共查询到20条相似文献,搜索用时 15 毫秒
1.
Roman M. Fedorov 《Selecta Mathematica, New Series》2010,16(2):241-266
We study some non-highest weight modules over an affine Kac–Moody algebra
[^(\mathfrak g)]{\hat{\mathfrak g}} at the non-critical level. Roughly speaking, these modules are non-commutative localizations of some non-highest weight “vacuum”
modules. Using free field realization, we embed some rings of differential operators in endomorphism rings of our modules.
These rings of differential operators act on a localization of the space of coinvariants of any
[^(\mathfrak g)]{\hat{\mathfrak g}}-module with respect to a certain level subalgebra. In a particular case this action is identified with the Casimir connection. 相似文献
2.
We prove a formula for the twining characters of certain Demazure modules, over a Borel subalgebra
\mathfrakb\mathfrak{b} of a finite dimensional complex semisimple Lie algebra
\mathfrakg\mathfrak{g}. This formula describes the twining character of the Demazure module by the w\omega-Demazure operator associated to an element of the Weyl group that is fixed by the Dynkin diagram automorphism w\omega of
\mathfrakg\mathfrak{g}. Our result is a refinement of the twining character formula for the irreducible highest weight
\mathfrakg\mathfrak{g}-modules of symmetric dominant integral highest weights, and also of the ordinary Demazure character formula. 相似文献
3.
We generalize a result of Kostant and Wallach concerning the algebraic integrability of the Gelfand-Zeitlin vector fields
to the full set of strongly regular elements in
\mathfrakg\mathfrakl \mathfrak{g}\mathfrak{l} (n, ℂ). We use decomposition classes to stratify the strongly regular set by subvarieties XD {X_\mathcal{D}} . We construct an étale cover
[^(\mathfrakg)]D {\hat{\mathfrak{g}}}_\mathcal{D} of XD {X_\mathcal{D}} and show that XD {X_\mathcal{D}} and
[^(\mathfrakg)]D {\hat{\mathfrak{g}}}_\mathcal{D} are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on XD {X_\mathcal{D}} to Hamiltonian vector fields on
[^(\mathfrakg)]D {\hat{\mathfrak{g}}}_\mathcal{D} and integrate these vector fields to an action of a connected, commutative algebraic group. 相似文献
4.
Alexander Premet 《Inventiones Mathematicae》2010,181(2):395-420
Let ${\mathfrak{g}}Let
\mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field
\mathbbK\mathbb{K} of characteristic 0. Let
\mathfrakg\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of
\mathfrakg{\mathfrak{g}} and
\mathfrakg\Bbbk=\mathfrakg\mathbbZ?\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where
\Bbbk\Bbbk is the algebraic closure of
\mathbbFp{\mathbb{F}}_{p}. Let
G\BbbkG_{\Bbbk} be a simple, simply connected algebraic
\Bbbk\Bbbk-group with
\operatornameLie(G\Bbbk)=\mathfrakg\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra
U(\mathfrakg\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for
\mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions
\Bbbk (\mathfrakg\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield
\Bbbk(\mathfrakg\Bbbk)G\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions
\mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield
\mathbbK(\mathfrakg)\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if
\mathfrakg{\mathfrak{g}} is of type B
n
, n≥3, D
n
, n≥4, E6, E7, E8 or F4. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the
above types. In other words, if
\mathfrakg{\mathfrak{g}} is of type B
n
, n≥3, D
n
, n≥4, E6, E7, E8 or F4, then the Lie field of
\mathfrakg{\mathfrak{g}} is more complicated than expected. 相似文献
5.
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. 相似文献
6.
Alexey V. Petukhov 《Transformation Groups》2011,16(4):1173-1182
Let
\mathfrakg \mathfrak{g} be a reductive Lie algebra and
\mathfrakk ì \mathfrakg \mathfrak{k} \subset \mathfrak{g} be a reductive in
\mathfrakg \mathfrak{g} subalgebra. A (
\mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-module M is a
\mathfrakg \mathfrak{g} -module for which any element m ∈ M is contained in a finite-dimensional
\mathfrakk \mathfrak{k} -submodule of M. We say that a (
\mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-module M is bounded if there exists a constant C
M
such that the Jordan-H?lder multiplicities of any simple finite-dimensional
\mathfrakk \mathfrak{k} -module in every finite-dimensional
\mathfrakk \mathfrak{k} -submodule of M are bounded by C
M
. In the present paper we describe explicitly all reductive in
\mathfraks\mathfrakln \mathfrak{s}{\mathfrak{l}_n} subalgebras
\mathfrakk \mathfrak{k} which admit a bounded simple infinite-dimensional (
\mathfraks\mathfrakln,\mathfrakk \mathfrak{s}{\mathfrak{l}_n},\mathfrak{k} )-module. Our technique is based on symplectic geometry and the notion of spherical variety. We also characterize the irreducible
components of the associated varieties of simple bounded (
\mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-modules. 相似文献
7.
8.
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
\mathfrakgx {\mathfrak{g}^x} in a semisimple Lie algebra
\mathfrakg \mathfrak{g} . 相似文献
9.
10.
D. V. Millionshchikov 《Mathematical Notes》2005,77(1):61-71
The cohomology H
\mathfrakg\mathfrak{g}
) of the tangent Lie algebra
\mathfrakg\mathfrak{g}
of the group G with coefficients in the one-dimensional representation
\mathfrakg\mathfrak{g}
\mathbbK\mathbb{K}
defined by
[(W)\tilde] \mathfrakg \tilde \Omega _\mathfrak{g}
of H
1(G/
\mathfrakg\mathfrak{g}
. 相似文献
11.
L. Magnin 《Algebras and Representation Theory》2010,13(6):723-753
For any complex 6-dimensional nilpotent Lie algebra
\mathfrakg,\mathfrak{g}, we compute the strain of all indecomposable 7-dimensional nilpotent Lie algebras which contain
\mathfrakg\mathfrak{g} by the adjoining a derivation method. We get a new determination of all 7-dimensional complex nilpotent Lie algebras, allowing
to check earlier results (some contain errors), along with a cross table intertwining nilpotent 6- and 7-dimensional Lie algebras. 相似文献
12.
By using the concept of weight graph associated to nonsplit complex nilpotent Lie algebras \mathfrakg\mathfrak{g}, we find necessary and sufficient conditions for a semidirect product \mathfrakg?? Ti\mathfrak{g}\overrightarrow{\oplus } T_{i} to be two-step solvable, where $T_{i}TT over \mathfrakg\mathfrak{g} which induces a decomposition of \mathfrakg\mathfrak{g} into one-dimensional weight spaces without zero weights. In particular we show that the semidirect product of such a Lie algebra with a maximal torus of derivations cannot be itself two-step solvable. We also obtain some applications to rigid Lie algebras, as a geometrical proof of the nonexistence of two-step nonsplit solvable rigid Lie algebras in dimensions n\geqslant 3n\geqslant 3. 相似文献
13.
M. T. Tarashchanskii 《Ukrainian Mathematical Journal》2011,62(9):1476-1486
We consider a relationship between two sets of extensions of a finite finitely additive measure μ defined on an algebra
\mathfrakB \mathfrak{B} of sets to a broader algebra
\mathfrakA \mathfrak{A} . These sets are the set ex S
μ
of all extreme extensions of the measure μ and the set H
μ
of all extensions defined as
l(A) = [^(m)]( h(A) ), A ? \mathfrakA \lambda (A) = \hat{\mu }\left( {h(A)} \right),\,\,\,A \in \mathfrak{A} , where [^(m)] \hat{\mu } is a quotient measure on the algebra
\mathfrakB