħg is a Stokes matrix for the dynamical KZ equations.

Banach space representations of Drinfeld–Jimbo algebras and their complex-analytic forms

We prove that every nondegenerate Banach space representation of the Drinfeld–Jimbo algebra Uq(g) of a semisimple complex Lie algebra g is finite dimensional when |q|≠1. As a corollary, we find an explicit form of the Arens–Michael envelope of Uq(g), which is similar to that of U(g) obtained by Joseph Taylor in 1970s. In the case when g=sl2, we also consider the representation theory of the corresponding analytic form, the Arens–Michael algebra U˜(sl2)ℏ (with eℏ=q) and show that it is simpler than for Uq(sl2). For example, all irreducible continuous representations of U˜(sl2)ℏ are finite dimensional for every admissible value of the complex parameter ℏ, while Uq(sl2) has a topologically irreducible infinite-dimensional representation when |q|=1 and q is not a root of unity.

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].

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.

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.

ON THE SINGULARITIES OF MISHCHENKO–FOMENKO SYSTEMS

To each complex semisimple Lie algebra \( \mathfrak{g} \) and regular element a ∈ \( \mathfrak{g} \)reg, one associates a Mishchenko–Fomenko subalgebra \( \mathcal{F} \)a ⊆ ℂ[\( \mathfrak{g} \)]. This subalgebra amounts to a completely integrable system on the Poisson variety \( \mathfrak{g} \), and as such has a bifurcation diagram Σa ⊆ Spec(\( \mathcal{F} \)a). We prove that Σa has codimension one in Spec(\( \mathcal{F} \)a) if a ∈ \( \mathfrak{g} \)reg is not nilpotent, and that it has codimension one or two if a ∈ \( \mathfrak{g} \)reg is nilpotent. In the nilpotent case, we show each of the possible codimensions to be achievable. Our results significantly sharpen existing estimates of the codimension of Σa.

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).

Remarks on the derived center of small quantum groups

Let $\mathsf{u}_q(\mathfrak{g})$ be the small quantum group associated with a complex semisimple Lie algebra $\mathfrak{g}$ and a primitive root of unity q, satisfying certain restrictions. We establish the equivalence between three different actions of $\mathfrak{g}$ on the center of $\mathsf{u}_q(\mathfrak{g})$ and on the higher derived center of $\mathsf{u}_q(\mathfrak{g})$. Based on the triviality of this action for $\mathfrak{g} = \mathfrak{sl}_2, \mathfrak{sl}_3, \mathfrak{sl}_4$, we conjecture that, in finite type A, central elements of the small quantum group $\mathsf{u}_q(\mathfrak{sl}_n)$ arise as the restriction of central elements in the big quantum group $\mathsf{U}_q(\mathfrak{sl}_n)$. We also study the role of an ideal $\mathsf{z}_{\mathrm{Hig}}$ known as the Higman ideal in the center of $\mathsf{u}_q(\mathfrak{g})$. We show that it coincides with the intersection of the Harish-Chandra center and its Fourier transform, and compute the dimension of $\mathsf{z}_{\mathrm{Hig}}$ in type A. As an illustration we provide a detailed explicit description of the derived center of $\mathsf{u}_q(\mathfrak{sl}_2)$ and its various symmetries.

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.

Nilpotent orbits and mixed gradings of semisimple Lie algebras

Let σ be an involution of a complex semisimple Lie algebra g and g=g0⊕g1 the related Z2-grading. We study relations between nilpotent G0-orbits in g0 and the respective G-orbits in g. If e∈g0 is nilpotent and {e,h,f}⊂g0 is an sl2-triple, then the semisimple element h yields a Z-grading of g. Our main tool is the combined Z×Z2-grading of g, which is called a mixed grading. We prove, in particular, that if eσ is a regular nilpotent element of g0, then the weighted Dynkin diagram of eσ, D(eσ), has only isolated zeros. It is also shown that if G⋅eσ∩g1≠∅, then the Satake diagram of σ has only isolated black nodes and these black nodes occur among the zeros of D(eσ). Using mixed gradings related to eσ, we define an inner involution σˇ such that σ and σˇ commute. Here we prove that the Satake diagrams for both σˇ and σσˇ have isolated black nodes.

Projective and Whittaker functors on category [formula omitted

We show that the Whittaker functor on a regular block of the BGG-category O of a semisimple complex Lie algebra can be obtained by composing a translation to the wall functor with Soergel and Miličić's equivalence between the category of Whittaker modules and a singular block of O. We show that the Whittaker functor is a quotient functor that commutes with all projective functors and endomorphisms between them.

Deformations of adjoint orbits for semisimple Lie algebras and Lagrangian submanifolds

We give a coadjoint orbit's diffeomorphic deformation between the classical semisimple case and the semi-direct product given by a Cartan decomposition. The two structures admit the Hermitian symplectic form defined in a semisimple complex Lie algebra. We provide some applications such as the constructions of Lagrangian submanifolds.

The Full Symmetric Toda Flow and Intersections of Bruhat Cells

In this short note we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements $w$, $w'$ in the Weyl group $W(\mathfrak g)$, the corresponding real Bruhat cell $X_w$ intersects with the dual Bruhat cell $Y_{w'}$ iff $w\prec w'$ in the Bruhat order on $W(\mathfrak g)$. Here $\mathfrak g$ is a normal real form of a semisimple complex Lie algebra $\mathfrak g_\mathbb C$. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations.

Contravariant forms on Whittaker modules

Let g \mathfrak {g} be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g \mathfrak {g} -modules Y ( χ , η ) Y(\chi , \eta ) introduced by Kostant. We prove that the set of all contravariant forms on Y ( χ , η ) Y(\chi , \eta ) forms a vector space whose dimension is given by the cardinality of the Weyl group of g \mathfrak {g} . We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M ( χ , η ) M(\chi , \eta ) introduced by McDowell.

Centrally generated primitive ideals of [formula omitted] for exceptional types

Let g be a complex semisimple Lie algebra, let b be a Borel subalgebra of g, let n be the nilradical of b, and let U(n) be the universal enveloping algebra of n. We study primitive ideals of U(n). Almost all primitive ideals are centrally generated, i.e., are generated by their intersections with the center Z(n) of U(n). We present an explicit characterization of the centrally generated primitive ideals of U(n) in terms of the Dixmier map and the Kostant cascade in the case when g is a simple algebra of exceptional type. (For classical simple Lie algebras, a similar characterization was obtained by Ivan Penkov and the first author.) As a corollary, we establish a classification of centrally generated primitive ideals of U(n) for an arbitrary semisimple algebra g.

Quasitriangular coideal subalgebras of Uq(g) in terms of generalized Satake diagrams

Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. According to G. Letzter, S. Kolb and M. Balagovi\'c the fixed-point subalgebra $\mathfrak{k} = \mathfrak{g}^\theta$ has a quantum counterpart $B$, a coideal subalgebra of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ possessing a universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of Satake diagrams. In the present work we extend this construction to generalized Satake diagrams, combinatorial data first considered by A. Heck. A generalized Satake diagram naturally defines a semisimple automorphism $\theta$ of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. It also defines a subalgebra $\mathfrak{k}\subset \mathfrak{g}$ satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$, but not necessarily a fixed-point subalgebra. The subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a universal K-matrix in the sense of Kolb and Balagovi\'c. We conjecture that all such coideal subalgebras of $U_q(\mathfrak{g})$ arise from generalized Satake diagrams in this way.