Finite-dimensional Semisimple Lie Algebra Research Articles

On Arkhipov?s and Enright?s functors

We give a description of Arkhipov’s and (Joseph’s and Deodhar-Mathieu’s versions of) Enright’s endofunctors on the category Open image in new window associated with a fixed triangular decomposition of a complex finite-dimensional semi-simple Lie algebra, in terms of (co)approximation functors with respect to suitably chosen injective (resp. projective) modules. We establish some new connections between these functors, for example we show that Arkhipov’s and Joseph’s functors are adjoint to each other. We also give several proofs of braid relations for Arkhipov’s and Enright’s functors.

Representations of twisted multi-loop Lie algebras

This paper describes finite-dimensional irreducible representations of “twisted multi-loop Lie algebras.” These representations are given in terms of the representations of finite-dimensional semisimple Lie algebras.

Category 𝒪: Quivers and endomorphism rings of projectives

We describe an algorithm for computing quivers of category O \mathcal O of a finite dimensional semisimple Lie algebra. The main tool for this is Soergel’s description of the endomorphism ring of the antidominant indecomposable projective module of a regular block as an algebra of coinvariants. We give explicit calculations for root systems of rank 1 and 2 for regular and singular blocks and also quivers for regular blocks for type A 3 A_3 . The main result in this paper is a necessary and sufficient condition for an endomorphism ring of an indecomposable projective object of O \mathcal O to be commutative. We give also an explicit formula for the socle of a projective object with a short proof using Soergel’s functor V \mathbb V and finish with a generalization of this functor to Harish-Chandra bimodules and parabolic versions of category O \mathcal O .

An Algorithm to Compute the Canonical Basis of an Irreducible Module Over a Quantized Enveloping Algebra

AbstractThe paper describes an algorithm to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for any module that is constructed as a submodule of a tensor product of modules with known canonical bases.

Invariants of semisimple Lie algebras acting on associative algebras

If g is a Lie algebra of derivations of an associative algebra R, then the subalgebra of invariants is the set R g = {r E R | δ(r) = 0 for all δ ∈ g}. In this paper, we study the relationship between the structure of R 9 and the structure of R, where g is a finite dimensional semisimple Lie algebra over a field of characteristic zero acting finitely on R, when R is semiprime.

Branching rules of semi-simple Lie algebras using affine extensions

We present a closed formula for the branching coefficients of an embedding p in g of two finite-dimensional semi-simple Lie algebras. The formula is based on the untwisted affine extension of p. It leads to an alternative proof of a simple algorithm for the computation of branching rules which is an analog of the Racah-Speiser algorithm for tensor products. We present some simple applications and describe how integral representations for branching coefficients can be obtained. In the last part we comment on the relation of our approach to the theory of NIM-reps of the fusion rings of WZW models with chiral algebra g_k. In fact, it turns out that for these models each embedding p in g induces a NIM-rep at level k to infinity. In cases where these NIM-reps can be be extended to finite level, we obtain a Verlinde-like formula for branching coefficients.

THE COINVARIANT ALGEBRA AND REPRESENTATION TYPES OF BLOCKS OF CATEGORY O

Let [Gfr ] be a finite-dimensional semisimple Lie algebra over the complex numbers. Let A be the finite-dimensional algebra of a (regular or singular) block of the BGG-category [Oscr ] . By results of Soergel, A has a combinatorial description in terms of a subalgebra C0 of the coinvariant algebra C. König and Mazorchuk have constructed an embedding from C0-mod into the category [Fscr ](Δ) of A-modules having a Verma flag. This is the main tool for the classification of [Fscr ] (Δ) into finite, tame and wild representation types presented here. As a consequence a classification of A-mod into finite, tame and wild representation types is obtained, thus re-proving a recent result of Futorny, Nakano and Pollack.

Lakshmibai–Seshadri Paths Fixed by a Diagram Automorphism

We prove that the Lakshmibai–Seshadri paths fixed by a diagram automorphism ω of a Kac–Moody algebra can be identified with the Lakshmibai–Seshadri paths for the orbit Lie algebra, by introducing ω-root operators. We also give certain formulas for the twining characters of integrable highest weight modules over a symmetrizable Kac–Moody algebra and of Demazure modules for a finite-dimensional semi-simple Lie algebra.

Invariant Lie Algebras and Lie Algebras with a Small Centroid

A subalgebra of a Lie algebra is said to be invariant if it is invariant under the action of some Cartan subalgebra of that algebra. A known theorem of Melville says that a nilpotent invariant subalgebra of a finite-dimensional semisimple complex Lie algebra has a small centroid. The notion of a Lie algebra with small centroid extends to a class of all finite-dimensional algebras. For finite-dimensional algebras of zero characteristic with semisimple derivations in a sufficiently broad class, their centroid is proved small. As a consequence, it turns out that every invariant subalgebra of a finite-dimensional reductive Lie algebra over an arbitrary definition field of zero characteristic has a small centroid.

On the Rigidity of Lie Lattices and Just Infinite Powerful Groups

A classical result of M. Gerstenhaber [5] states that finite-dimensional semisimple complex Lie algebras are rigid. One interpretation of this fact is as follows. Let L be a semisimple complex Lie algebra of dimension d and let B be a C-basis of L. Let L ( d ) ⊂ C d ċ ( d 2 ) denote the affine variety of all structure constants of C-Lie algebras of dimension d and let c ∈ L(d) denote the point corresponding to the structure constants of L with respect to B. Then there exists an open neighbourhood in the metric topology U ⊂ L(d) of c ∈ L(d) such that Lc* is isomorphic to L for all c* ∈ U, where Lc* denotes the Lie algebra defined by the structure constants c* ∈ L(d). Our aim is to generalize this result to semisimple Lie algebras over discrete valuation domains and to apply these results to powerful pro-p-groups.

Invariants of Real Forms of Affine Kac-Moody Lie Algebras

The theory of Vogan diagrams, which are Dynkin diagrams with an overlay of certain additional information, allows one to give a rapid classification of finite-dimensional real semisimple Lie algebras and to make use of this classification in practice. This paper develops a corresponding theory of Vogan diagrams for “almost compact” real forms of indecomposable nontwisted affine Kac–Moody Lie algebras. In this case also, the equivalence classes of Vogan diagrams correspond to the isomorphism classes of almost compact real forms. Although the real forms of such algebras had already been classified, the theory of Vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification.

The coradical filtration of U q(g)at roots of unity

We compute the coradical filtration and the group of Hopf algebra automorphisms of the non-restricted specialization of the quantized universal enveloping algebra of a finite-dimensional semisimple Lie algebra.

Isomorphisms of Finite Invariant for Enveloping Algebras, Semi-simple Case

Letgbe a finite dimensional semi-simple Lie algebra,U(g) its enveloping algebra, andHa finite subgroup ofAutU(g). LetAbe the invariant algebraUH. In this article, we prove that the Lie algebragis given (up to an isomorphism) by the algebraA. If we impose thatHis a finite subgroup of the adjoint group ofgacting on the enveloping algebraU(g), then the algebraAgives a unique group algebraC[H]. Ifg=sl2, then the groupHcan be recovered fromA.

Leibniz Representations of Lie Algebras

ALeibniz representationof the Lie algebra g is a vector spaceMequipped with two actions (left and right) [–,–]: g⊗M→Mand [–,–]:M⊗g→Mwhich satisfy the relations[formula]when one of the variables is inMand the two others are in g. In this paper we show that the categoryL(g) of finite dimensional Leibniz representations of a finite dimensional semi-simple Lie algebra is not semi-simple, but thatL(g) has global dimension 2. We give an explicit description of the extensions of simple objects and we obtain the description of the quiver ofL(g) (in the sense of Gabriel). It turns out thatL(g) is tame only for g=sl2. We give the complete list of indecomposable objects in that case.

Cartan decompositions and BGG-resolutions

Necessary and sufficient criteria are given for the existence of BGG-resolutions (finite resolutions of modules by finite direct sums of Weyl modules) for simple modules over quasi-hereditary algebras, which have strong exact Borel subalgebras and strong Δ-subalgebras. Our main technical tool is the existence of Cartan decompositions for these algebras. The results apply to simple objects in the BGG-categoryO of a finite-dimensional semisimple complex Lie algebra and to finite dimensional simple rational modules over simply connected semisimple algebraic groups.

The coradical filtration of U Q (S(2)) Q at roots of unity

Let u q (s[(2)) denote the quantum enveloping algebra of s((2) over a field k. In this note we show that if q is a root of unity, then the coradical filtration {H n}n ≥0 of u q (s[(2)) is the filtration by a certain degree function depending on the order of q 4 and on the characteristic of k. The degree function will be explicitly described in Section 2 for char s = 0 and in Section 3 for char k = s > 0. We use the fact that u q (g) is pointed for any finite-dimensional semisimple Lie algebra (proved in [R], [Ml] and remarked in [Tl]). The skew-primitives of u q (g[(n)) and u q (s[(n)) are described in [T2], based en [Tl]. The theorem in [T2] motivated the present result, although we did not need its full strength in the proof. The filtration of u q (g) over Q(q), where g is a finite dimensional complex simple Lie algebra, and q is an indeterminate over Q, is described in [CM]. The case of u q (s[(2)) with char k = 0 and where q is not a root of unity was first solved in [Ml].

A Module Induced from a Whittaker Module

In an earlier paper [On modules induced from Whittaker modules, J. Algebra 96 (1985)] we constructed a class of induced modules, over a finite-dimensional semisimple Lie algebra, which includes the Verma modules of Verma [Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968)] and the irreducible Whittaker modules of Kostant [On Whittaker vectors and representation theory, Invent. Math. 48 (1978)]. We proved that every module in this class has finite length and is irreducible most of the time. In this article we present a concrete example of this construction, over $\operatorname {sl} (3,C)$, showing that proper submodules can exist when the induced module is not a Verma module.

A module induced from a Whittaker module

In an earlier paper [On modules induced from Whittaker modules, J. Algebra 96 (1985)] we constructed a class of induced modules, over a finite-dimensional semisimple Lie algebra, which includes the Verma modules of Verma [Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968)] and the irreducible Whittaker modules of Kostant [On Whittaker vectors and representation theory, Invent. Math. 48 (1978)]. We proved that every module in this class has finite length and is irreducible most of the time. In this article we present a concrete example of this construction, over sl ⁡ ( 3 , C ) \operatorname {sl} (3,C) , showing that proper submodules can exist when the induced module is not a Verma module.

Semisimple Lie superalgebras which are not the direct sums of simple Lie superalgebras

Any finite-dimensional semisimple Lie algebra (LA) is known to be realised as a direct sum of simple LA. The situation is somewhat more complicates in the Lie superalgebra (LSA) case. A theorem of Kac states how to build all the semisimple LSA. The goal of this paper is to give some explicit examples arising from this theorem, such illustrations perhaps being better understood by physicists rather than an abstract statement; and to show the relevance of the Neveu-Schwarz and Raman superalgebras when generalising the theorem to the infinite case.

A homology vanishing theorem for Kac-Moody algebras with coefficients in the category [formula omitted

In this paper, we study the homology of a Kac-Moody Lie-algebra, associated to a symmetrizable generalized Cartan matrix, with coefficients in a module M in the category 0. We show that all the homology groups vanish if the Casimir operator acts as an automorphism on M. In fact we prove a slightly more general theorem (Theorem (1.2)). This is one of the main theorems of this paper. We further study the homology of g with coefficients in arbitrary Verma modules and also integrable highest weight modules. WC prove (Proposition (1.5)) that Hi(g, M(A)) =0 for all i>O and for all the Verma modules M(A) with highest weight ,? E h* such that i is not equal to wp p for any w E W, whereas Hi(g, M( wp p)) w /1 ip ‘(“j(h). This result in the case when g is a finite dimensional semi-simple Lie-algebra is due to Williams. If L(1) is an integrable highest weight module with i #O (2 is automatically dominant integral) then we have, using BGG resolution and Proposition (1.5), that Hi(g, L(;i)) = 0 for all ia 0. This is the content of our Theorem (1.7). In the case when g is finite dimensional, the integrable highest weight modules are precisely the finite dimensional irreducible modules. In this case, this theorem is well known and is due to Whitehead. As in the finite dimensional case, the situation when ,J = 0 (so that L(I1) is the trivial one dimensional module) is drastically different. We have recently proved [Ku; Theorem 1.61 that the homology of the commutator