共查询到20条相似文献,搜索用时 46 毫秒
1.
Denote by γ the Gauss measure on ℝ
n
and by ${\mathcal{L}}${\mathcal{L}} the Ornstein–Uhlenbeck operator. In this paper we introduce a Hardy space
\mathfrak h1g{{\mathfrak{h}}^1}{{\rm \gamma}} of Goldberg type and show that for each u in ℝ ∖ {0} and r > 0 the operator ( rI+ L) iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is unbounded from
\mathfrak h1g{{\mathfrak{h}}^1}{{\rm \gamma}} to L
1γ. This result is in sharp contrast both with the fact that ( rI+ L) iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is bounded from H
1γ to L
1γ, where H
1γ denotes the Hardy type space introduced in Mauceri and Meda (J Funct Anal 252:278–313, 2007), and with the fact that in the Euclidean case ( rI-D) iu(r{\mathcal{I}}-\Delta)^{iu} is bounded from the Goldberg space
\mathfrak h1\mathbb Rn{{\mathfrak{h}}^1}{{\mathbb{R}}^n} to L
1ℝ
n
. We consider also the case of Riemannian manifolds M with Riemannian measure μ. We prove that, under certain geometric assumptions on M, an operator T{\mathcal{T}}, bounded on L
2
μ, and with a kernel satisfying certain analytic assumptions, is bounded from H
1
μ to L
1
μ if and only if it is bounded from
\mathfrak h1m{{\mathfrak{h}}^1}{\mu} to L
1
μ. Here H
1
μ denotes the Hardy space introduced in Carbonaro et al. (Ann Sc Norm Super Pisa, 2009), and
\mathfrak h1m{{\mathfrak{h}}^1}{\mu} is defined in Section 4, and is equivalent to a space recently introduced by M. Taylor (J Geom Anal 19(1):137–190, 2009). The case of translation invariant operators on homogeneous trees is also considered. 相似文献
2.
Let G be a reductive algebraic group over an algebraically closed field K of characteristic zero. Let
p:\mathfrak gr ? X = \mathfrak gr// G \pi :{\mathfrak{g}^r} \to X = {\mathfrak{g}^r}//G be the categorical quotient where
\mathfrak g \mathfrak{g} is the adjoint representation of G and r is a suitably large integer (in general r ≥ 5, but for many cases r ≥ 3 or even r ≥ 2 suffices). We show that every automorphism φ of X lifts to a map
F:\mathfrak gr ? \mathfrak gr \Phi :{\mathfrak{g}^r} \to {\mathfrak{g}^r} commuting with π. As an application we consider the action of φ on the Luna stratification of X. 相似文献
3.
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 | / |
m {{\mathfrak{B}} \left/ {\mu } \right.} of the classes of μ-equivalence and
h:\mathfrakA ? \mathfrakB | / |
m h:\mathfrak{A} \to {{\mathfrak{B}} \left/ {\mu } \right.} is a homomorphism extending the canonical homomorphism
\mathfrakB \mathfrak{B} to
\mathfrakB | / |
m {{\mathfrak{B}} \left/ {\mu } \right.} . We study the properties of extensions from H
μ
and present necessary and sufficient conditions for the existence of these extensions, as well as the conditions under which
the sets ex S
μ
and H
μ
coincide. 相似文献
5.
Let e be a nilpotent element of a complex simple Lie algebra $ \mathfrak{g} Let e be a nilpotent element of a complex simple Lie algebra
\mathfrakg \mathfrak{g} . The weighted Dynkin diagram of e, D(e) \mathcal{D}(e) , is said to be divisible if D(e) | / |
2 {{{\mathcal{D}(e)}} \left/ {2} \right.} is again a weighted Dynkin diagram. The corresponding pair of nilpotent orbits is said to be friendly. In this paper we classify
the friendly pairs and describe some of their properties. Any subalgebra
\mathfraks\mathfrakl3 \mathfrak{s}{\mathfrak{l}_3} in
\mathfrakg \mathfrak{g} gives rise to a friendly pair; such pairs are called A2-pairs. If Gx is the lower orbit in an A2-pair, then
x ? [ \mathfrakgx,\mathfrakgx ] x \in \left[ {{\mathfrak{g}^x},{\mathfrak{g}^x}} \right] , i.e., x is reachable. We also show that
\mathfrakgx {\mathfrak{g}^x} has other interesting properties. Let
\mathfrakgx = ?i \geqslant 0\mathfrakgx(i) {\mathfrak{g}^x} = { \oplus_{i \geqslant 0}}{\mathfrak{g}^x}(i) be the
\mathbbZ - \textgrading \mathbb{Z} - {\text{grading}} determined by a characteristic of x. We prove that
\mathfrakgx {\mathfrak{g}^x} is generated by the Levi subalgebra
\mathfrakgx(0) {\mathfrak{g}^x}(0) and two elements of
\mathfrakgx(1) {\mathfrak{g}^x}(1) . In particular, the nilpotent radical of
\mathfrakgx {\mathfrak{g}^x} is generated by the subspace
\mathfrakgx(1) {\mathfrak{g}^x}(1) . 相似文献
6.
Let
\mathfrak a \mathfrak{a} be a finite-dimensional Lie algebra and
Y( \mathfrak a ) Y\left( \mathfrak{a} \right) the
\mathfrak a \mathfrak{a} invariant subalgebra of its symmetric algebra
S( \mathfrak a ) S\left( \mathfrak{a} \right) under adjoint action. Recently there has been considerable interest in studying situations when
Y( \mathfrak a ) Y\left( \mathfrak{a} \right) may be polynomial on index
\mathfrak a \mathfrak{a} generators, for example if
\mathfrak a \mathfrak{a} is a biparabolic or a centralizer
\mathfrak gx {\mathfrak{g}^x} in a semisimple Lie algebra
\mathfrak g \mathfrak{g} . 相似文献
7.
Let V be a 2 m-dimensional symplectic vector space over an algebraically closed field K. Let $ \mathfrak{B}_n^{(f)} Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let
\mathfrakBn(f) \mathfrak{B}_n^{(f)} be the two-sided ideal of the Brauer algebra
\mathfrakBn( - 2m ) {\mathfrak{B}_n}\left( { - 2m} \right) over K generated by e
1
e
3⋯
e
2f-1 where 0 ≤ f ≤ [n/2]. Let HTf ?n \mathcal{H}\mathcal{T}_f^{ \otimes n} be the subspace of partial-harmonic tensors of valence f in V
⊗n
. In this paper we prove that dimHTf ?n \mathcal{H}\mathcal{T}_f^{ \otimes n} and dim
\textEn\textdK\textSp(V)( V ?n \mathord | / |
\vphantom V ?n V ?n V ?n\mathfrakBn(f) ) {\text{En}}{{\text{d}}_{K{\text{Sp}}(V)}}\left( {{{{V^{ \otimes n}}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}} {{V^{ \otimes n}}}}} \right.} {{V^{ \otimes n}}}}\mathfrak{B}_n^{(f)}} \right) are both independent of K, and the natural homomorphism from
\mathfrakBn( - 2m ) \mathord | / |
\vphantom ( - 2m ) \mathfrakBn(f) \mathfrakBn(f) {\mathfrak{B}_n}{{\left( { - 2m} \right)} \mathord{\left/{\vphantom {{\left( { - 2m} \right)} {\mathfrak{B}_n^{(f)}}}} \right.} {\mathfrak{B}_n^{(f)}}} to
\textEn\textdK\textSp(V)( V ?n \mathord | / |
\vphantom V ?n V ?n V ?n\mathfrakBn(f) ) {\text{En}}{{\text{d}}_{K{\text{Sp}}(V)}}\left( {{{{V^{ \otimes n}}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}} {{V^{ \otimes n}}}}} \right.} {{V^{ \otimes n}}}}\mathfrak{B}_n^{(f)}} \right) is always surjective. We show that HTf ?n \mathcal{H}\mathcal{T}_f^{ \otimes n} has a Weyl filtration and is isomorphic to the dual of
V ?n\mathfrakBn(f) \mathord | / |
\vphantom V ?n\mathfrakBn(f) V V ?n\mathfrakBn( f + 1 ) {{{{V^{ \otimes n}}\mathfrak{B}_n^{(f)}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}\mathfrak{B}_n^{(f)}} V}} \right.} V}^{ \otimes n}}\mathfrak{B}_n^{\left( {f + 1} \right)} as an
\textSp(V) - ( \mathfrakBn( - 2m ) \mathord | / |
\vphantom ( - 2m ) \mathfrakBn( f + 1 ) \mathfrakBn( f + 1 ) ) {\text{Sp}}(V) - \left( {{\mathfrak{B}_n}{{\left( { - 2m} \right)} \mathord{\left/{\vphantom {{\left( { - 2m} \right)} {\mathfrak{B}_n^{\left( {f + 1} \right)}}}} \right.} {\mathfrak{B}_n^{\left( {f + 1} \right)}}}} \right) -bimodule. We obtain an
\textSp(V) - \mathfrakBn {\text{Sp}}(V) - {\mathfrak{B}_n} -bimodules filtration of V
⊗n
such that each successive quotient is isomorphic to some
?( l) ?zg,l\mathfrakBn \nabla \left( \lambda \right) \otimes {z_{g,\lambda }}{\mathfrak{B}_n} with λ ⊢ n 2g, ℓ(λ)≤m and 0 ≤ g ≤ [n/2], where ∇(λ) is the co-Weyl module associated to λ and z
g,λ is an explicitly constructed maximal vector of weight λ. As a byproduct, we show that each right
\mathfrakBn {\mathfrak{B}_n} -module
zg,l\mathfrakBn {z_{g,\lambda }}{\mathfrak{B}_n} is integrally defined and stable under base change. 相似文献
8.
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
9.
We investigate splitting number and reaping number for the structure ( ω)
ω
of infinite partitions of ω. We prove that
\mathfrak rd £ non( M), non( N),\mathfrak d{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and
\mathfrak sd 3 \mathfrak b{\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
\mathfrak rd < add( M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and
\mathfrak sd < cof( M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants
\mathfrak rpair{\mathfrak{r}_{pair}} and
\mathfrak spair{\mathfrak{s}_{pair}} . We also study the relation between
\mathfrak rpair, \mathfrak spair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that
cov( M), cov( N) £ \mathfrak rpair £ \mathfrak sd,\mathfrak r{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and
\mathfrak s £ \mathfrak spair £ non( M), non( N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} . 相似文献
10.
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
Ann R( M) ì \mathfrak p{{\rm Ann}_R(M) \subset \mathfrak{p}} a submodule U ì M{U \subset M} with
Ann R( U)=\mathfrak p.{{\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/\mathfrak p)=0{H^d_{\mathfrak{q}/\mathfrak{p}}(R/\mathfrak{p})=0} for all prime ideals
\mathfrak p ì \mathfrak q \subsetneq \mathfrak m{\mathfrak{p} \subset \mathfrak{q} \subsetneq \mathfrak{m}} where
d = dim( R/\mathfrak p){d = {\rm {dim}}(R/\mathfrak{p})}. 相似文献
11.
A complete Boolean algebra
\mathbb B{\mathbb{B}}satisfies property (( h/ 2p)){(\hbar)}iff each sequence x in
\mathbb B{\mathbb{B}}has a subsequence y such that the equality lim sup z
n
= lim sup y
n
holds for each subsequence z of y. This property, providing an explicit definition of the a posteriori convergence in complete Boolean algebras with the sequential
topology and a characterization of sequential compactness of such spaces, is closely related to the cellularity of Boolean
algebras. Here we determine the position of property (( h/ 2p)){(\hbar)}with respect to the hierarchy of conditions of the form κ-cc. So, answering a question from Kurilić and Pavlović (Ann Pure Appl Logic 148(1–3):49–62, 2007), we show that ${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}is not a theorem of ZFC and that there is no cardinal
\mathfrak k{\mathfrak{k}}, definable in ZFC, such that ${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}is a theorem of ZFC. Also, we show that the set { k: each k-cc c.B.a. has (( h/ 2p) ) }{\{ \kappa : {\rm each}\, \kappa{\rm -cc\, c.B.a.\, has}\, (\hbar ) \}}is equal to
[0, \mathfrak h){[0, \mathfrak{h})}or
[0, \mathfrak h]{[0, {\mathfrak h}]}and that both values are consistent, which, with the known equality
{k: each c.B.a. having (( h/ 2p) ) has the k-cc } = [\mathfrak s, ¥){{\{\kappa : {\rm each\, c.B.a.\, having }\, (\hbar )\, {\rm has\, the}\, \kappa {\rm -cc } \} =[{\mathfrak s}, \infty )}}completes the picture. 相似文献
12.
The entanglement characteristics of two qubits are encoded in the invariants of the adjoint action of the group SU(2) ⊗ SU(2)
on the space of density matrices
\mathfrak P+ {\mathfrak{P}_{+} } , defined as the space of 4 × 4 positive semidefinite Hermitian matrices. The corresponding ring
\text C[ \mathfrak P+ ] \textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} of polynomial invariants is studied. A special integrity basis for
\text C[ \mathfrak P+ ] \textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} is described, and the constraints on its elements imposed by the positive semidefiniteness of density matrices are given
explicitly in the form of polynomial inequalities. The suggested basis is characterized by the property that the minimum number
of invariants, namely, two primary invariants of degree 2, 3 and one secondary invariant of degree 4 appearing in the Hironaka
decomposition of
\text C[ \mathfrak P+ ] \textSU( 2 ) ?\textSU ?( 2 ) {\text{C}}{\left[ {{\mathfrak{P}_{+} }} \right]^{{\text{SU}}\left( {2} \right) \otimes {\text{SU}} \otimes \left( {2} \right)}} , are subject to the polynomial inequalities. Bibliography: 32 titles. 相似文献
13.
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
\mathfrak g\mathfrak l \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
[^(\mathfrak g)] D {\hat{\mathfrak{g}}}_\mathcal{D} of XD {X_\mathcal{D}} and show that XD {X_\mathcal{D}} and
[^(\mathfrak g)] 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
[^(\mathfrak g)] D {\hat{\mathfrak{g}}}_\mathcal{D} and integrate these vector fields to an action of a connected, commutative algebraic group. 相似文献
14.
We study a type of left-invariant structure on Lie groups or, equivalently, on Lie algebras. We introduce obstructions to
the existence of a hypo structure, namely the five-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3).
The choice of a splitting
\mathfrak g* = V1 ? V2 {\mathfrak{g}^*} = {V_1} \oplus {V_2} , and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions
for the existence of a hypo structure with a fixed almost-contact form. For nonunimodular Lie algebras, we derive an obstruction
to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that
admit a hypo structure. 相似文献
15.
Let H*( Be ) {H^*}\left( {{\mathcal{B}_e}} \right) be the total Springer representation of W for the nilpotent element e in a simple Lie algebra
\mathfrak g \mathfrak{g} . Let Λ
i
V denote the ith exterior power of the reflection representation V of W. The focus of this paper is on the algebra of W-invariants in
H*( Be ) ?L*V {H^*}\left( {{\mathcal{B}_e}} \right) \otimes {\Lambda^*}V 相似文献
16.
To any field
\Bbb K \Bbb K of characteristic zero, we associate a set
(\mathbb K) (\mathbb{K}) and a group
G0(\Bbb K) {\cal G}_0(\Bbb K) . Elements of
(\mathbb K) (\mathbb{K}) are equivalence classes of families of Lie polynomials subject to associativity relations. Elements of
G0(\Bbb K) {\cal G}_0(\Bbb K) are universal automorphisms of the adjoint representations of Lie bialgebras over
\Bbb K \Bbb K . We construct a bijection between
(\mathbb K)× G0(\Bbb K) (\mathbb{K})\times{\cal G}_0(\Bbb K) and the set of quantization functors of Lie bialgebras over
\Bbb K \Bbb K . This construction involves the following steps.? 1) To each element v \varpi of
(\mathbb K) (\mathbb{K}) , we associate a functor
\frak a?\operatorname Shv(\frak a) \frak a\mapsto\operatorname{Sh}^\varpi(\frak a) from the category of Lie algebras to that of Hopf algebras;
\operatorname Shv(\frak a) \operatorname{Sh}^\varpi(\frak a) contains
U\frak a U\frak a .? 2) When
\frak a \frak a and
\frak b \frak b are Lie algebras, and
r\frak a\frak b ? \frak a?\frak b r_{\frak a\frak b} \in\frak a\otimes\frak b , we construct an element
? v ( r\frak a\frak b) {\cal R}^{\varpi} (r_{\frak a\frak b}) of
\operatorname Shv(\frak a)?\operatorname Shv(\frak b) \operatorname{Sh}^\varpi(\frak a)\otimes\operatorname{Sh}^\varpi(\frak b) satisfying quasitriangularity identities; in particular,
? v( r\frak a\frak b) {\cal R}^\varpi(r_{\frak a\frak b}) defines a Hopf algebra morphism from
\operatorname Shv(\frak a) * \operatorname{Sh}^\varpi(\frak a)^* to
\operatorname Shv(\frak b) \operatorname{Sh}^\varpi(\frak b) .? 3) When
\frak a = \frak b \frak a = \frak b and
r\frak a ? \frak a?\frak a r_\frak a\in\frak a\otimes\frak a is a solution of CYBE, we construct a series
r v( r\frak a) \rho^\varpi(r_\frak a) such that
? v(r v( r\frak a)) {\cal R}^\varpi(\rho^\varpi(r_\frak a)) is a solution of QYBE. The expression of
r v( r\frak a) \rho^\varpi(r_\frak a) in terms of
r\frak a r_\frak a involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing
statements for cohomologies arising from universal algebras for the solutions of CYBE.? 4) We define the quantization of a
Lie bialgebra
\frak g \frak g as the image of the morphism defined by ? v(r v( r)) {\cal R}^\varpi(\rho^\varpi(r)) , where
r ? \mathfrak g ?\mathfrak g* r \in \mathfrak{g} \otimes \mathfrak{g}^* .<\P> 相似文献
17.
Let R be a noetherian ring,
\mathfrak a{\mathfrak{a}} an ideal of R, and M an R-module. We prove that for a finite module M, if
H i\mathfraka( M){{\rm H}^{i}_{\mathfrak{a}}(M)} is minimax for all i ≥ r ≥ 1, then
H i\mathfraka( M){{\rm H}^{i}_{\mathfrak{a}}(M)} is artinian for i ≥ r. A local–global principle for minimax local cohomology modules is shown. If
H i\mathfraka( M){{\rm H}^{i}_{\mathfrak{a}}(M)} is coatomic for i ≤ r ( M finite) then
H i\mathfraka( M){{\rm H}^{i}_{\mathfrak{a}}(M)} is finite for i ≤ r. We give conditions for a module which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing
theorems are proved for local cohomology modules. 相似文献
18.
By using the concept of weight graph associated to nonsplit complex nilpotent Lie algebras \mathfrak g\mathfrak{g}, we find necessary and sufficient conditions for a semidirect product \mathfrak g? ? 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
19.
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 ) | / |
I {{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.} always divides dim V. For
\mathfrakg = \mathfraks\mathfrakln \mathfrak{g} = \mathfrak{s}{\mathfrak{l}_n} , we use a theorem of Joseph on Goldie fields of primitive quotients of
U( \mathfrakg ) U\left( \mathfrak{g} \right) to establish the equality
\textrk( U( \mathfrakg ) | / |
I ) = dimV {\text{rk}}\left( {{{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.}} \right) = \dim V . We show that this equality continues to hold for
\mathfrakg \ncong \mathfraks\mathfrakln \mathfrak{g} \ncong \mathfrak{s}{\mathfrak{l}_n} provided that the Goldie field of
U( \mathfrakg ) | / |
I {{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.} is isomorphic to a Weyl skew-field and use this result to disprove Joseph’s version of the Gelfand–Kirillov conjecture formulated
in the mid-1970s. 相似文献
20.
In this paper, it is shown that the dual
[(\text Qord)\tilde]\mathfrak A \widetilde{\text{Qord}}\mathfrak{A} of the quasiorder lattice of any algebra
\mathfrak A \mathfrak{A} is isomorphic to a sublattice of the topology lattice
á( \mathfrak A ) \Im \left( \mathfrak{A} \right) . Further, if
\mathfrak A \mathfrak{A} is a finite algebra, then
[(\text Qord)\tilde]\mathfrak A @ á( \mathfrak A ) \widetilde{\text{Qord}}\mathfrak{A} \cong \Im \left( \mathfrak{A} \right) . We give a sufficient condition for the lattices
[(\text Con)\tilde]\mathfrak A\text, [(\text Qord)\tilde]\mathfrak A \widetilde{\text{Con}}\mathfrak{A}{\text{,}} \widetilde{\text{Qord}}\mathfrak{A} , and
á( \mathfrak A ) \Im \left( \mathfrak{A} \right) . to be pairwise isomorphic. These results are applied to investigate topology lattices and quasiorder lattices of unary algebras. 相似文献
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