共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
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
4.
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
6.
We introduce a spanning set of Beilinson–Lusztig–MacPherson type, { A( j, r)}
A,j
, for affine quantum Schur algebras
S\vartriangle( n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and construct a linearly independent set { A( j)}
A,j
for an associated algebra
[^( K)] \vartriangle( n){{{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n)} . We then establish explicitly some multiplication formulas of simple generators
E\vartriangleh,h+1( 0){E^\vartriangle_{h,h+1}}(\mathbf{0}) by an arbitrary element A( j) in
[^( K)] \vartriangle( n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle(n)}} via the corresponding formulas in
S\vartriangle( n, r){{{\boldsymbol{\mathcal S}}_\vartriangle(n, r)}} , and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the Ringel–Hall
algebras
\mathfrak H\vartriangle( n){{{\boldsymbol{\mathfrak H}_\vartriangle(n)}}} associated with cyclic quivers. This allows us to use the triangular relation between monomial and PBW type bases for
\mathfrak H\vartriangle( n){{\boldsymbol{\mathfrak H}}_\vartriangle}(n) established in Deng and Du (Adv Math 191:276–304, 2005) to derive similar triangular relations for
S\vartriangle( n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and
[^( K)] \vartriangle( n){{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n) . Using these relations, we then show that the subspace
\mathfrak A\vartriangle( n){{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)} of
[^( K)] \vartriangle( n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} spanned by { A( j)}
A,j
contains the quantum enveloping algebra
U\vartriangle( n){{{\mathbf U}_\vartriangle}(n)} of affine type A as a subalgebra. As an application, we prove that, when this construction is applied to quantum Schur algebras S( n, r){\boldsymbol{\mathcal S}(n,r)} , the resulting subspace
\mathfrak A\vartriangle( n){{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}} is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of
\mathfrak gln{\mathfrak{gl}_n} . We conjecture that
\mathfrak A\vartriangle( n){{{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}}} is a subalgebra of
[^( K)] \vartriangle( n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} . 相似文献
7.
This paper continues the study of associative and Lie deep matrix algebras,
DM( X,\mathbb K){\mathcal{DM}}(X,{\mathbb{K}}) and
\mathfrak gld( X,\mathbb K){\mathfrak{gld}}(X,{\mathbb{K}}), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras,
BDM( X,\mathbb K){\mathcal{BDM}}(X,{\mathbb{K}}) and
\mathfrak bld( X,\mathbb K){\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,
\mathfrak bld( X,\mathbb K){\mathfrak{bld}}(X,{\mathbb{K}}) is shown to be semisimple. The Lie algebra
\mathfrak bld( X,\mathbb K){\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
\mathfrak sln{\mathfrak{{sl}_n}}’s. We classify all associative bilinear forms on
\mathfrak sl2\mathfrak d{\mathfrak{sl}_2\mathfrak{d}} (a natural depth analogue of
\mathfrak sl2{\mathfrak{{sl}_2}}) and
\mathfrak bld{\mathfrak{bld}}. 相似文献
8.
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
9.
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 quantum loop algebra
Uv( L\mathfrak sln){U_v({\bf L}\mathfrak{sl}_n)} in the K-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra
ü
v
([^(\mathfrak sl)] n){(\widehat{\mathfrak{sl}}_n)} in the K-theory of the affine version of Laumon spaces. 相似文献
10.
Let
\mathfrak g \mathfrak{g} be a reductive Lie algebra and
\mathfrak k ì \mathfrak g \mathfrak{k} \subset \mathfrak{g} be a reductive in
\mathfrak g \mathfrak{g} subalgebra. A (
\mathfrak g,\mathfrak k \mathfrak{g},\mathfrak{k} )-module M is a
\mathfrak g \mathfrak{g} -module for which any element m ∈ M is contained in a finite-dimensional
\mathfrak k \mathfrak{k} -submodule of M. We say that a (
\mathfrak g,\mathfrak k \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
\mathfrak k \mathfrak{k} -module in every finite-dimensional
\mathfrak k \mathfrak{k} -submodule of M are bounded by C
M
. In the present paper we describe explicitly all reductive in
\mathfrak s\mathfrak ln \mathfrak{s}{\mathfrak{l}_n} subalgebras
\mathfrak k \mathfrak{k} which admit a bounded simple infinite-dimensional (
\mathfrak s\mathfrak ln,\mathfrak k \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 (
\mathfrak g,\mathfrak k \mathfrak{g},\mathfrak{k} )-modules. 相似文献
11.
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
12.
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
\mathfrak sln{\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
\mathfrak sln[ 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
\mathfrak Mn,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
\mathfrak gln{\mathfrak{gl}_n} naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on
\mathfrak Mn,d{\mathfrak{M}_{n,d}} is the image of a noncommutative power sum in Z. 相似文献
13.
We consider a semigroup
FP\textfin+ ( \mathfrak S\textfin( \mathbb N ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) defined as a finitary factor power of a finitary symmetric group of countable order. It is proved that all automorphisms
of
FP\textfin+ ( \mathfrak S\textfin( \mathbb N ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) are induced by permutations from
\mathfrak S( \mathbb N ) \mathfrak{S}\left( \mathbb{N} \right) . 相似文献
15.
Let ( X m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of ${\mathbb{R}^{m}}Let (X
m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of
\mathbbRm{\mathbb{R}^{m}} . The Legendrian Low conjecture formulated by Natário and Tod says that two events x, y ? X{x, y \in X} are causally related if and only if the Legendrian link of spheres
\mathfrakSx, \mathfrakSy{{\mathfrak{S}_x,\,\mathfrak{S}_y}} whose points are light geodesics passing through x and y is non-trivial in the contact manifold of all light geodesics in X. The Low conjecture says that for m = 2 the events x, y are causally related if and only if
\mathfrakSx, \mathfrakSy{{\mathfrak{S}_x,\,\mathfrak{S}_y}} is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements
hold for any globally hyperbolic (X
m+1, g) such that a cover of its Cauchy surface is diffeomorphic to an open domain in
\mathbbRm{\mathbb{R}^{m}} . 相似文献
16.
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
17.
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
18.
A string is a pair ( L, \mathfrak m){(L, \mathfrak{m})} where L ? [0, ¥]{L \in[0, \infty]} and \mathfrak m{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrak m{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L2(\mathfrak m){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator - D\mathfrakmDx{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as f¢(x) + z ò0L f(y) d\mathfrakm(y) = 0, x ? \mathbb R, f¢(0-) = 0.f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0. 相似文献
19.
We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal
Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ 2≃ℂ. We describe all representations of
\mathfrak g\mathfrak{g}
via vector fields in J
0ℝ 2=ℝ 3( x, y, u), which project to the canonical representation, and find their algebra of scalar differential invariants. 相似文献
20.
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
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