Kostant's problem,Goldie rank and the Gelfand-Kirillov conjecture

Let g be a complex semisimple Lie algebra andU(g) its enveloping algebra. GivenM a simpleU(g) module, letL(M, M) denote the subspace of ad g finite elements of Hom(M, M). Kostant has asked if the natural homomorphism ofU(g) intoL(M, M) is surjective. Here the question is analysed for simple modules with a highest weight vector. This has a negative answer if g admits roots of different length ([7], 6.5). Here general conditions are obtained under whichU(g)/AnnM andL(M, M) have the same ring of fractions—in particular this is shown to always hold if g has only typeA _n factors. Combined with [21], this provides a method for determining the Goldie ranks for the primitive quotients ofU(g). Their precise form is given in typeA _n (Cartan notation) for which the generalized Gelfand-Kirillov conjecture for primitive quotients is also established.This paper was written while the author was a guest of the Institute for Advanced Studies, the Hebrew University of Jerusalem and on leave of absence from the Centre Nationale de la Recherche Scientifique     

Lifting via cocycle deformation

We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such lifting.     

Finite-dimensional pointed Hopf algebras with alternating groups are trivial

It is shown that Nichols algebras over alternating groups \mathbb A_m{\mathbb A_m} (m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to \mathbb A_m{\mathbb A_m} is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups \mathbb S_m{\mathbb S_m} are all infinite-dimensional, except maybe those related to the transpositions considered in Fomin and Kirillov (Progr Math 172:146–182, 1999), and the class of type (2, 3) in \mathbb S₅{\mathbb S_5}. We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra \mathfrak B(X, q){\mathfrak B(X, \bf q)} is infinite dimensional, q an arbitrary cocycle.     

Les sous-algebres de Cartan reelles et la frontiere d'une orbite ouverte dans une variete de drapeaux

We show how the non compact imaginary roots of a non compact real semi-simple Lie algebra with respect to a Cartan subalgebra to allows us, alike the real roots of, to give a complete classification of the G-conjugacy classes of Cartan subalgebras of if G_c is a complex connected group whose algebra is the complexified of, if B is a Borel subgroup of G_c and G the analytic subgroup of G_c corresponding to the subalgebra of, we determine the G-orbits of codimension one in the boundary of an open G-orbit of the complex flag manifold G_c/B. If is a maximally compact Cartan subalgebra of contained in, we show how the imaginary non compact simple roots of allows us to determine such orbits.     

On self-adjoint subloops of moufang loops

In this paper we give a combinatorial rule to compute the composition of two convolution products of endomorphisms of a free associative algebra and deduce the construction of a subalgebra of QB _n (the group algebra of Hyperoctahedral group) which contains the descent algebra X#?. We also deduce a proof of the multiplication rule in the algebra ∑QB _n- Finally, we generalize this construction to other wreath products of symmetric groups by abelian groups.     

On the homological finiteness properties of some modules over metabellian Lie algebras

We characterise the modulesB of homological typeFP _m over a finitely generated Lie algebraL such thatL is a split extension of an abelian idealA and an abelian subalgebraQ andA acts trivially onB. The characterisation is in terms of the invariant Δ introduced by R. Bryant and J. Groves and is a Lie algebra version of the generalisation [K 4, conjecture 1] of the still openFP _m -Conjecture for metabelian groups [Bi-G, Conjecture p. 367]. The casem=1 of our main result is treated separately, as there the characterisation is proved without restrictions on the type of the extension.     

Construction of representation-finite selfinjective artin algebras of classB n andC n

In this paper, we discuss the representation-finite selfinjective artin algebras of classB _n andC _n and obtain the following main results: For any fieldk, let Λ be a representation-finite selfinjective artin algebras of classB _n orC _n overk.

On finite products of soluble groups

Let the finite groupG =AB be the product of two soluble subgroupsA andB, and letπ be a set of primes. We investigate under which conditions for the maximal normalπ-subgroups ofA, B andG the following holds:O _π (G) ∩O _π (G) ⊆O _π (G). The second author would like to thank the Department of Mathematics of the University of Mainz and the Mathematische Forschungsinstitut Oberwolfach for their excellent hospitality during the preparation of this paper.     

Non-weight representations of Cartan type S Lie algebras

For the two Cartan type S subalgebras of the Witt algebra 𝒲_n, called Lie algebras of divergence-zero vector fields, we determine all module structures on the universal enveloping algebra of their Cartan subalgebra 𝔥_n. We also give all submodules of these modules.     

Generalized Poisson brackets and lie algebras of type H in characteristic 0

The Lie algebra of Cartan type H which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials . We show in this paper that these generalizations of Cartan type H algebras are isomorphic to certain generalizations of the classical algebra of Poisson brackets, and that it can be generalized further. In turn, these algebras can be recast in a form that is an adaption of a class of Lie algebras of characteristic p that was defined in 1958 be R. Block. A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, find their derivations, and determine all possible isomorphisms between two of these algebras. Received December 20, 1996; in final form September 15, 1997     

Cartan determinants and Shapovalov forms

We compute the determinant of the Gram matrix of the Shapovalov form on weight spaces of the basic representation of an affine Kac-Moody algebra of ADE type (possibly twisted). As a consequence, we obtain explicit formulae for the determinants of the Cartan matrices of p-blocks of the symmetric group and its double cover, and of the associated Hecke algebras at roots of unity. Received: 20 November 2001 / Revised version: 7 February 2002 / Published online: 27 June 2002 Second author partially supported by the NSF (grant no. DMS-9900134).     

Casimir Operators and Monodromy Representations of Generalised Braid Groups

Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections ∇_κ on h with values in any finite-dimensional g-module V and simple poles on the root hyperplanes. The corresponding monodromyre presentation of the braid group B_g of type g is a deformation of the action of (afinite extension of) W on V. The residues of ∇_κ are the Casimirs κ_α of the subalgebra ssl^α₂ ⊂ g corresponding to the roots of g. The irreducibility of a subspace U ⊂= V under the κ_α implies that, for generic values of the parameter, the braid group B_g acts irreducibly on U. Answering a question of Knutson and Procesi, we show that these Casimirs act irreducibly on the weight spaces of all simple g-modules if g = sl₃ but that this is not the case if g ≇ sl₂,sl₃. We use this to disprove a conjecture of Kwon and Lusztig stating the irreducibility of quantum Weyl group actions of Artin’s braid group B_n on the zero-weight spaces of all simple Usl_n-modules for n≥4. Finally, we study their reducibility of the action of the Casimirs on the zero-weight spaces of self-dual g-modules and obtain complete classification results for g = sl_n and g₂.     

On the structure of graded <Emphasis Type="Italic">λ</Emphasis>-Hopf algebras

Let G be an abelian group, B the G-graded λ-Hopf algebra with A being a bicharacter on G. By introducing some new twisted algebras （coalgebras）, we investigate the basic properties of the graded antipode and the structure for B. We also prove that a G-graded λ-Hopf algebra can be embedded in a usual Hopf algebra. As an application, it is given that if G is a finite abelian group then the graded antipode of a finite dimensional G-graded A-Hopf algebra is invertible.     

Multiplier Hopf Algebras in Categories and the Biproduct Construction

Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct. Presented by K. Goodearl.     

On Poset Boolean Algebras

Let (P,≤) be a partially ordered set. The poset Boolean algebra of P, denoted F(P), is defined as follows: The set of generators of F(P) is {x _p  : p∈P}, and the set of relations is {x _p ⋅x _q =x _p  : p≤q}. We say that a Boolean algebra B is well-generated, if B has a sublattice G such that G generates B and (G,≤ ^B |G) is well-founded. A well-generated algebra is superatomic. THEOREM 1. Let (P,≤) be a partially ordered set. The following are equivalent. (i) P does not contain an infinite set of pairwise incomparable elements, and P does not contain a subset isomorphic to the chain of rational numbers, (ii) F(P) is superatomic, (iii) F(P) is well-generated. The equivalence (i) ⇔ (ii) is due to M. Pouzet. A partially ordered set W is well-ordered, if W does not contain a strictly decreasing infinite sequence, and W does not contain an infinite set of pairwise incomparable elements. THEOREM 2. Let F(P) be a superatomic poset algebra. Then there are a well-ordered set W and a subalgebra B of F(W), such that F(P) is a homomorphic image of B. This is similar but weaker than the fact that every interval algebra of a scattered chain is embeddable in an ordinal algebra. Remember that an interval algebra is a special case of a poset algebra. This revised version was published online in June 2006 with corrections to the Cover Date.     

Extending involutions on Frobenius algebras

 Let A be a central simple algebra of degree n over a field of characteristic different from 2 and let B ? A be a maximal commutative subalgebra. We show that if there is an involution on A that preserves B and such that the socle of each local component of B is a homogeneous C ₂ -module for this action, then B is a Frobenius algebra. For a fixed commutative Frobenius algebra B of finite dimension n equipped with an involution σ, we characterize the central simple algebras A of degree n that contain B and carry involutions extending σ. Received: 29 October 2001 / Revised version: 2 February 2002     

The initial algebra of maximal minors of a generalized hankel matrix

One deals with a certain generalization of the (generic) Hankel matrix. Fixing a field K, for a suitable diagonal term order on the polynomial ring B over K generated by the entries of this matrix, one considers the K-subalgebra A ? B generated by the set of initial terms of the the maximal minors of the matrix. The algebra A detects the underlying combinatorics in terms of certain generalized standard tableaux and these give a way of showing that A is normal. A more involved technique also yields that the Rees algebra of the ideal of B generated by the generators of A is normal. In particular, both algebras are Cohen-Macaulay. We also present a Gröbner basis for the presentation ideal of A and compute some numerical invariants using the simplicial complex associated to A.     

A basis for the symplectic group branching algebra

The symplectic group branching algebra, B\mathcal {B}, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp_2n−2(ℂ) in each finite-dimensional irreducible representation of Sp_2n (ℂ). By describing on B\mathcal {B} an ASL structure, we construct an explicit standard monomial basis of B\mathcal {B} consisting of Sp_2n−2(ℂ) highest weight vectors. Moreover, B\mathcal {B} is known to carry a canonical action of the n-fold product SL₂×⋯×SL₂, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi algebras we describe a deformation of Spec(B)\mathrm{Spec}(\mathcal {B}) into an explicitly described toric variety.     

Finiteness properties of chevalley groups overFq[t]

LetG be a simple Chevalley group of rankn and Γ=G( ). Then the finiteness length of Γ shall be determined by studying the action of Γ on the Bruhat-Tits buildingX ofG . This is always possible provided that certain subcomplexes of the links of simplices inX are spherical. As a consequence, one obtains that Γ is of typeF _n−1 but not of typeFP _n ifG is of typeA _n, B_n, C_n orD _n andq≥2²ⁿ⁻¹.     

K-theory of twisted differential operators on flag varieties

Let ? be a semisimple Lie algebra over k, an algebraically closed field of characteristic zero, and let ?⊂? be a Cartan subalgebra inside a Borel subalgebra of ?. Let ? be the enveloping algebra of ?. For μ∈? ^* let M(μ) denote the corresponding Verma module and let ? _μ = ?/ Ann  M(μ). Let W be the Weyl group and let W ⁰ _μ be the stabiliser of μ in W. We prove the following theorem, which affirms a conjecture of T.J. Hodges. Oblatum 16-XII-1994