共查询到20条相似文献,搜索用时 62 毫秒
1.
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
2.
Aleksander Tsymbaliuk 《Selecta Mathematica, New Series》2010,16(2):173-200
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\mathfraksln){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
([^(\mathfraksl)]n){(\widehat{\mathfrak{sl}}_n)} in the K-theory of the affine version of Laumon spaces. 相似文献
3.
Reflection equation algebras and related
Uq(\mathfrak g){U{_q}(\mathfrak g)} -comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or
equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of
such so called ‘covariantized’ algebras, in particular concerning their centres, invariants, and characters. The locally finite
part
Fl(Uq (\mathfrak g)){F_l(U{_q} (\mathfrak g))} of
Uq(\mathfrak g){U{_q}(\mathfrak g)} with respect to the left adjoint action is a special example of a covariantized algebra. Generalising Noumi’s construction
of quantum symmetric pairs we define a coideal subalgebra B
f
of
Uq(\mathfrak g){U{_q}(\mathfrak g)} for each character f of a covariantized algebra. We show that for any character f of
Fl(Uq(\mathfrak g)){F_l(U{_q}(\mathfrak g))} the centre Z(B
f
) canonically contains the representation ring
Rep(\mathfrak g){{\rm Rep}(\mathfrak g)} of the semisimple Lie algebra
\mathfrak g{\mathfrak g} . We show moreover that for
\mathfrak g = \mathfrak sln(\mathbb C){\mathfrak g = {\mathfrak sl}_n(\mathbb C)} such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit
realisations of
Rep(\mathfrak sln(\mathbb C)){{\rm Rep}({\mathfrak sl}_n(\mathbb C))} inside
Uq(\mathfrak sln(\mathbb C)){U_q({\mathfrak sl}_n(\mathbb C))} . As an example we discuss the solutions of the reflection equation corresponding to the Grassmannian manifold Gr(m,2m) of m-dimensional subspaces in
\mathbb C2m{{\mathbb C}^{2m}}. 相似文献
4.
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
5.
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
6.
The motivation for this paper comes from the Halperin–Carlsson conjecture for (real) moment-angle complexes. We first give
an algebraic combinatorics formula for the M?bius transform of an abstract simplicial complex K on [m]={1,…,m} in terms of the Betti numbers of the Stanley–Reisner face ring k(K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle
complexes
ZK(\mathbb D, \mathbb S)\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}, including the moment-angle complex ZK\mathcal{Z}_{K} and the real moment-angle complex
\mathbbRZK\mathbb{R}\mathcal {Z}_{K} as special examples. We show that
H*(ZK(\mathbb D, \mathbb S);k)H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k}) has the same graded k-module structure as Tor
k[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for ZK\mathcal{Z}_{K} (resp.
\mathbb RZK\mathbb{ R}\mathcal{Z}_{K}) under the restriction of the natural T
m
-action on ZK\mathcal{Z}_{K} (resp. (ℤ2)
m
-action on
\mathbb RZK\mathbb{ R}\mathcal{Z}_{K}). 相似文献
7.
Spiros A. Argyros Irene Deliyanni Andreas G. Tolias 《Israel Journal of Mathematics》2011,181(1):65-110
We provide a characterization of the Banach spaces X with a Schauder basis (e
n
)
n∈ℕ which have the property that the dual space X* is naturally isomorphic to the space L
diag(X) of diagonal operators with respect to (e
n
)
n∈ℕ. We also construct a Hereditarily Indecomposable Banach space $
\mathfrak{X}
$
\mathfrak{X}
D with a Schauder basis (e
n
)
n∈ℕ such that $
\mathfrak{X}
$
\mathfrak{X}
*D is isometric to L
diag($
\mathfrak{X}
$
\mathfrak{X}
D) with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every T ∈ L
diag($
\mathfrak{X}
$
\mathfrak{X}
D) is of the form T = λI + K, where K is a compact operator. 相似文献
8.
George E. Andrews 《Aequationes Mathematicae》1987,33(2-3):230-250
In this paper we consider a problem posed by W Mills D Robbins and H Rumsey for a certain plane partition generating functionZ n (x, m) The special caseZ n (1,m) is the generating function that arose in the weak Macdonald conjecture Mills—Robbins—Rumsey conjectured thatZ n (2,m) also possesses a nice finite product representation Their conjecture is proved as Theorem 1 The method of proof resembles that of the evaluation ofZ n (1,m) given previously Many results for the3 F 2 hypergeometric function are required including Whipple's theorem, the Pfaff-Saalschutz summation and contiguous relations In passing we note that our Lemma 2 provides a new and simpler representation ofZ n (2,m) as a determinant $$Z_n (2,m) = \det \left( {\delta _{ij} + \sum\limits_{i = 0}^l {\left( {\begin{array}{*{20}c} {m + j + t} \\ t \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {m + i} \\ {m + t} \\ \end{array} } \right)} } \right)_{0 \leqq ij \leqq n - 1} $$ Conceivably this new representation may provide new interpretations of the combinatorial significance ofZ n (2,m) In the final analysis, one would like a combinatorial explanation ofZ n (2,m) that would provide an algorithmic proof of the Mills Robbins—Rumsey conjecture 相似文献
9.
Boris Feigin Michael Finkelberg Igor Frenkel Leonid Rybnikov 《Selecta Mathematica, New Series》2011,17(2):337-361
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 calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand–Tsetlin subalgebra of U(gl
n
) and formulate a conjectural answer for the small quantum cohomology rings in terms of certain commutative shift of argument
subalgebras of U(gl
n
). 相似文献
10.
George E. Andrews 《Aequationes Mathematicae》1987,33(1):230-250
In this paper we consider a problem posed by W Mills D Robbins and H Rumsey for a certain plane partition generating functionZ n (x, m) The special caseZ n (1,m) is the generating function that arose in the weak Macdonald conjecture Mills—Robbins—Rumsey conjectured thatZ n (2,m) also possesses a nice finite product representation Their conjecture is proved as Theorem 1 The method of proof resembles that of the evaluation ofZ n (1,m) given previously Many results for the3 F 2 hypergeometric function are required including Whipple's theorem, the Pfaff-Saalschutz summation and contiguous relations In passing we note that our Lemma 2 provides a new and simpler representation ofZ n (2,m) as a determinant $$Z_n (2,m) = \det \left( {\delta _{ij} + \sum\limits_{t = 0}^1 {\left( {\mathop {m + j + t}\limits_t } \right)} \left( {\mathop {m + t}\limits_{m + t} } \right)} \right)_{0 \leqq ij \leqq n - 1} $$ Conceivably this new representation may provide new interpretations of the combinatorial significance ofZ n (2,m) In the final analysis, one would like a combinatorial explanation ofZ n (2,m) that would provide an algorithmic proof of the Mills Robbins—Rumsey conjecture 相似文献
11.
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 )