Finite-dimensional Semisimple Lie Algebra Research Articles

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.

Twisted Calabi–Yau property of right coideal subalgebras of quantized enveloping algebras

Suppose that U is the quantized enveloping algebra of some finite-dimensional semisimple Lie algebra g and C is a right coideal subalgebra of U such that the group-like elements contained in C form a group. Then C is Artin–Schelter regular and twisted Calabi–Yau. The Nakayama automorphism of C is also determined if C is contained in the Borel part U⩾0.

ON A FORMULA FOR THE PI-EXPONENT OF LIE ALGEBRAS

We prove that one of the conditions in Zaicev's formula for the PI-exponent and in its natural generalization for the Hopf PI-exponent, can be weakened. Using the modification of the formula, we prove that if a finite-dimensional semisimple Lie algebra acts by derivations on a finite-dimensional Lie algebra over a field of characteristic 0, then the differential PI-exponent coincides with the ordinary one. Analogously, the exponent of polynomial G-identities of a finite-dimensional Lie algebra with a rational action of a connected reductive affine algebraic group G by automorphisms, coincides with the ordinary PI-exponent. In addition, we provide a simple formula for the Hopf PI-exponent and prove the existence of the Hopf PI-exponent itself for H-module Lie algebras whose solvable radical is nilpotent, assuming only the H-invariance of the radical, i.e. under weaker assumptions on the H-action, than in the general case. As a consequence, we show that the analog of Amitsur's conjecture holds for G-codimensions of all finite-dimensional Lie G-algebras whose solvable radical is nilpotent, for an arbitrary group G.

Derivations, Gradings, Actions of Algebraic Groups, and Codimension Growth of Polynomial Identities

Suppose a finite dimensional semisimple Lie algebra \(\mathfrak g\) acts by derivations on a finite dimensional associative or Lie algebra A over a field of characteristic 0. We prove the \(\mathfrak g\)-invariant analogs of Wedderburn—Mal’cev and Levi theorems, and the analog of Amitsur’s conjecture on asymptotic behavior for codimensions of polynomial identities with derivations of A. It turns out that for associative algebras the differential PI-exponent coincides with the ordinary one. Also we prove the analog of Amitsur’s conjecture for finite dimensional associative algebras with an action of a reductive affine algebraic group by automorphisms and anti-automorphisms or graded by an arbitrary Abelian group. In addition, we provide criteria for G-, H- and graded simplicity in terms of codimensions.

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.

Rigidity of Tilting Modules

Let $U_q$ denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that $q$ is a complex root of unity of odd order and that $U_q$ is %the quantum group version obtained via Lusztig's $q$-divided powers construction. We prove that all regular projective (tilting) modules for $U_q$ are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl modules for $U_q$. On the other hand, we give examples of non-rigid indecomposable tilting modules as well as non-rigid Weyl modules. These examples are for type $B_2$ and in this case as well as for type $A_2$ we calculate explicitly the Loewy structure for all regular Weyl modules. We also demonstrate that these results carry over to the modular case when the highest weights in question are in the so-called Jantzen region. At the same time we show by examples that as soon as we leave this region non-rigid tilting modules do occur.

Lie algebras and algebras of associative type

In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕αeG A α graded by some group G, over a field of characteristic zero, has a nonzero component A 1 (where 1 stands for the identity element of G), and A 1 is a semisimple associative algebra. Let B = ⊕αeG B α be a finite-dimensional semisimple Lie algebra over a prime field F p , and let B be graded by a commutative group G. If B = F p ⊗ ℤ A L , where A L is the commutator algebra of a ℤ-algebra A = ⊕αeG A α ; if ℚ ◯ ℤ A is an algebra of associative type, then the 1-component of the algebra K ◯ ℤ B, where K stands for the algebraic closure of the field F p , is the sum of some algebras of the form gl(n i ,K).

Nonzero Infinitesimal Blocks of Category $\boldsymbol{\mathcal{O}}_{\textbf{\emph{S}}}$

Category \(\mathcal{O}\) is a nice category of modules for a finite dimensional semisimple Lie algebra \(\mathfrak{g}\) It was first introduced by Bernstein–Gelfand–Gelfand in 1976. In the early 1980’s, Rocha–Caridi introduced parabolic category \(\mathcal{O}_S\), where S is a subset of simple roots. Category \(\mathcal{O}_S\) is a generalization of ordinary category O. These are highest weight categories that decompose into certain subcategories, called infinitesimal blocks. An infinitesimal block contains at most finitely many simple modules, and some contain only the zero module. The representation type of the infinitesimal blocks of category \(\mathcal{O}_S\) has been studied by Futorny–Nakano–Pollack, Brustle–Konig–Mazorchuk, and Boe–Nakano. The representation type of the singular blocks of category \(\mathcal{O}_S\) is still generally unknown, though Boe–Nakano classified certain of these. Understanding when a singular block is zero is an important step to understanding the singular blocks in general. In this work, we will answer this question. It is given in terms of nilpotent orbits of \(\mathfrak{g}\).

Poisson homology in degree 0 for some rings of symplectic invariants

Let g be a finite-dimensional semi-simple Lie algebra, h a Cartan subalgebra of g , and W its Weyl group. The group W acts diagonally on V : = h ⊕ h ∗ , as well as on C [ V ] . The purpose of this article is to study the Poisson homology of the algebra of invariants C [ V ] W endowed with the standard symplectic bracket. To begin with, we give general results about the Poisson homology space in degree 0, denoted by HP 0 ( C [ V ] W ) , in the case where g is of type B n − C n or D n , results which support Alev's conjecture. Then we are focusing the interest on the particular cases of ranks 2 and 3, by computing the Poisson homology space in degree 0 in the cases where g is of type B 2 ( so 5 ), D 2 ( so 4 ), then B 3 ( so 7 ), and D 3 = A 3 ( so 6 ≃ sl 4 ). In order to do this, we make use of a functional equation introduced by Y. Berest, P. Etingof and V. Ginzburg. We recover, by a different method, the result established by J. Alev and L. Foissy, according to which the dimension of HP 0 ( C [ V ] W ) equals 2 for B 2 . Then we calculate the dimension of this space and we show that it is equal to 1 for D 2 . We also calculate it for the rank 3 cases, we show that it is equal to 3 for B 3 − C 3 and 1 for D 3 = A 3 .

P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras

For any finite-dimensional semisimple Lie algebra g, a ℤ+-graded vertex algebra is construsted on the vacuum representation Vk(ĝ[θ]) of ĝ[θ], which is a one-dimentional central extension of θ-invariant subspace on the loop algebra Lg = g ⊗ ℂ((t1/p)).

Hom-Lie algebra structures on semi-simple Lie algebras

Hom-Lie algebras can be considered as a deformation of Lie algebras. In this note, we prove that the hom-Lie algebra structures on finite-dimensional simple Lie algebras are trivial. We find when a finite-dimensional semi-simple Lie algebra admits non-trivial hom-Lie algebra structures and the isomorphic classes of non-trivial hom-Lie algebras are determined.

Frobenius–Schur indicators for semisimple Lie algebras

Let g be a finite-dimensional complex semisimple Lie algebra, and let V be a finite-dimensional complex representation of g . We give a closed formula for the mth Frobenius–Schur indicator, m > 1 , of V in representation-theoretic terms. We deduce that the indicators take integer values, and that for a large enough m, the mth indicator of V equals the dimension of the zero weight space of V. For the classical complex Lie algebras sl ( n ) , so ( 2 n ) , so ( 2 n + 1 ) and sp ( 2 n ) , this is the case for m greater or equal to 2 n − 1 , 4 n − 5 , 4 n − 3 and 2 n + 1 , respectively.

On functors associated to a simple root

Associated to a simple root of a finite-dimensional complex semisimple Lie algebra, there are several endofunctors (defined by Arkhipov, Enright, Frenkel, Irving, Jantzen, Joseph, Mathieu, Vogan and Zuckerman) on the BGG category O . We study their relations, compute cohomologies of their derived functors and describe the monoid generated by Arkhipov's and Joseph's functors and the monoid generated by Irving's functors. It turns out that the endomorphism rings of all elements in these monoids are isomorphic. We prove that the functors give rise to an action of the singular braid monoid on the bounded derived category of O 0 . We also use Arkhipov's, Joseph's and Irving's functors to produce new generalized tilting modules.

On Whittaker modules over a class of algebras similar to U(sl 2)

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A 1 and finding interesting families of new noetherian rings, a class of algebras similar to U(sl 2) was introduced and studied by Smith. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained. But it seems that not much has been done on the part of nonweight modules. In this paper, we generalize Kostant’s results on the Whittaker model for the universal enveloping algebras U(g) of finite dimensional semisimple Lie algebras g to Smith’s algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith’s algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.

Good grading polytopes

Let 𝔤 be a finite-dimensional semisimple Lie algebra over ℂ and e ∈ 𝔤 a nilpotent element. Elashvili and Kac have recently classified all good ℤ-gradings for e. We instead consider good ℝ-gradings, which are naturally parameterized by an open convex polytope in a Euclidean space arising from the reductive part of the centralizer of e in 𝔤. As an application, we prove that the isomorphism type of the finite W-algebra attached to a good ℝ-grading for e is independent of the particular choice of good grading.

BGP-reflection functors and cluster combinatorics

We define Bernstein–Gelfand–Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the “truncated simple reflections” on the set of almost positive roots Φ ≥ − 1 associated with a finite dimensional semi-simple Lie algebra. Combining this with the tilting theory in cluster categories developed in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054], we give a unified interpretation via quiver representations for the generalized associahedra associated with the root systems of all Dynkin types (simply laced or non-simply laced). This confirms the Conjecture 9.1 in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054] for all Dynkin types.

Lie algebras generated by bounded linear operators on Hilbert spaces

It is proved that the operator Lie algebra ε ( T , T ∗ ) generated by a bounded linear operator T on Hilbert space H is finite-dimensional if and only if T = N + Q , N is a normal operator, [ N , Q ] = 0 , and dim A ( Q , Q ∗ ) < + ∞ , where ε ( T , T ∗ ) denotes the smallest Lie algebra containing T , T ∗ , and A ( Q , Q ∗ ) denotes the associative subalgebra of B ( H ) generated by Q , Q ∗ . Moreover, we also give a sufficient and necessary condition for operators to generate finite-dimensional semi-simple Lie algebras. Finally, we prove that if ε ( T , T ∗ ) is an ad-compact E-solvable Lie algebra, then T is a normal operator.

Number of representations providing noiseless subsystems

This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free (DF) subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function ${f}_{d}(n)$ which is the fraction of all $d$-dimensional quantum systems which preserve $n$ bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.

Weak Hopf algebras corresponding to Cartan matrices

We replace the group of grouplike elements of the quantized enveloping algebra Uq(g) of a finite dimensional semisimple Lie algebra g by some regular monoid and get the weak Hopf algebra wqd(g). It is a subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of wqd(g) and determine the group of weak Hopf algebra automorphisms of wqd(g) when q is not a root of unity.

Quantum Groups and Double Quiver Algebras

For a finite dimensional semisimple Lie algebra ${\frak{g}}$ and a root $q$ of unity in a field $k,$ we associate to these data a double quiver $\bar{\cal{Q}}.$ It is shown that a restricted version of the quantized enveloping algebras $U_q(\frak g)$ is a quotient of the double quiver algebra $k\bar{\cal Q}.$