Complex Semisimple Lie Algebra Research Articles

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.

The Bogomolov multiplier of Lie algebras

In this paper, we extend the notion of the Bogomolov multipliers and the CP-extensions to Lie algebras. Then, we compute the Bogomolov multipliers for Abelian, Heisenberg and nilpotent Lie algebras of class at most 6. Finally, we compute the Bogomolov multipliers of complex simple and semisimple Lie algebras.

Derived invariants of the fixed ring of enveloping algebras of semisimple Lie algebras

Let \({\mathfrak {g}}\) be a semisimple complex Lie algebra, and let W be a finite subgroup of \({\mathbb {C}}\)-algebra automorphisms of the enveloping algebra \(U({\mathfrak {g}})\). We show that the derived category of \(U({\mathfrak {g}})^W\)-modules determines isomorphism classes of both \({\mathfrak {g}}\) and W. Our proofs are based on the geometry of the Zassenhaus variety of the reduction modulo \(p\gg 0\) of \({\mathfrak {g}}.\) Specifically, we use non-existence of certain étale coverings of its smooth locus.

GEOMETRY OF REGULAR HESSENBERG VARIETIES

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. For a regular element $x$ in $\mathfrak{g}$ and a Hessenberg space $H\subseteq \mathfrak{g}$, we consider a regular Hessenberg variety $X(x,H)$ in the flag variety associated with $\mathfrak{g}$. We take a Hessenberg space so that $X(x,H)$ is irreducible, and show that the higher cohomology groups of the structure sheaf of $X(x,H)$ vanish. We also study the flat family of regular Hessenberg varieties, and prove that the scheme-theoretic fibers over the closed points are reduced. We include applications of these results as well.

Path developments and tail asymptotics of signature for pure rough paths

Solutions to linear controlled differential equations can be expressed in terms of global iterated path integrals along the driving path. This collection of iterated integrals encodes essentially all information about the underlying path. While upper bounds for iterated path integrals are well known, lower bounds are much less understood, and it is known only relatively recently that some types of asymptotics for the n-th order iterated integral can be used to recover some intrinsic quantitative properties of the path, such as the length for C1 paths.In the present paper, we investigate the simplest type of rough paths (the rough path analogue of line segments), and establish uniform upper and lower estimates for the tail asymptotics of iterated integrals in terms of the local variation of the underlying path. Our methodology, which we believe is new for this problem, involves developing paths into complex semisimple Lie algebras and using the associated representation theory to study spectral properties of Lie polynomials under the Lie algebraic development.

Weights, Weyl-Equivariant Maps and a Rank Conjecture

In this note, given a pair where is a complex semisimple Lie algebra and is a dominant integral weight of where is the real span of the coroots inside a fixed Cartan subalgebra, we associate an SU(2) and Weyl equivariant smooth map where is the configuration space of regular triples in and m, n depend on the initial data We conjecture that, for any the rank of is at least the rank of a collinear configuration in X (collinear when viewed as an ordered r-tuple of points in with r being the rank of ). A stronger conjecture is also made using the singular values of a matrix representing This work is a generalization of the Atiyah-Sutcliffe problem to a Lie-theoretic setting.

A Kazhdan-Lusztig Algorithm for Whittaker Modules

We study a category of Whittaker modules over a complex semisimple Lie algebra by realizing it as a category of twisted $\mathcal {D}$ -modules on the associated flag variety using Beilinson–Bernstein localization. The main result of this paper is the development of a geometric algorithm for computing the composition multiplicities of standard Whittaker modules. This algorithm establishes that these multiplicities are determined by a collection of polynomials we refer to as Whittaker Kazhdan–Lusztig polynomials. In the case of trivial nilpotent character, this algorithm specializes to the usual algorithm for computing multiplicities of composition factors of Verma modules using Kazhdan–Lusztig polynomials.

Kostant–Toda lattices and the universal centralizer

To each complex semisimple Lie algebra g decorated with appropriate data, one may associate two completely integrable systems. One is the well-studied Kostant–Toda lattice, while the second is an integrable system defined on the universal centralizer Zg of g. These systems are similar in that each exploits and closely reflects the invariant theory of g, as developed by Chevalley, Kostant, and others. One also has Kostant’s description of level sets in the Kostant–Toda lattice, which turns out to suggest deeper similarities between the two integrable systems in question. Some more recent works allude to a strong connection between the two systems, e.g. Bezrukavnikov et al. (2005), Teleman (2014, 2018).We study relationships between the two aforementioned integrable systems, partly to understand and contextualize the similarities mentioned above. Our main result is a canonical open embedding of a flow-invariant open dense subset of the Kostant–Toda lattice into Zg. Secondary results include some qualitative features of the integrable system on Zg.

On the Fibres of Mishchenko-Fomenko Systems

This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra \mathfrak{g} . Their theory associates a maximal Poisson-commutative subalgebra of \mathbb{C}[\mathfrak{g}] to each regular element a\in\mathfrak{g} , and one can assemble free generators of this subalgebra into a moment map F_a:\mathfrak{g}\rightarrow\mathbb{C}^b . This leads one to pose basic structural questions about F_a and its fibres, e.g. questions concerning the singular points and irreducible components of such fibres. We examine the structure of fibres in Mishchenko-Fomenko systems, building on the foundation laid by Bolsinov, Charbonnel-Moreau, Moreau, and others. This includes proving that the critical values of F_a have codimension 1 or 2 in \mathbb{C}^b , and that each codimension is achievable in examples. Our results on singularities make use of a subalgebra \mathfrak{b}^a\subseteq\mathfrak{g} , defined to be the intersection of all Borel subalgebras of \mathfrak{g} containing a . In the case of a non-nilpotent a\in\mathfrak{g}_{\mathrm{reg}} and an element x\in\mathfrak{b}^a , we prove the following: x+[\mathfrak{b}^a,\mathfrak{b}^a] lies in the singular locus of F_a^{-1}(F_a(x)) , and the fibres through points in \mathfrak{b}^a form a \text{rank}(\mathfrak{g}) -dimensional family of singular fibres. We next consider the irreducible components of our fibres, giving a systematic way to construct many components via Mishchenko-Fomenko systems on Levi subalgebras \mathfrak{l}\subseteq\mathfrak{g} . In addition, we obtain concrete results on irreducible components that do not arise from the aforementioned construction. Our final main result is a recursive formula for the number of irreducible components in F_a^{-1}(0) , and it generalizes a result of Charbonnel-Moreau. Illustrative examples are included at the end of this paper.

Braided module categories via quantum symmetric pairs

Let ${\mathfrak g}$ be a finite dimensional complex semisimple Lie algebra. The finite dimensional representations of the quantized enveloping algebra $U_q({\mathfrak g})$ form a braided monoidal category $O_{int}$. We show that the category of finite dimensional representations of a quantum symmetric pair coideal subalgebra $B_{c,s}$ of $U_q({\mathfrak g})$ is a braided module category over an equivariantization of $O_{int}$. The braiding for $B_{c,s}$ is realized by a universal K-matrix which lies in a completion of $B_{c,s}\otimes U_q({\mathfrak g})$. We apply these results to describe a distinguished basis of the center of $B_{c,s}$.

AN EXPLICIT ISOMORPHISM BETWEEN QUANTUM AND CLASSICAL $$ \mathfrak{s}{\mathfrak{l}}_n $$

Let \( \mathfrak{g} \) be a complex semisimple Lie algebra. Although the quantum group \( {U}_{\hslash \mathfrak{g}} \) is known to be isomorphic, as an algebra, to the undeformed enveloping algebra \( U\mathfrak{g}\left\llbracket \hslash \right\rrbracket \), no such isomorphism is known when \( \mathfrak{g}\ne \mathfrak{s}{\mathfrak{l}}_2 \). In this paper, we construct an explicit isomorphism for \( \mathfrak{g}=\mathfrak{s}{\mathfrak{l}}_n \), for every n ≥ 2, which preserves the standard flag of type A. We conjecture that this isomorphism quantizes the Poisson diffeomorphism of Alekseev and Meinrenken [2].

Value sets of folding polynomials over finite fields

Let $k$ be a positive integer that is relatively prime to the order of the Weyl group of a semisimple complex Lie algebra $\mf{g}$. We find the cardinality of the value sets of the folding polynomials $P_\mf{g}^k(\mb{x}) \in \Z[\mb{x}]$ of arbitrary rank $n \geq 1$, over finite fields. We achieve this by using a characterization of their fixed points in terms of exponential sums.

THE STRUCTURE OF Q-W ALGEBRAS

We suggest two explicit descriptions of the Poisson q-W algebras which are Poisson algebras of regular functions on certain algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra mathfrak{g} . To obtain the first description we introduce certain projection operators which are analogous to the quasi-classical versions of the so-called Zhelobenko and extremal projection operators. As a byproduct we obtain some new formulas for natural coordinates on Bruhat cells in algebraic groups.

Serre type relations for complex semisimple Lie algebras associated to positive definite quasi-Cartan matrices

Cartan matrices play an important role in the classification of complex semi-simple Lie algebras. The well-known Serre's Theorem states that every finite dimensional complex semisimple Lie algebra g can be constructed from a Cartan matrix A by using generators and relations. We generalize Serre's Theorem by associating to each positive definite quasi-Cartan matrix a complex semi-simple Lie algebra, and we prove that two positive definite quasi-Cartan matrices are equivalent if and only if its corresponding Lie algebras are isomorphic. This work complements the results obtained by Barot and Rivera in [1].

The center of small quantum groups II: singular blocks

We generalize to the case of singular blocks the result in \cite{BeLa} that describes the center of the principal block of a small quantum group in terms of sheaf cohomology over the Springer resolution. Then using the method developed in \cite{LQ1}, we present a linear algebro-geometric approach to compute the dimensions of the singular blocks and of the entire center of the small quantum group associated with a complex semisimple Lie algebra. A conjectural formula for the dimension of the center of the small quantum group at an $l$th root of unity is formulated in type A.

Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category

We show that for any singular dominant integral weight λ of a complex semisimple Lie algebra g, the endomorphism algebra B of any projective-injective module of the parabolic BGG category Oλp is a symmetric algebra (as conjectured by Khovanov) extending the results of Mazorchuk and Stroppel for the regular dominant integral weight. Moreover, the endomorphism algebra B is equipped with a homogeneous (non-degenerate) symmetrizing form. In the appendix, there is a short proof due to K. Coulembier and V. Mazorchuk showing that the endomorphism algebra Bλp of the basic projective-injective module of Oλp is a symmetric algebra.

Cohomology in Singular Blocks for a Quantum Group at a Root of Unity

Let Uζ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra mathfrak {g} and a root of unity ζ. When L, L′ are irreducible Uζ-modules having regular highest weights, the dimension of text {Ext}^{n}_{U_{zeta }}(L,L^{prime }) can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of Uζ. This paper shows for L, L′ irreducible modules in a singular block that dim text {Ext}^{n}_{U_{zeta }}(L,L^{prime }) is explicitly determined using the coefficients of parabolic Kazhdan-Lusztig polynomials. This also computes the corresponding cohomology for q-Schur algebras and many generalized q-Schur algebras. The result depends on a certain parity vanishing property which we obtain from the Kazhdan-Lusztig correspondence and a Koszul grading of Shan-Varagnolo-Vasserot for the corresponding affine Lie algebra.

Logarithmic conformal field theories of type B n, ℓ = 4 and symplectic fermions

There are important conjectures about logarithmic conformal field theories (LCFT’s), which are constructed as a kernel of screening operators acting on the vertex algebra of the rescaled root lattice of a finite-dimensional semisimple complex Lie algebra. In particular, their representation theory should be equivalent to the representation theory of an associated small quantum group. This article solves the case of the rescaled root lattice Bn/2 as a first working example beyond A1/p. We discuss the kernel of short screening operators, its representations, and graded characters. Our main result is that this vertex algebra is isomorphic to a well-known example: The even part of n pairs of symplectic fermions. In the screening operator approach, this vertex algebra appears as an extension of the vertex algebra associated with A1n/2, which are n copies of the even part of one pair of symplectic fermions. The new long screenings give the new global Cn-symmetry. The extension is due to a degeneracy in this particular case: Rescaled long roots still have an even integer norm. For the associated quantum group of divided powers, the first author has previously encountered matching degeneracies: It contains the small quantum group of type A1n and the Lie algebra Cn. Recent results by Farsad, Gainutdinov, and Runkel on symplectic fermions suggest finally the conjectured category equivalence to this quantum group. We also study the other degenerate cases of a quantum group, giving extensions of LCFT’s of type Dn, D4, A2 with larger global symmetry Bn, F4, G2.

Reducible good representations of semisimple Lie algebras Ar and Br Part I

Given a semisimple (preferably simple) complex Lie algebra $L$, we consider the monoid $\Gamma=\Gamma(L)$ of equivalence classes of the finite dimensional reducible complex representations of $L$. Here $\Gamma$ is identified with the lattice of the corresponding highest weights. (This equips $\Gamma$ with the monoid structure.) For $\pi\in\Gamma$ one considers the symmetric algebra $\displaystyle S(\pi)=\bigoplus_{n=0}^{\infty}S^n(\pi)$ (here regarded as a representation). The elements of $\Gamma$ ``occurring'' in $S(\pi)$ -- i.e., which are the highest weights of some irreducible component of the representation $S(\pi)$ -- form a subsemigroup $M(\pi)$ of $\Gamma$. Such a $M(\pi)$ has a naturally defined rank $r(\pi)$ with $1\leq r(\pi)\leq r = \text{rank of }L$. In this paper we give a classification, for all the simple $L=A_r$ and $L=B_r$ of all the $\pi$ with $r(\pi)< r$.

Parabolic subgroup orbits on finite root systems

Oshima's Lemma describes the orbits of parabolic subgroups of irreducible finite Weyl groups on crystallographic root systems. This note generalises that result to all root systems of finite Coxeter groups, and provides a self contained proof, independent of the representation theory of semisimple complex Lie algebras.