Slices for biparabolics of index 1

Let \mathfraka \mathfrak{a} be an algebraic Lie subalgebra of a simple Lie algebra \mathfrakg \mathfrak{g} with index \mathfraka \mathfrak{a}  ≤ rank \mathfrakg \mathfrak{g} . Let Y( \mathfraka ) Y\left( \mathfrak{a} \right) denote the algebra of \mathfraka \mathfrak{a} invariant polynomial functions on \mathfraka^* {\mathfrak{a}^*} . An algebraic slice for \mathfraka \mathfrak{a} is an affine subspace η + V with h ? \mathfraka^* \eta \in {\mathfrak{a}^*} and V ì \mathfraka^* V \subset {\mathfrak{a}^*} subspace of dimension index \mathfraka \mathfrak{a} such that restriction of function induces an isomorphism of Y( \mathfraka ) Y\left( \mathfrak{a} \right) onto the algebra R[η + V] of regular functions on η + V. Slices have been obtained in a number of cases through the construction of an adapted pair (h, η) in which h ? \mathfraka h \in \mathfrak{a} is ad-semisimple, η is a regular element of \mathfraka^* {\mathfrak{a}^*} which is an eigenvector for h of eigenvalue minus one and V is an h stable complement to ( \textad  \mathfraka )h \left( {{\text{ad}}\;\mathfrak{a}} \right)\eta in \mathfraka^* {\mathfrak{a}^*} . The classical case is for \mathfrakg \mathfrak{g} semisimple [16], [17]. Yet rather recently many other cases have been provided; for example, if \mathfrakg \mathfrak{g} is of type A and \mathfraka \mathfrak{a} is a “truncated biparabolic” [12] or a centralizer [13]. In some of these cases (in particular when the biparabolic is a Borel subalgebra) it was found [13], [14], that η could be taken to be the restriction of a regular nilpotent element in \mathfrakg \mathfrak{g} . Moreover, this calculation suggested [13] how to construct slices outside type A when no adapted pair exists. This article makes a first step in taking these ideas further. Specifically, let \mathfraka \mathfrak{a} be a truncated biparabolic of index one. (This only arises if \mathfrakg \mathfrak{g} is of type A and \mathfraka \mathfrak{a} is the derived algebra of a parabolic subalgebra whose Levi factor has just two blocks whose sizes are coprime.) In this case it is shown that the second member of an adapted pair (h, η) for \mathfraka \mathfrak{a} is the restriction of a particularly carefully chosen regular nilpotent element of \mathfrakg \mathfrak{g} . A by-product of our analysis is the construction of a map from the set of pairs of coprime integers to the set of all finite ordered sequences of ±1.     

Parabolic Actions in type A and their Eigenslices

The main theme of this paper is that many of the remarkable properties of invariant theory pertaining to semisimple Lie algebras carry over to parabolic subalgebras even though the latter have less structure. This includes the polynomiality of the invariant subalgebra of the symmetric algebra of a (truncated) parabolic subalgebra, the existence of a slice to the regular coadjoint orbits and the construction of maximal Poisson commutative polynomial subalgebras by "shift of argument". The first of these properties was established for most parabolics in [FJ1]. Here the existence of a slice to (most) regular coadjoint orbits is established for parabolics in type A which are invariant under the Dynkin diagram involution. In a subsequent paper [JL] maximal Poisson commutative polynomial subalgebras are described for those (truncated) parabolics in sl(n) having index n - 1.     

THE CLASSIFICATION OF CONVEX ORDERS ON AFFINE ROOT SYSTEMS

We classify all total orders with a convex property on the positive root system of an arbitrary untwisted affine Lie algebra g. Such total orders are called convex orders and are used to construct convex bases of Poincaré-Birkhoff-Witt type of the upper triangular subalgebra U_q ⁺ of the quantized universal enveloping algebra U_q (g).     

Kac–Moody groups and cluster algebras

Let Q be a finite quiver without oriented cycles, let Λ be the associated preprojective algebra, let g be the associated Kac–Moody Lie algebra with Weyl group W, and let n be the positive part of g. For each Weyl group element w, a subcategory Cw of mod(Λ) was introduced by Buan, Iyama, Reiten and Scott. It is known that Cw is a Frobenius category and that its stable category is a Calabi–Yau category of dimension two. We show that Cw yields a cluster algebra structure on the coordinate ring C[N(w)] of the unipotent group N(w):=N∩(w−1N₋w). Here N is the pro-unipotent pro-group with Lie algebra the completion of n. One can identify C[N(w)] with a subalgebra of , the graded dual of the universal enveloping algebra U(n) of n. Let S^? be the dual of Lusztig?s semicanonical basis S of U(n). We show that all cluster monomials of C[N(w)] belong to S^?, and that S^?∩C[N(w)] is a C-basis of C[N(w)]. Moreover, we show that the cluster algebra obtained from C[N(w)] by formally inverting the generators of the coefficient ring is isomorphic to the algebra C[Nw] of regular functions on the unipotent cell Nw of the Kac–Moody group with Lie algebra g. We obtain a corresponding dual semicanonical basis of C[Nw]. As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.     

Slice regular functions on real alternative algebras

In this paper we develop a theory of slice regular functions on a real alternative algebra A. Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in A and we prove a Cauchy integral formula for slice functions of class C¹.     

Generalized Verma modules over some Block algebras

In this paper, a class of generalized Verma modules M(V) over some Block type Lie algebra ℬ(G) are constructed, which are induced from nontrivial simple modules V over a subalgebra of ℬ(G). The irreducibility of M(V) is determined.      

Cohomologies of infinite-dimensional Borel algebras

Let G be an affine Lie algebra and B be its Borel subalgebra. The chohomologies Hⁿ(B, B) of the adjoint representation are studied. The main result is as follows: If the algebra G is untwisted andn 2, then Hⁿ(B, B)=0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1278–1283, September, 1990.     

Verma type modules for toroidal lie algebras

We study the structure of imaginary Verma modules induced from the"natural"Borel subalgebra of a toroidal Lie algebra. In particular, we establish a criterion of irreducibility for imaginary Verma modules and describe their submodules and irreducible quotients. We also describe the structure of Verma type modules in the case of sl(2)-toroidal Lie algebra over two variables.     

Integration of the lifting formulas and the cyclic homology of the algebras of differential operators

We integrate the Lifting cocycles Y_2n+1, Y_2n+3, Y_2n+5,? ([Sh1,2]) \Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,([\rm Sh1,2]) on the Lie algebra Dif_n of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle l \lambda on an n-dimensional complex manifold M in the sense of Gelfand--Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin--Tsygan theorem [FT1]:¶¶ H^·_Lie(\frak g\frak l^fin_￥(Dif_n);\Bbb C) = ù^·(Y_2n+1, Y_2n+3, Y_2n+5,? ) H^\bullet_{\rm Lie}({\frak g}{\frak l}^{\rm fin}_\infty({\rm Dif}_n);{\Bbb C}) = \wedge^\bullet(\Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,) .     

Projective functors and the restriction of a Verma module to a subalgebra of Levi type

We observe that the restriction of a Verma module over a semi-simple Lie algebra to a subalgebra of Levi type may be viewed as a projective functor. By simple arguments we prove that this restriction can be decomposed into a direct sum of standard indecomposables in the category O. For the restriction problem from sl(n+1) to gl(n) we describe the complete answer. We study the properties of the modules with Verma flag also and prove that any module with Verma flag is a submodule of some projective.     

Symmetric Pairs for Quantized Enveloping Algebras

Let θ be an involution of a semisimple Lie algebra g, let g^θ denote the fixed Lie subalgebra, and assume the Cartan subalgebra of g has been chosen in a suitable way. We construct a quantum analog of U(g^θ) which can be characterized as the unique subalgebra of the quantized enveloping algebra of g which is a maximal right coideal that specializes to U(g^θ).     

Affine Slice for the Coadjoint Action of a Class of Biparabolic Subalgebras of a Semisimple Lie Algebra

In this article, we give a simple explicit construction of an affine slice for the coadjoint action of a certain class of biparabolic (also called seaweed) subalgebras of a semisimple Lie algebra over an algebraically closed field of characteristic zero. In particular, this class includes all Borel subalgebras.     

Lie algebras associated to systems of Dyson–Schwinger equations

We consider systems of combinatorial Dyson–Schwinger equations in the Connes–Kreimer Hopf algebra HI of rooted trees decorated by a set I. Let H(S) be the subalgebra of HI generated by the homogeneous components of the unique solution of this system. If it is a Hopf subalgebra, we describe it as the dual of the enveloping algebra of a Lie algebra g(S) of one of the following types:

• 1. 

  g(S) is an associative algebra of paths associated to a certain oriented graph.

• 2. 

  Or g(S) is an iterated extension of the Faà di Bruno Lie algebra.

• 3. 

  Or g(S) is an iterated extension of an infinite-dimensional abelian Lie algebra.

We also describe the character groups of H(S).     

A Dual Approach to Constrained Interpolationfrom a Convex Subset of Hilbert Space

Many interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationP_K(x) to anyxXfrom the setKC∩A⁻¹(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andbY. The main point of this paper is to show thatP_K(x)isidenticaltoP_C(x+A*y)—the best approximationto a certain perturbationx+A*yofx—from the convexsetCor from a certain convex extremal subsetC_bofC. Thelatter best approximation is generally much easier to computethan the former. Prior to this, the result had been known onlyin the case of a convex cone or forspecialdata sets associatedwith a closed convex set. In fact, we give anintrinsic characterizationof those pairs of setsCandA⁻¹(b) for which this canalways be done. Finally, in many cases, the best approximationP_C(x+A*y) can be obtained numerically from existingalgorithms or from modifications to existing algorithms. Wegive such an algorithm and prove its convergence     

Polynomiality of invariants, unimodularity and adapted pairs

Let \mathfraka \mathfrak{a} be a finite-dimensional Lie algebra and Y( \mathfraka ) Y\left( \mathfrak{a} \right) the \mathfraka \mathfrak{a} invariant subalgebra of its symmetric algebra S( \mathfraka ) S\left( \mathfrak{a} \right) under adjoint action. Recently there has been considerable interest in studying situations when Y( \mathfraka ) Y\left( \mathfrak{a} \right) may be polynomial on index \mathfraka \mathfrak{a} generators, for example if \mathfraka \mathfrak{a} is a biparabolic or a centralizer \mathfrakg^x {\mathfrak{g}^x} in a semisimple Lie algebra \mathfrakg \mathfrak{g} .     

Local systems in simple finitary lie algebras

A Lie subalgebra L of gl_k(V) is said to be finitary if it consists of elements of finite rank. We show that every simple finitary Lie algebra over a field of characteristic ≠2, 3, 5, 7 has a local system consisting of perfect central extensions of finite-dimensional simple Lie algebras.     

BOTT–BOREL–WEIL THEORY AND BERNSTEIN–GEL’FAND–GEL’FAND RECIPROCITY FOR LIE SUPERALGEBRAS

The main focus of this paper is Bott–Borel–Weil (BBW) theory for basic classical Lie superalgebras. We take a purely algebraic self-contained approach to the problem. A new element in this study is twisting functors, which we use in particular to prove that the top of the cohomology groups of BBW theory for generic weights is described by the recently introduced star action. We also study the algebra of regular functions, related to BBW theory. Then we introduce a weaker form of genericness, relative to the Borel subalgebra and show that the virtual BGG reciprocity of Gruson and Serganova becomes an actual reciprocity in the relatively generic region. We also obtain a complete solution of BBW theory for \( \mathfrak{o}\mathfrak{s}\mathfrak{p} \)(m|2), D(2, 1; α), F(4) and G(3) with distinguished Borel subalgebra. Furthermore, we derive information about the category of finite-dimensional \( \mathfrak{o}\mathfrak{s}\mathfrak{p} \)(m|2)-modules, such as BGG-type resolutions and Kostant homology of Kac modules and the structure of projective modules.     

On the Quantum KPRV Determinants for Semisimple and Affine Lie Algebras

Let g be a semisimple or affine Lie algebra and U _q (g) its quantized enveloping algebra. Extending earlier work, the KPRV determinant for an admissible integrable U _q (g) module V relative to a parabolic subalgebra pg is defined and shown to be nonzero. These determinants had previously been evaluated for g semisimple and p a Borel subalgebra. The present results can be used to extend this to g affine as will be shown in a subsequent publication.For a parabolic subalgebra the evaluation of these determinants is much more difficult. For appropriate overalgebras of the primitive quotients of the enveloping algebra U(g) defined by one-dimensional representations of p, these determinants had been calculated for g semisimple. However the quantum case is interesting because it is unnecessary to pass to overalgebras and besides for U(g):g affine, it is not even clear how these determinants should be defined. Here for g semisimple, the degrees of the determinants are computed and shown to depend on being the same type of functions as in the enveloping algebra case; yet in a different fashion. Some special cases (in type A ₄) are computed explicity. Here, as in the Borel case, the determinants take a remarkably simple form and notably can be expressed as a product of linear factors. However compared to the enveloping algebra case one finds additional factors corresponding to what are called quantum zeros and whose origin remains unknown.     

Homomorphisms and composition operators on algebras of analytic functions of bounded type

Let U and V be convex and balanced open subsets of the Banach spaces X and Y, respectively. In this paper we study the following question: given two Fréchet algebras of holomorphic functions of bounded type on U and V, respectively, that are algebra isomorphic, can we deduce that X and Y (or X^* and Y^*) are isomorphic? We prove that if X^* or Y^* has the approximation property and H_wu(U) and H_wu(V) are topologically algebra isomorphic, then X^* and Y^* are isomorphic (the converse being true when U and V are the whole space). We get analogous results for H_b(U) and H_b(V), giving conditions under which an algebra isomorphism between H_b(X) and H_b(Y) is equivalent to an isomorphism between X^* and Y^*. We also obtain characterizations of different algebra homomorphisms as composition operators, study the structure of the spectrum of the algebras under consideration and show the existence of homomorphisms on H_b(X) with pathological behaviors.