共查询到20条相似文献,搜索用时 46 毫秒
1.
A. I. Molev 《Selecta Mathematica, New Series》2005,12(1):1-38
Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including
the quantized algebra of functions on GLN and the Yangian for
$$ \mathfrak{g}\mathfrak{l}_{{N}} $$ . We prove a version of this theorem for the twisted Yangians
$$ {\text{Y(}}\mathfrak{g}_{N} {\text{)}} $$associated with the orthogonal and symplectic Lie algebras
$$ \mathfrak{g}_{N} = \mathfrak{o}_{N} {\text{ or }}\mathfrak{s}\mathfrak{p}_{N} $$. This gives rise to representations of
the twisted Yangian
$$ {\text{Y}}{\left( {\mathfrak{g}_{{N - M}} } \right)} $$ on the space of homomorphisms
$$ {\text{Hom}}_{{\mathfrak{g}_{M} }} {\left( {W,V} \right)} $$, where W and V are finite-dimensional irreducible modules over
$$ \mathfrak{g}_{{M}} {\text{ and }}\mathfrak{g}_{{N}} $$, respectively. In the symplectic case these representations turn
out to be irreducible and we identify them by calculating the corresponding Drinfeld polynomials.We also apply the quantum
Sylvester theorem to realize the twisted Yangian as a projective limit of certain centralizers in universal enveloping algebras. 相似文献
2.
A. I. Molev 《Selecta Mathematica, New Series》2006,12(1):1-38
Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including
the quantized algebra of functions on GL
N
and the Yangian for
$$ \mathfrak{g}\mathfrak{l}_{{N}} $$ . We prove a version of this theorem for the twisted Yangians
$$ {\text{Y(}}\mathfrak{g}_{N} {\text{)}} $$associated with the orthogonal and symplectic Lie algebras
$$ \mathfrak{g}_{N} = \mathfrak{o}_{N} {\text{ or }}\mathfrak{s}\mathfrak{p}_{N} $$. This gives rise to representations of
the twisted Yangian
$$ {\text{Y}}{\left( {\mathfrak{g}_{{N - M}} } \right)} $$ on the space of homomorphisms
$$ {\text{Hom}}_{{\mathfrak{g}_{M} }} {\left( {W,V} \right)} $$, where W and V are finite-dimensional irreducible modules over
$$ \mathfrak{g}_{{M}} {\text{ and }}\mathfrak{g}_{{N}} $$, respectively. In the symplectic case these representations turn
out to be irreducible and we identify them by calculating the corresponding Drinfeld polynomials.We also apply the quantum
Sylvester theorem to realize the twisted Yangian as a projective limit of certain centralizers in universal enveloping algebras. 相似文献
3.
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
4.
Daniele Mundici 《Milan Journal of Mathematics》2011,79(2):643-656
In a recent paper, F. Boca investigates the AF algebra
\mathfrakA{{\mathfrak{A}}} associated with the Farey-Stern-Brocot sequence. We show that
\mathfrakA{{\mathfrak{A}}} coincides with the AF algebra
\mathfrakM1{{\mathfrak{M_{1}}}} introduced by the present author in 1988. As proved in that paper (Adv. Math., vol.68.1), the K
0-group of
\mathfrakA{\mathfrak{A}} is the lattice-ordered abelian group M1{\mathcal{M}_{1}} of piecewise linear functions on the unit interval, each piece having integer coefficients, with the constant 1 as the distinguished
order unit. Using the elementary properties of M1{\mathcal{M}_{1}} we can give short proofs of several results in Boca’s paper. We also prove many new results: among others,
\mathfrakA{{\mathfrak{A}}} is a *-subalgebra of Glimm universal algebra, tracial states of
\mathfrakA{{\mathfrak{A}}} are in one-one correspondence with Borel probability measures on the unit real interval, all primitive ideals of
\mathfrakA{{\mathfrak{A}}} are essential. We describe the automorphism group of
\mathfrakA{{\mathfrak{A}}} . For every primitive ideal I of
\mathfrakA{{{\mathfrak{A}}}} we compute K
0(I) and
K0(\mathfrakA/I){{K_{0}(\mathfrak{A}/I)}}. 相似文献
5.
It is shown how, the classification of those Lie superalgebras having
\mathfraku2{\mathfrak{u}_2} as its underlying Lie algebra, and further restricted by the fact that the representation of
\mathfraku2{\mathfrak{u}_2} into its odd module is the adjoint representation, leads to at least ten different natural superhomogeneous models for Minkowski
superspacetime. 相似文献
6.
S. Ole Warnaar 《Acta Mathematica》2009,203(2):269-304
A new q-binomial theorem for Macdonald polynomials is employed to prove an A
n
analogue of the celebrated Selberg integral. This confirms the
\mathfrakg = An \mathfrak{g} ={\rm{A}}_{n} case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra
\mathfrakg \mathfrak{g} . 相似文献
7.
Christopher Kennedy 《Algebras and Representation Theory》2011,14(6):1187-1202
This paper continues the study of associative and Lie deep matrix algebras,
DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) and
\mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras,
BDM(X,\mathbbK){\mathcal{BDM}}(X,{\mathbb{K}}) and
\mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}), are defined and studied for an infinite set X. The global structures of these two algebras are studied, devising a depth grading on both as well as determining their ideal
lattices. In particular,
\mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) is shown to be semisimple. The Lie algebra
\mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) possesses a deep Cartan decomposition and is locally finite with every finite subalgebra naturally enveloped by a semi-direct
product of
\mathfraksln{\mathfrak{{sl}_n}}’s. We classify all associative bilinear forms on
\mathfraksl2\mathfrakd{\mathfrak{sl}_2\mathfrak{d}} (a natural depth analogue of
\mathfraksl2{\mathfrak{{sl}_2}}) and
\mathfrakbld{\mathfrak{bld}}. 相似文献
8.
Boris Feigin Michael Finkelberg Andrei Negut Leonid Rybnikov 《Selecta Mathematica, New Series》2011,17(3):573-607
Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety
of GL
n
. We construct the action of the Yangian of
\mathfraksln{\mathfrak{sl}_n} in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric
deformation of the universal enveloping algebra of the universal central extension of
\mathfraksln[s±1,t]{\mathfrak{sl}_n[s^{\pm1},t]}) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine
Yangian in the fixed point basis of cohomology. This basis is an affine analog of the Gelfand-Tsetlin basis. The affine analog
of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology
ring of the moduli space
\mathfrakMn,d{\mathfrak{M}_{n,d}} of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is naturally embedded into the cohomology ring of certain affine Laumon space. It is the image
of the center Z of the Yangian of
\mathfrakgln{\mathfrak{gl}_n} naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on
\mathfrakMn,d{\mathfrak{M}_{n,d}} is the image of a noncommutative power sum in Z. 相似文献
9.
Julianna Tymoczko 《Journal of Algebraic Combinatorics》2012,35(4):611-632
Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic
operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and
\mathfrak sl2\mathfrak {sl}_{2}. Building on a result of Kuperberg, Khovanov–Kuperberg found a recursive algorithm giving a bijection between standard Young
tableaux of shape 3×n and irreducible webs for
\mathfraksl3\mathfrak{sl}_{3} whose boundary vertices are all sources. 相似文献
10.
Hiroaki Minami 《Archive for Mathematical Logic》2010,49(4):501-518
We investigate splitting number and reaping number for the structure (ω)
ω
of infinite partitions of ω. We prove that
\mathfrakrd £ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and
\mathfraksd 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency
\mathfrakrd < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and
\mathfraksd < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants
\mathfrakrpair{\mathfrak{r}_{pair}} and
\mathfrakspair{\mathfrak{s}_{pair}} . We also study the relation between
\mathfrakrpair, \mathfrakspair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that
cov(M),cov(N) £ \mathfrakrpair £ \mathfraksd,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and
\mathfraks £ \mathfrakspair £ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} . 相似文献
11.
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
12.
Clément de Seguins Pazzis 《Archiv der Mathematik》2010,95(4):333-342
When
\mathbbK{\mathbb{K}} is an arbitrary field, we study the affine automorphisms of
Mn(\mathbbK){{\rm M}_n(\mathbb{K})} that stabilize
GLn(\mathbbK){{\rm GL}_n(\mathbb{K})}. Using a theorem of Dieudonné on maximal affine subspaces of singular matrices, this is easily reduced to the known case
of linear preservers when n > 2 or # ${\mathbb{K} > 2}${\mathbb{K} > 2}. We include a short new proof of the more general Flanders theorem for affine subspaces of
Mp,q(\mathbbK){{\rm M}_{p,q}(\mathbb{K})} with bounded rank. We also find that the group of affine transformations of
M2(\mathbbF2){{\rm M}_2(\mathbb{F}_2)} that stabilize
GL2(\mathbbF2){{\rm GL}_2(\mathbb{F}_2)} does not consist solely of linear maps. Using the theory of quadratic forms over
\mathbbF2{\mathbb{F}_2}, we construct explicit isomorphisms between it, the symplectic group
Sp4(\mathbbF2){{\rm Sp}_4(\mathbb{F}_2)} and the symmetric group
\mathfrakS6{\mathfrak{S}_6}. 相似文献
13.
We propose a notion of algebra of twisted chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess
families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra
of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an
affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of “smallest”
such modules are irreducible
[^(\mathfrakg)]{{\hat{{\mathfrak{g}}}}} -modules, and all irreducible
\mathfrakg{{\mathfrak{g}}} -integrable
[^(\mathfrakg)]{{\hat{{\mathfrak{g}}}}} -modules at the critical level arise in this way. 相似文献
14.
15.
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
16.
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
17.
Giorgia Bellomonte Camillo Trapani Salvatore Triolo 《Mediterranean Journal of Mathematics》2010,7(1):63-74
The existence of extensions of a positive linear functional ω defined on a dense *-subalgebra
\mathfrakA0{\mathfrak{A}_0} of a topological *-algebra
\mathfrakA{\mathfrak{A}}, satisfying certain regularity conditions, is examined. The main interest is focused on the case where ω is nonclosable and sufficient conditions for the existence of an absolutely convergent extension of ω are given. 相似文献
18.
Martín Mombelli 《Mathematische Zeitschrift》2010,266(2):319-344
We develop some techniques for studying exact module categories over some families of pointed finite-dimensional Hopf algebras.
As an application we classify exact module categories over the tensor category of representations of the small quantum groups
uq(\mathfraksl2){u_q(\mathfrak{sl}_2)}. 相似文献
19.
Helmut Zöschinger 《Archiv der Mathematik》2010,95(3):225-231
Let
(R,\mathfrak m){(R,\mathfrak m)} be a noetherian, local ring with completion [^(R)]{\hat{R}} . We show that R ì [^(R)]{R \subset \hat{R}} satisfies the condition Going up if and only if there exists to every artinian R-module M with
AnnR(M) ì \mathfrakp{{\rm Ann}_R(M) \subset \mathfrak{p}} a submodule U ì M{U \subset M} with
AnnR(U)=\mathfrakp.{{\rm {Ann}}_R(U)=\mathfrak{p}.} This is further equivalent to R being formal catenary, to α(R) = 0 and to
Hd\mathfrakq/\mathfrakp(R/\mathfrakp)=0{H^d_{\mathfrak{q}/\mathfrak{p}}(R/\mathfrak{p})=0} for all prime ideals
\mathfrakp ì \mathfrakq \subsetneq \mathfrakm{\mathfrak{p} \subset \mathfrak{q} \subsetneq \mathfrak{m}} where
d = dim(R/\mathfrakp){d = {\rm {dim}}(R/\mathfrak{p})}. 相似文献
20.
Alexander Premet 《Transformation Groups》2011,16(3):857-888
Let
U( \mathfrakg,e ) U\left( {\mathfrak{g},e} \right) be the finite W-algebra associated with a nilpotent element e in a complex simple Lie algebra
\mathfrakg = \textLie(G) \mathfrak{g} = {\text{Lie}}(G) and let I be a primitive ideal of the enveloping algebra
U( \mathfrakg ) U\left( \mathfrak{g} \right) whose associated variety equals the Zariski closure of the nilpotent orbit (Ad G) e. Then it is known that
I = \textAn\textnU( \mathfrakg )( Qe ?U( \mathfrakg,e )V ) I = {\text{An}}{{\text{n}}_{U\left( \mathfrak{g} \right)}}\left( {{Q_e}{ \otimes_{U\left( {\mathfrak{g},e} \right)}}V} \right) for some finite dimensional irreducible
U( \mathfrakg,e ) U\left( {\mathfrak{g},e} \right) -module V, where Q
e
stands for the generalised Gelfand–Graev
\mathfrakg \mathfrak{g} -module associated with e. The main goal of this paper is to prove that the Goldie rank of the primitive quotient
U( \mathfrakg )