The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite W-algebras

We address two problems with the structure and representation theory of finite W-algebras associated with general linear Lie algebras. Finite W-algebras can be defined using either Kostant's Whittaker modules or a quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of finite W-algebras. The second main result is a parameterization of finite families of irreducible Gelfand-Tsetlin modules using Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Gelfand-Tsetlin modules for finite W-algebras.     

Schubert varieties and short braidedness

LetW be a finite Weyl group. We give a characterization of those elements ofW whose reduced expressions avoid substrings of the formsts wheres andt are noncommuting generators. We give as an application a family of singular Schubert varieties.Supported in part by a NSF postdoctoral fellowship     

Symplectic leaves in real banach Lie–Poisson spaces

We present several large classes of real Banach Lie–Poisson spaces whose characteristic distributions are integrable, the integral manifolds being symplectic leaves just as in finite dimensions. We also investigate when these leaves are embedded submanifolds or when they have K?hler structures. Our results apply to the real Banach Lie–Poisson spaces provided by the self-adjoint parts of preduals of arbitrary W*-algebras, as well as of certain operator ideals. Received: April 2004 Accepted: September 2004     

Commutative quotients of finite W-algebras

In this paper we reduce the problem of 1-dimensional representations for the finite W-algebras and Humphreys' conjecture on small representations of reduced enveloping algebras to the case of rigid nilpotent elements in exceptional Lie algebras. We use Katsylo's results on sections of sheets to determine the Krull dimension of the largest commutative quotient of the finite W-algebra U(g,e).     

Finitely generated simple algebras: A question of B. I. Plotkin

In his recent series of lectures, Prof. B. I. Plotkin discussed geometrical properties of the variety of associativeK-algebras. In particular, he studied geometrically noetherian and logically noetherian algebras and, in this connection, he asked whether there exist uncountably many simpleK-algebras with a fixed finite number of generators. We answer this question in the affirmative using both crossed product constructions and HNN extensions of division rings. Specifically, we show that there exist uncountably many nonisomorphic 4-generator simple Ore domains, and also uncountably many nonisomorphic division algebras having 2 generators as a division algebra. The first author is grateful to Professor B. I. Plotkin for communicating this problem to him and for stimulating conversations.     

Ein globales Ergebnis für Bifurkationsaufgaben mit schwach differenzierbaren Operatoren

In this paper we are dealing with positive linear functionals on W-algebras. We introduce the notion of a positive linear functional with ∑-property (see Definition 1.1). It is shown that each positive linear functional on a W-algebra possesses the ∑-property. This fact allows to give a short proof of UHLMANN's conjecture on unitary mixing in the state space of a W-algebra. In proving our main theorem (see Theorem 1.2.) we obtain some results on positive linear functionals and orthoprojections which are useful in other context, too.     

Frobenius manifolds from regular classical W-algebras

We obtain polynomial Frobenius manifolds from classical W-algebras associated to regular nilpotent elements in simple Lie algebras using the related opposite Cartan subalgebras.     

A Hecke Algebra Quotient and Some Combinatorial Applications

Let (W, S) be a Coxeter group associated to a Coxeter graph which has no multiple bonds. Let H be the corresponding Hecke Algebra. We define a certain quotient \-H of H and show that it has a basis parametrized by a certain subset W _cof the Coxeter group W. Specifically, W _cconsists of those elements of W all of whose reduced expressions avoid substrings of the form sts where s and t are noncommuting generators in S. We determine which Coxeter groups have finite W _cand compute the cardinality of W _cwhen W is a Weyl group. Finally, we give a combinatorial application (which is related to the number of reduced expressions for w W _cof an exponential formula of Lusztig which utilizes a specialization of a subalgebra of \-H.     

I n -symmetrical Heyting algebras

In [11, p. 210] A. Monteiro suggested the possibility of generalizing the results of L. Iturrioz [4] to theI _n -symmetrical Heyting algebras which he namedI _n -algebras. We prove that these algebras are semi-simple. We characterize simple algebras and their subalgebras. Finally, we determine the structure of theI _n -algebra with a finite set of free generators and we give an answer to one of the problems posed by A. Monteiro.Presented by W. Taylor.Some of the results of this paper were presented at the Annual Meeting of the Union Matemática Argentina (September, 1987) ([14]).     

Lie p-algebras of finite p-subalgebra rank

We say that a Lie p-algebra L has finite p-subalgebra rank if the minimal number of generators required to generate every finitely generated p-subalgebra is uniformly bounded by some integer r. This paper is concerned with the following problem: does L being of finite p-subalgebra rank force ad(L) to be finite-dimensional? Although this seems unlikely in general, we show that this is indeed the case for Lie p-algebras in a large class including all locally, residually, and virtually soluble Lie p-algebras.     

Generalized Vertex Algebras Generated by Parafermion-Like Vertex Operators

It is proved that for any vector space W, any set of parafermion-like vertex operators on W in a certain canonical way generates a generalized vertex algebra in the sense of Dong and Lepowsky with W as a natural module. As an application, generalized vertex algebras are constructed from the Lepowsky–Wilson Z-algebras of any nonzero level.     

Chordal Coxeter groups

A solution of the isomorphism problem is presented for the class of Coxeter groups W that have a finite set of Coxeter generators S such that the underlying graph of the presentation diagram of the system (W,S) has the property that every cycle of length at least four has a chord. As an application, we construct counterexamples to two conjectures concerning the isomorphism problem for Coxeter groups.      

The Haagerup problem on subadditive weights on W*-algebras. II

In 1975 U. Haagerup has posed the following question: Whether every normal subadditive weight on a W*-algebra is σ-weakly lower semicontinuous? In 2011 the author has positively answered this question in the particular case of abelian W*-algebras and has presented a general form of normal subadditive weights on these algebras. Here we positively answer this question in the case of finite-dimensional W*-algebras. As a corollary, we give a positive answer for subadditive weights with some natural additional condition on atomic W*-algebras. We also obtain the general form of such normal subadditive weights and norms for wide class of normed solid spaces on atomic W*-algebras.     

Resolutions of 2 and 3 Dimensional Rings of Invariants for Cyclic Groups

Let G be the cyclic group of order n, and suppose F is a field containing a primitive nth root of unity. We consider the ring of invariants F[W] ^G of a three dimensional representation W of G where G ? SL(W). We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gröbner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of F[W] ^G . The case where W is any two dimensional representation of G is also handled.     

Reelle Jordanalgebren mit endlicher Spur

We study real Jordan algebras of arbitrary dimension which admit an associative, positive definite trace form and have suitable continuity properties. We construct certain completions until we arrive at a class of Jordan algebras corresponding to associative W^*-algebras of finite type. For these Jordan algebras we derive some basic properties concerning positive elements and idempotents. In contrast to other related investigations exceptional Jordan algebras are not excluded.     

Die struktur endlicher schwach abgeschlossener Jordanalgebren Teil II. Diskrete Jordanalgebren

In the preceding note [6] we reduced the study of continuous finite weakly closed Jordan algebras to real associative W^*-algebras of type II₁. Here we treat the remaining case of discrete finite weakly closed Jordan algebras and describe them completely by finite dimensional simple formally real Jordan algebras and by simple formally real Jordan algebras of quadratic forms of real Hilbert spaces. Jacobsons theory of Jordan algebras with minimum condition combined with W^*-algebra techniques constitutes an essential tool in the proof.     

Infinitesimal Generators of C *-Algebras

We develop the method introduced previously, to construct infinitesimal generators on locally compact group C ^*-algebras and on tensor product of C ^*-algebras. It is shown in particular that there is a C ^* -algebra A such that the C ^*-tensor product of A and an arbitrary C ^*-algebra B can have a non-approximately inner strongly one parameter group of ^*-automorphisms.     

The Algebraic K-Theory of Operator Algebras

We the study the algebraic K-theory of C *-algebras, forgetting the topology. The main results include a proof that commutative C*-algebras are K-regular in all degrees (that is, all theirN ^T K _iand extensions of the Fischer-Prasolov Theorem comparing algebraic and topological K-theory with finite coefficients.     

Homotopy invariance of AF-embeddability

We prove that AF-embeddability is a homotopy invariant in the class of separable exact C ^*-algebras. This work was inspired by Spielberg's work on homotopy invariance of AF-embeddability and Dadarlat's serial works on AF-embeddability of residually finite dimensional C ^*-algebras. Submitted: February 2002.