共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
3.
Christopher Kennedy 《Algebras and Representation Theory》2011,14(2):317-339
A deep matrix algebra,
DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}), is a unital associative algebra over a field
\mathbbK\mathbb{K} with basis all deep matrix units,
\mathfrake(h,k)\mathfrak{e}(h,k), indexed by pairs of elements h and k taken from a free monoid generated by a set X. After briefly describing the construction of
DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}), we determine necessary and sufficient conditions for constructing representations for
DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}). With these conditions in place, we define null modules and give three canonical examples of such. A classification of general
null modules is then given in terms of the canonical examples along with their submodules and quotients. In the final section,
additional examples of natural actions for
DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}) are given and their submodules determined depending on the cardinality of the set X. 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.
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
6.
7.
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
8.
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
9.
Miroslav Jerković 《The Ramanujan Journal》2012,27(3):357-376
Exact sequences of Feigin–Stoyanovsky’s type subspaces for affine Lie algebra
\mathfraksl(l+1,\mathbbC)[\tilde]\mathfrak{sl}(l+1,\mathbb{C})^{\widetilde{}} lead to systems of recurrence relations for formal characters of those subspaces. By solving the corresponding system for
\mathfraksl(3,\mathbbC)[\tilde]\mathfrak{sl}(3,\mathbb{C})^{\widetilde{}}, we obtain a new family of character formulas for all Feigin–Stoyanovsky’s type subspaces at the general level. 相似文献
10.
V. G. Puzarenko 《Algebra and Logic》2010,49(4):340-353
\mathfrakc \mathfrak{c} -Universal semilattices
\mathfrakA \mathfrak{A} of the power of the continuum (of an upper semilattice of m-degrees ) on admissible sets are studied. Moreover, it is shown that a semilattice of
\mathbbH\mathbbF( \mathfrakM ) \mathbb{H}\mathbb{F}\left( \mathfrak{M} \right) -numberings of a finite set is
\mathfrakc \mathfrak{c} -universal if
\mathfrakM \mathfrak{M} is a countable model of a c-simple theory. 相似文献
11.
I.V. Arzhantsev E. A. Makedonskii A. P. Petravchuk 《Ukrainian Mathematical Journal》2011,63(5):827-832
Let W
n
(
\mathbb K {\mathbb K} ) be the Lie algebra of derivations of the polynomial algebra
\mathbb K {\mathbb K} [X] :=
\mathbb K {\mathbb K} [x
1,…,x
n
]over an algebraically closed field
\mathbb K {\mathbb K} of characteristic zero. A subalgebra
L í Wn(\mathbbK) L \subseteq {W_n}(\mathbb{K}) is called polynomial if it is a submodule of the
\mathbb K {\mathbb K} [X]-module W
n
(
\mathbb K {\mathbb K} ). We prove that the centralizer of every nonzero element in L is abelian, provided that L is of rank one. This fact allows one to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one. 相似文献
12.
We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms:
Hence the Lie algebras \(\chi (\sinh {u})\) and χ(eu + e??2u) are slowly linearly growing Lie algebras with average growth rates \(\frac {3}{2}\) and \(\frac {4}{3}\) respectively. 相似文献
- 1)the first one is between the characteristic Lie algebra \(\chi (\sinh {u})\) of the sinh-Gordon equation \(u_{xy}=\sinh {u}\) and the non-negative part \({\mathcal {L}}({\mathfrak {sl}}(2,{\mathbb {C}}))^{\ge 0}\) of the loop algebra of \({\mathfrak {sl}}(2,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{1}^{(1)}\)$$\chi(\sinh{u})\cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(2,{\mathbb{C}}))^{\ge 0}={\mathfrak{s}\mathfrak{l}}(2, {\mathbb{C}}) \otimes {\mathbb{C}}[t]. $$
- 2)the second isomorphism is for the Tzitzeica equation uxy = eu + e??2uwhere \({\mathcal {L}}({\mathfrak {sl}}(3,{\mathbb {C}}), \mu )=\bigoplus _{j \in {\mathbb {Z}}}{\mathfrak {g}}_{j (\text {mod} \; 2)} \otimes t^{j}\) is the twisted loop algebra of the simple Lie algebra \({\mathfrak {sl}}(3,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{2}^{(2)}\).$$\chi(e^{u}{+}e^{-2u}) \cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(3,{\mathbb{C}}), \mu)^{\ge0}=\bigoplus_{j = 0}^{+\infty}{\mathfrak{g}}_{j (\text{mod} \; 2)} \otimes t^{j}, $$
13.
14.
If
\mathfrakA{\mathfrak{A}} is a unital weak-* closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property
\mathbbA1(1){\mathbb{A}_1(1)}, then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for
a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We
also show that the multiplier algebra of a complete NP space has
\mathbbA1(1){\mathbb{A}_1(1)}, and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies,
in particular, to all unital weak-* closed subalgebras of H
∞ acting on Hardy space or on Bergman space. 相似文献
15.
S. V. Hudzenko 《Ukrainian Mathematical Journal》2010,62(7):1158-1162
We consider a semigroup
FP\textfin+ ( \mathfrakS\textfin( \mathbbN ) ) 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+ ( \mathfrakS\textfin( \mathbbN ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) are induced by permutations from
\mathfrakS( \mathbbN ) \mathfrak{S}\left( \mathbb{N} \right) . 相似文献
16.
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
17.
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
18.
Let G be a finite soluble group and
F\mathfrakX(G) {\Phi_\mathfrak{X}}(G) an intersection of all those maximal subgroups M of G for which
G