Complex Semisimple Lie Algebra Research Articles

Hermitian real forms of contragredient Lie superalgebras

For a complex semisimple Lie algebra g with Hermitian real form gR=kR+pR, there exists a positive system of roots such that the adjoint k-representation on p stabilizes the positive and negative root spaces. In this article, we extend this result to contragredient Lie superalgebras g, and study the number of irreducible components of the k-representation. We also discuss the complex structure on gR/kR.

Factoriality for the reductive Zassenhaus variety and quantum enveloping algebra

Let U(g) be the enveloping algebra of a finite dimensional reductive Lie algebra g over an algebraically closed field of prime characteristic. Let Uϵ,P(s:) be the simply connected quantum enveloping algebra at the root of unity ϵ, of a complex semi-simple finite dimensional Lie algebra s:. We show, by similar proofs, that the centers of both are factorial. While the first result was established by R. Tange [32] (by different methods), the second one confirms a conjecture in [4]. We also provide a general criterion for the factoriality of the centers of enveloping algebras in prime characteristic.

Category $\mathcal O$ for quantum groups

In this paper we study the BGG-categories \mathcal O_q associated to quantum groups. We prove that many properties of the ordinary BGG-category \mathcal O for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when q is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for \mathcal O and for finite dimensional U_q -modules we are able to determine all irreducible characters as well as the characters of all indecomposable tilting modules in \mathcal O_q . As a consequence of these results we are able to recover also a known result, namely that the generic quantum case behaves like the classical category \mathcal O .

Extremal vectors of the Verma modules of the Lie algebra B 2 in Poincaré-Birkhoff-Witt basis

The aim of this work is to present a full set of expressions for the extremal vectors in a Verma module over the B2 complex semisimple Lie algebra in the Poincare-Birkhoff-Witt basis.

Restrictions of Symmetric Invariants of a Complex Semisimple Lie Algebra and Some Applications

Let 𝔤 be a finite-dimensional complex semisimple Lie algebra and σ an arbitrary semisimple automorphism of 𝔤. Let 𝔱 be a Cartan subalgebra of 𝔨 = 𝔤σ and 𝔥 =Z 𝔤(𝔱) be the centralizer of 𝔱 in 𝔤. Then 𝔥 is a σ-invariant Cartan subalgebra of 𝔤 and 𝔱 = 𝔥σ. Let W(𝔤, 𝔥) be the Weyl group. One knows that Δ(𝔤, 𝔱), the set of roots of 𝔤 in 𝔱, is also a root system. It is proved that the corresponding Weyl group W(𝔤, 𝔱) is isomorphic to W(𝔤, 𝔥)σ, which is the subgroup of W(𝔤, 𝔥) consisting of those elements commuting with σ. It is also shown that the image of the restriction map S(𝔥*) W(𝔤, 𝔥) → S(𝔱*) W(𝔨, 𝔱), where S(𝔥*) and S(𝔱*) are the polynomial algebras on 𝔥 and 𝔱, respectively, is exactly S(𝔱*) W(𝔤, 𝔱). Based on the above result, we also get a complete classification of the pairs (𝔤, σ) such that 𝔤σ is noncohomologous to zero in 𝔤.

On Ky Fan's result on eigenvalues and real singular values of a matrix

Ky Fan's result states that the real parts of the eigenvalues of an n×n complex matrix A is majorized by the real singular values of A. The converse was established independently by Amir-Moéz and Horn, and Mirsky. We extend the results in the context of complex semisimple Lie algebras. The real semisimple case is also discussed. The complex skew symmetric case and the symplectic case are explicitly worked out in terms of inequalities. The symplectic case and the odd dimensional skew symmetric case can be stated in terms of weak majorization. The even dimensional skew symmetric case involves Pfaffian.

Curvature spectra of simple Lie groups

The Killing form \beta\ of a real (or complex) semisimple Lie group G is a left-invariant pseudo-Riemannian (or, respectively, holomorphic) Einstein metric. Let \Omega\ denote the multiple of its curvature operator, acting on symmetric 2-tensors, with the factor chosen so that \Omega\beta=2\beta. The result of Meyberg [8], describing the spectrum of \Omega\ in complex simple Lie groups G, easily implies that 1 is not an eigenvalue of \Omega\ in any real or complex simple Lie group G except those locally isomorphic to SU(p,q), or SL(n,R), or SL(n,C) or, for even n only, SL(n/2,H), where p\ge q\ge0 and p+q=n>2. Due to the last conclusion, on simple Lie groups G other the ones just listed, nonzero multiples of the Killing form \beta\ are isolated among left-invariant Einstein metrics. Meyberg's theorem also allows us to understand the kernel of \Lambda, which is another natural operator. This in turn leads to a proof of a known, yet unpublished, fact: namely, that a semisimple real or complex Lie algebra with no simple ideals of dimension 3 is essentially determined by its Cartan three-form.

KRULL DIMENSION OF AFFINOID ENVELOPING ALGEBRAS OF SEMISIMPLE LIE ALGEBRAS

AbstractUsing Beilinson–Bernstein localisation, we give another proof of Levasseur's theorem on the Krull dimension of the enveloping algebra of a complex semisimple Lie algebra. The proof also extends to the case of affinoid enveloping algebras.

Contragredient supersymmetric pairs

The symmetric pairs of complex semisimple Lie algebras can be characterized by commuting involutions. We present its analogue on the contragredient Lie superalgebras, so that their symmetric pairs correspond to some commuting automorphisms. As an application, we classify the symmetric pairs of the exceptional Lie superalgebras F(4) and G(3).

The geometric meaning of Zhelobenko operators

Let \( \mathfrak{g} \) be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, \( \mathfrak{b} \) a Borel subalgebra of \( \mathfrak{g} \), \( \mathfrak{h}\subset \mathfrak{b} \) the Cartan sublagebra, and N ⊂ G the unipotent subgroup corresponding to the nilradical \( \mathfrak{n}\subset \mathfrak{b} \). We show that the explicit formula for the extremal projection operator for \( \mathfrak{g} \) obtained by Asherova, Smirnov, and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence \( N\times \mathfrak{h}\to \mathfrak{b} \) given by the restriction of the adjoint action. Simple geometric proofs of formulas for the “classical” counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.

On finite W-algebras for Lie algebras and superalgebras

The finiteW-algebras are certain associative algebras associated to a complex semi-simple or reductive Lie algebra g and a nilpotent element e of g. Due to recent results of I. Losev, A. Premet and others, finite W-algebras play a very important role in description of primitive ideals. In the full generality, the finiteW-algebras were introduced by A. Premet. It is a result of B. Kostant that for a regular nilpotent (principal) element e, the finiteW-algebra coincides with the center of U(g). Premet’s definition makes sense for a simple Lie superalgebra g = g¯0 g¯1 in the case when g¯0 is reductive, g admits an invariant super-symmetric bilinear form, and e is an even nilpotent element. We show that certain results of A. Premet can be generalized for classical Lie superalgebras. We consider the case when e is an even regular nilpotent element. The associated finiteW-algebra is called principal. Kostant’s result does not hold in this case. This is joint work with V. Serganova.

Lie algebras and higher torsion in p-groups

The primary aim of this paper is to study exceptional torsion in the integral cohomology of a family of p-groups associated to p-adic Lie algebras. A spectral sequence Er⁎,⁎[g] is defined for any Lie algebra g which models the Bockstein spectral sequence of the corresponding group in characteristic p.This spectral sequence is then studied for complex semisimple Lie algebras like sln(C) and the results there are transferred to the corresponding p-group via the intermediary arithmetic Lie algebra defined over Z. The results obtained this way for a fixed Lie algebra scheme like sln(−) hold in a range in the corresponding Bockstein spectral sequence for all but finitely many primes depending on the chosen range.Over C, it is shown that E1⁎,⁎[g]=H⁎(g,U(g)⁎)=H⁎(ΛBG) where U(g)⁎ is the (filtered) dual of the universal enveloping algebra of g equipped with the dual adjoint action and ΛBG is the free loop space of the classifying space of an associated compact, connected real form Lie group G to g.When passing to characteristic p, in the corresponding Bockstein spectral sequence, a char 0 to char p phase transition is observed. For example, it is shown that the algebra E1⁎,⁎[sl2[Fp]] requires at least 17 generators unlike its characteristic zero counterpart which only requires two.

A Multiplicity Formula for Isotropy Representations of Symmetric Pairs

Let 𝔤 be a complex semisimple Lie algebra and θ an involutory automorphism of 𝔤. Let 𝔤 = 𝔨 ⊕ 𝔭 be the decomposition of 𝔤 where 𝔨 and 𝔭 are the respective 1 and −1 eigenspaces of θ in 𝔤. Let l be the difference of the rank of 𝔤 and that of 𝔨. The spin module S of 𝔰𝔬(𝔭) becomes a 𝔨-module via the isotropy representation ν: 𝔨 → 𝔰𝔬(𝔭). Assume that the 𝔨-module S is primary of type V ρ n . Then we prove that the multiplicity of the isotropy representation ν in V ρ n ⊗ V ρ n equals l. Let C(𝔭) be the Clifford algebra over 𝔭 and i: 𝔰𝔬(𝔭) → C(𝔭) be the natural inclusion. Let φ: 𝔨 → C(𝔭) be the composition of ν and i. We will describe the structure of the subalgebra E of C(𝔭) generated by the image of 𝔨 and also the structure of the centralizer of E in C(𝔭).

SYMMETRY INSIGHTS FOR DESIGN OF SUPERCOMPUTER NETWORK TOPOLOGIES: ROOTS AND WEIGHTS LATTICES

We present a family of networks whose local interconnection topologies are generated by the root vectors of a semi-simple complex Lie algebra. Cartan classification theorem of those algebras ensures those families of interconnection topologies to be exhaustive. The global arrangement of the network is defined in terms of integer or half-integer weight lattices. The mesh or torus topologies that network millions of processing cores, such as those in the IBM BlueGene series, are the simplest member of that category. The symmetries of the root systems of an algebra, manifested by their Weyl group, lends great convenience for the design and analysis of hardware architecture, algorithms and programs.

Some comments on Weyl’s complete reducibility theorem

In this note we discuss a purely algebraic proof of Weyl’s theorem that all finite-dimensional representations of a complex semisimple Lie algebra are completely reducible. We give a simple and direct proof which is elementary in the sense that it does not use cohomology, and which is a synthesis of the older proofs of Casimir ‐ van der Waerden and of Brauer.

Homogeneous right coideal subalgebras of quantized enveloping algebras

For a quantized enveloping algebra of a complex semisimple Lie algebra with deformation parameter not a root of unity, we classify all homogeneous right coideal subalgebras. Any such right coideal subalgebra is determined uniquely by a triple consisting of two elements of the Weyl group and a subset of the set of simple roots satisfying some natural conditions. The essential ingredients of the proof are the Lusztig automorphisms and the classification of homogeneous right coideal subalgebras of the Borel Hopf subalgebras of quantized enveloping algebras obtained previously by H.-J. Schneider and the first-named author.

On visual representations of Lie algebras: the type Al between the old and the new

This paper is about some graph isomorphisms between the Auslander-Reiten quivers of the path algebras of quivers with underly- ing Dynkin diagrams of type Al, and the weight diagrams or the weight graphs relative to some basic (adjoint) representations of a semi-simple complex Lie algebra associated to the same Dynkin diagrams. The cru- cial point is that, in the attempt to establish the above relationships, we constructed the oriented weight diagrams and the oriented weight graphs with respect to particular orientations on the Dynkin diagrams of type Al. Our main goal is to furnish another application of weight theory and visual representations.

Weyl groups of fine gradings on matrix algebras, octonions and the Albert algebra

Given a grading Γ:A=⊕g∈GAg on a nonassociative algebra A by an abelian group G, we have two subgroups of Aut(A): the automorphisms that stabilize each component Ag (as a subspace) and the automorphisms that permute the components. By the Weyl group of Γ we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the automorphism group of the root system, i.e., the so-called extended Weyl group. A grading is called fine if it cannot be refined. We compute the Weyl groups of all fine gradings on matrix algebras, octonions and the Albert algebra over an algebraically closed field (of characteristic different from 2 in the case of the Albert algebra).

Maximal eigenvalues of a Casimir operator and multiplicity-free modules

Let g \mathfrak g be a finite-dimensional complex semisimple Lie algebra and b \mathfrak b a Borel subalgebra. Then g \mathfrak g acts on its exterior algebra ∧ g \wedge \mathfrak g naturally. We prove that the maximal eigenvalue of the Casimir operator on ∧ g \wedge \mathfrak g is one third of the dimension of g \mathfrak g , that the maximal eigenvalue m i m_i of the Casimir operator on ∧ i g \wedge ^i\mathfrak g is increasing for 0 ≤ i ≤ r 0\le i\le r , where r r is the number of positive roots, and that the corresponding eigenspace M i M_i is a multiplicity-free g \mathfrak g -module whose highest weight vectors correspond to certain ad-nilpotent ideals of b \mathfrak {b} . We also obtain a result describing the set of weights of the irreducible representation of g \mathfrak g with highest weight a multiple of ρ \rho , where ρ \rho is one half the sum of positive roots.

Cartan automorphisms and Vogan superdiagrams

The real forms of complex semisimple Lie algebras are characterized by Cartan involutions and Vogan diagrams. We extend these notions to the Cartan automorphisms and Vogan superdiagrams, and show that they characterize the real forms of complex contragredient Lie superalgebras.