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高被引文献（top10）：

Record 1 of 10

Author(s): Altafini, C

Title: Controllability of quantum mechanical systems by root space decomposition of su(N)

Source: JOURNAL OF MATHEMATICAL PHYSICS, 43 (5): 2051-2062 MAY 2002

Abstract: The controllability property of the unitary propagator of an N-level quantum mechanical system subject to a single control field is described using the structure theory of semisimple Lie algebras. Sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases. (C) 2002 American Institute of Physics.

Times Cited: 45

DOI: 10.1063/1.1467611

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Record 2 of 10

Author(s): Beilinson, A; Ginzburg, V; Soergel, W

Title: Koszul duality patterns in representation theory

Source: JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, 9 (2): 473-527 APR 1996

Abstract: The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to representation theory. The paper consists of three parts relatively independent of each other.

The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain Z-graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show that the block of the Bernstein-Gelfand-Gelfand category O that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category O again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain categories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras.

Times Cited: 151

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Record 3 of 10

Author(s): KHOROSHKIN, SM; TOLSTOY, VN

Title: UNIVERSAL R-MATRIX FOR QUANTIZED (SUPER)ALGEBRAS

Source: COMMUNICATIONS IN MATHEMATICAL PHYSICS, 141 (3): 599-617 1991

Abstract: For quantum deformations of finite-dimensional contragredient Lie (super)algebras we give an explicit formula for the universal R-matrix. This formula generalizes the analogous formulae for quantized semisimple Lie algebras obtained by M. Rosso, A. N. Kirillov, and N. Reshetikhin, Ya. S. Soibelman, and S. Z. Levendorskii. Our approach is based on careful analysis of quantized rank 1 and 2 (super)algebras, a combinatorial structure of the root systems and algebraic properties of q-exponential functions. We don't use quantum Weyl group.

Times Cited: 142

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Record 4 of 10

Author(s): MARKUSHEVICH, DG; OLSHANETSKY, MA; PERELOMOV, AM

Title: DESCRIPTION OF A CLASS OF SUPERSTRING COMPACTIFICATIONS RELATED TO SEMISIMPLE LIE-ALGEBRAS

Source: COMMUNICATIONS IN MATHEMATICAL PHYSICS, 111 (2): 247-274 1987

Times Cited: 45

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Record 5 of 10

Author(s): HOTTA, R; KASHIWARA, M

Title: THE INVARIANT HOLONOMIC SYSTEM ON A SEMISIMPLE LIE-ALGEBRA

Source: INVENTIONES MATHEMATICAE, 75 (2): 327-358 1984

Times Cited: 51

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Record 6 of 10

Author(s): BERNSTEIN, JN; GELFAND, SI

Title: TENSOR-PRODUCTS OF FINITE AND INFINITE DIMENSIONAL REPRESENTATIONS OF SEMISIMPLE LIE-ALGEBRAS

Source: COMPOSITIO MATHEMATICA, 41 (2): 245-285 1980

Times Cited: 97

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Record 7 of 10

Author(s): JOSEPH, A

Title: PREPARATION THEOREM FOR PRIME SPECTRUM OF A SEMISIMPLE LIE-ALGEBRA

Source: JOURNAL OF ALGEBRA, 48 (2): 241-289 1977

Times Cited: 45

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Record 8 of 10

Author(s): BORHO, W; JANTZEN, JC

Title: PRIMITIVE IDEALS IN ENVELOPES OF SEMISIMPLE LIE-ALGEBRA

Source: INVENTIONES MATHEMATICAE, 39 (1): 1-53 1977

Times Cited: 85

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Record 9 of 10

Author(s): OLSHANETSKY, MA; PERELOMOV, AM

Title: COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS CONNECTED WITH SEMISIMPLE LIE-ALGEBRAS

Source: INVENTIONES MATHEMATICAE, 37 (2): 93-108 1976

Times Cited: 118

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Record 10 of 10

Author(s): VERMA, DN

Title: STRUCTURE OF CERTAIN INDUCED REPRESENTATIONS OF COMPLEX SEMISIMPLE LIE ALGEBRAS

Source: BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 74 (1): 160-& 1968

Times Cited: 51

最新文献（2009）：

Record 1 of 15

Author(s): Bolsinov, AV (Bolsinov, A. V.); Oshemkov, AA (Oshemkov, A. A.)

Title: Bi-Hamiltonian structures and singularities of integrable systems

Source: REGULAR & CHAOTIC DYNAMICS, 14 (4-5): 431-454 AUG 2009

Abstract: A Hamiltonian system on a Poisson manifold M is called integrable if it possesses sufficiently many commuting first integrals f (1), aEuro broken vertical bar f (s) which are functionally independent on M almost everywhere. We study the structure of the singular set K where the differentials df (1), aEuro broken vertical bar, df (s) become linearly dependent and show that in the case of bi-Hamiltonian systems this structure is closely related to the properties of the corresponding pencil of compatible Poisson brackets. The main goal of the paper is to illustrate this relationship and to show that the bi-Hamiltonian approach can be extremely effective in the study of singularities of integrable systems, especially in the case of many degrees of freedom when using other methods leads to serious computational problems. Since in many examples the underlying bi-Hamiltonian structure has a natural algebraic interpretation, the technology developed in this paper allows one to reformulate analytic and topological questions related to the dynamics of a given system into pure algebraic language, which leads to simple and natural answers.

Times Cited: 0

DOI: 10.1134/S1560354709040029

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Record 2 of 15

Author(s): Meshcheryakov, V (Meshcheryakov, V.)

Title: On the coincidence of two manifolds associated with the Calogero model

Source: JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS, 15 (3): 393-403 JUL 2009

Abstract: Let R be a root system, for example, the root system associated to a semisimple Lie algebra. In [1], V. A. Golubeva and V. P. Lexin constructed two algebraic manifolds (Bethe and Dunkl manifolds) using the "universal" Dunkl operators. These manifolds were defined as subsets of the complex space a", (N) of dimension equal to the number of roots of the root system under consideration.

The first manifold (Bethe manifold) is characterized by the following property: the Laplace operator constructed by means of Dunkl operators coincides with the "universal" Hamiltonian of the Calogero model. The second one (Dunkl manifold) is characterized by the property: the "universal" Dunkl operators commute.

In this paper, the manifolds associated with the irreducible root system of Coxeter type are considered. We give their construction supposing that these manifolds are embedded in a", (N/2). A theoremon the coincidence of Bethe and Dunkl manifolds is proved.

Times Cited: 0

DOI: 10.1007/s10883-009-9070-3

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Record 3 of 15

Author(s): Alekseevsky, DV (Alekseevsky, D. V.); Medori, C (Medori, C.); Tomassini, A (Tomassini, A.)

Title: Homogeneous para-Kahler Einstein manifolds

Source: RUSSIAN MATHEMATICAL SURVEYS, 64 (1): 1-43 JAN-FEB 2009

Abstract: A para-Kahler manifold can be defined as a pseudo-Riemannian manifold (M, g) with a parallel skew-symmetric para-complex structure K, that is, a parallel field of skew-symmetric endomorphisms with K-2 = Id or, equivalently, as a symplectic manifold (M, w) with a bi-Lagrangian structure L+/-, that is, two complementary integrable Lagrangian distributions. A homogeneous manifold M = G / H of a semisimple Lie group G admits an invariant para-Kahler structure (g, K) if and only if it is a covering of the adjoint orbit. Ad(G)h of a semisimple element h. A description is given of all invariant para-Kahler structures (g, K) on such a homogeneous manifold. With the use of a para-complex analogue of basic formulae of Kahler geometry it is proved that any invariant para-complex structure K on M = G/H defines a unique para-Kahler Einstein structure (g, K) with given non-zero scalar curvature. An explicit formula for the Einstein metric g is given. A survey of recent results on para-complex geometry is included.

Times Cited: 0

DOI: 10.1070/RM2009v064n01ABEH004591

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Record 4 of 15

Author(s): Parshall, BJ (Parshall, Brian J.); Scott, LL (Scott, Leonard L.)

Title: SOME Z/2-GRADED REPRESENTATION THEORY

Source: QUARTERLY JOURNAL OF MATHEMATICS, 60 (3): 327-351 SEP 2009

Abstract: In representation theory, the existence of a Z(+)-grading on a related finite dimensional algebra often plays an important role. For example, such a grading arises from the Koszul structure of the finite dimensional algebra representing the principal block of the BGG category O associated to a complex semisimple Lie algebra. But Koszul gradings in positive characteristic have proved elusive. For example, except for small values of a positive integer n, it is not known if the Schur algebra S(n, n) has such a Koszul grading, assuming the characteristic p of the base field satisfies p >= n, though this grading would suffice to establish Lusztig's character formula for these algebras. (And even though the character formula is known for p sufficiently large [H. Andersen, J. Jantzen and W. Soergel, Representations of Quantum Groups at a pth Root of Unity and of Semisimple Groups in Characteristic p, Asterique, Vol. 220, 1994], it is not known if the Schur algebra is Koszul for p sufficiently large.) This paper introduces Z/2-gradings on quasi-hereditary algebras, and shows that these gradings are almost as useful as a full Z(+)-grading, while being possibly much easier to find. We define the notion of a Z/2-based Kazhdan-Lusztig theory, which appears to be more flexible than, and generalizes, the notion of a Kazhdan-Lusztig theory (as first defined in [E. Cline, B. Parshall and L. Scott, Abstract Kazhdan-Lusztig theories, Tohoku Math. J. 45 (1993), 511-534]). However, its existence suffices, as was the case with the original notion, to establish character formulas in the standard settings, determine Ext(n)-groups, and show that homological duals behave well. Finally, we present some suggestive symmetric group examples involving Schur algebras which were an outgrowth of this work.

Times Cited: 0

DOI: 10.1093/qmath/han014

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Record 5 of 15

Author(s): Juteau, D (Juteau, Daniel)

Title: DECOMPOSITION NUMBERS FOR PERVERSE SHEAVES

Source: ANNALES DE L INSTITUT FOURIER, 59 (3): 1177-1229 2009

Abstract: The purpose of this article is to set foundations for decomposition numbers of perverse sheaves, to give some methods to calculate them in simple cases, and to compute them concretely in two situations: for a simple (Kleinian) surface singularity, and for the closure of the minimal non-trivial nilpotent orbit in a simple Lie algebra.

This work has applications to modular representation theory, for Weyl groups using the nilpotent cone of the corresponding semisimple Lie algebra, and for reductive algebraic group schemes using the affine Grassmannian of the Langlands dual group.

Times Cited: 0

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Record 6 of 15

Author(s): Ondrus, M (Ondrus, Matthew); Wiesner, E (Wiesner, Emilie)

Title: WHITTAKER MODULES FOR THE VIRASORO ALGEBRA

Source: JOURNAL OF ALGEBRA AND ITS APPLICATIONS, 8 (3): 363-377 JUN 2009

Abstract: Whittaker modules have been well studied in the setting of complex semisimple Lie algebras. Their definition can easily be generalized to certain other Lie algebras with triangular decomposition, including the Virasoro algebra. We define Whittaker modules for the Virasoro algebra and obtain analogues to several results from the classical setting, including a classification of simple Whittaker modules by central characters and composition series for general Whittaker modules.

Times Cited: 0

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Record 7 of 15

Author(s): Levy, P (Levy, Paul); McNinch, G (McNinch, George); Testerman, DM (Testerman, Donna M.)

Title: Nilpotent subalgebras of semisimple Lie algebras

Source: COMPTES RENDUS MATHEMATIQUE, 347 (9-10): 477-482 MAY 2009

Abstract: Let g be the Lie algebra of a semisimple linear algebraic group. Under mild conditions on the characteristic of the underlying field, one can show that any subalgebra of g consisting of nilpotent elements is contained in some Borel subalgebra. In this Note, we provide examples for each semisimple group G and for each of the torsion primes for G of nil subalgebras not lying ill any Borel subalgebra of g. To cite this article: P Levy et al., C R. Acad. Sci. Paris, Ser. 1347 (2009). (C) 2009 Published by Elsevier Masson SAS on behalf of Academie des sciences.

Times Cited: 0

DOI: 10.1016/j.crma.2009.03.015

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Record 8 of 15

Author(s): Bulois, M (Bulois, Michael)

Title: Irreducible Components of the Nilpotent Commuting Variety of a Symmetric Semisimple Lie Algebra

Source: ANNALES DE L INSTITUT FOURIER, 59 (1): 37-80 2009

Abstract: Let theta be an involution of the finite dimensional semisimple Lie algebra g and g = f circle plus p be the associated Cartan decomposition. The nilpotent commuting variety of (g,theta) consists in pair of nilpotent elements (x,y) of p such that [x,y] = 0. It is conjectured that this variety is equidimensional and that its irreducible components are indexed by the orbits of p distinguished elements. This conjecture was established by A. Premet in the case (g X g,theta) where theta (x,y) = (y,x). In this work we prove the conjecture in a significant number of other cases.

Times Cited: 0

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Record 9 of 15

Author(s): Heckenberger, I (Heckenberger, Istvan); Joseph, A (Joseph, Anthony)

Title: On the Left and Right Brylinski-Kostant Filtrations

Source: ALGEBRAS AND REPRESENTATION THEORY, 12 (2-5): 417-442 OCT 2009

Abstract: Let g be a complex semisimple Lie algebra, b a Borel subalgebra, and h subset of b a Cartan subalgebra. Let V be a finite dimensional simple U(g) module. Based on a principal s-triple (e, h, f) and following work of Kostant, Brylinski (J Amer Math Soc 2(3): 517-533, 1989) defined a filtration F-e for all weight subspaces V-mu of V and calculated the dimensions of the graded subspaces for mu dominant. In Joseph et al. (J Amer Math Soc 13(4): 945-970, 2000) these dimensions were calculated for all mu. Let delta M(0) be the finite dual of the Verma module of highest weight 0. It identifies with the global functions on a Weyl group translate of the open Bruhat cell and so inherits a natural degree filtration. On the other hand, up to an appropriate shift of weights, there is a unique U(b) module embedding of V into delta M(0) and so the degree filtration induces a further filtration F on each weight subspace V-mu. A casual reading of Joseph et al. (J Amer Math Soc 13(4): 945-970, 2000) might lead one to believe that F and F-e coincide. However this is quite false. Rather one should view F-e as coming from a left action of U(n) and then there is a second (Brylinski-Kostant) filtration F-e' coming from a right action. It is Fe' which may coincide with F. In this paper the above claim is made precise. First it is noted that F is itself not canonical, but depends on a choice of variables. Then it is shown that a particular choice can be made to ensure that F = F-e'. An explicit form for the unique left U(b) module embedding V (sic) delta M(0) is given using the right action of U(n). This is used to give a purely algebraic proof of Brylinski's main result in Brylinski (J Amer Math Soc 2(3): 517-533, 1989) which is much simpler than Joseph et al. (J Amer Math Soc 13(4): 945-970, 2000). It is noted that the dimensions of the graded subspaces of gr(Fe) V-mu and gr(Fe') V-mu can be very different. Their interrelation may involve the Kashiwara involution. Indeed a combinatorial formula for multiplicities in tensor products involving crystal bases and the Kashiwara involution is recovered. Though the dimensions for the graded subspaces of gr(Fe')V(mu) are determined by polynomial degree, their values remain unknown.

Times Cited: 0

DOI: 10.1007/s10468-009-9159-0

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Record 10 of 15

Author(s): Krotz, B (Kroetz, Bernhard); Schlichtkrull, H (Schlichtkrull, Henrik)

Title: Normal forms for real quadratic forms

Source: ARCHIV DER MATHEMATIK, 92 (2): 129-136 FEB 2009

Abstract: We investigate the non-diagonal normal forms of a quadratic form on R-n, in particular for n = 3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of R-5.

Times Cited: 0

DOI: 10.1007/s00013-008-2925-0

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Record 11 of 15

Author(s): Alekseevsky, DV (Alekseevsky, Dimitri V.); Medori, C (Medori, Costantino); Tomassini, A (Tomassini, Adriano)

Title: Para-Kahler Einstein metrics on homogeneous manifolds

Source: COMPTES RENDUS MATHEMATIQUE, 347 (1-2): 69-72 JAN 2009

Abstract: A para-Kahler structure on a manifold M is a pair (g, K) where g is a pseudo-Riemannian metric and K is a parallel field of skew-symmetric endomorphisms with K-2 = Id. We give a description of all invariant para-Kahler structures (g, K) on homogeneous manifolds M = G/H of semisimple Lie groups G. To cite this article: D. V. Alekseevsky et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). (C) 2008 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Times Cited: 0

DOI: 10.1016/j.crma.2008.11.016

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Record 12 of 15

Author(s): Campoamor-Stursberg, R (Campoamor-Stursberg, R.); Low, SG (Low, S. G.)

Title: Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

Source: JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 42 (6): Art. No. 065205 FEB 13 2009

Abstract: Given a semidirect product g = s (circle plus) over right arrow tau of semisimple Lie algebras s and solvable algebras r, we construct polynomial operators in the enveloping algebra U( g) of g that commute with r and transform like the generators of s, up to a functional factor that turns out to be a Casimir operator of r. Such operators are said to generate a virtual copy of s in U( g), and allow us to compute the Casimir operators of g in a closed form, using the classical formulae for the invariants of s. The behavior of virtual copies with respect to contractions of Lie algebras is analyzed. Applications to the class of Hamilton algebras and their inhomogeneous extensions are given.

Times Cited: 0

Article Number: 065205

DOI: 10.1088/1751-8113/42/6/065205

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Record 13 of 15

Author(s): Mazorchuk, V (Mazorchuk, Volodymyr); Ovsienko, S (Ovsienko, Serge); Stroppel, C (Stroppel, Catharina)

Title: QUADRATIC DUALS, KOSZUL DUAL FUNCTORS, AND APPLICATIONS

Source: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 361 (3): 1129-1172 2009

Abstract: This paper studies quadratic and Koszul duality for modules over positively graded categories. Typical examples are modules over a path algebra, which is graded by the path length, of a not necessarily finite quiver with relations. We present a very general definition of quadratic and Koszul duality functors backed up by explicit examples. This generalizes the work of Beilinson, Ginzburg, and Soergel, 1996, in two substantial ways: We work in the setup of graded categories, i.e. we allow infinitely many idempotents and also de. ne a "Koszul" duality functor for not necessarily Koszul categories. As an illustration of the techniques we reprove the Koszul duality (Ryom-Hansen, 2004) of translation and Zuckerman functors for the classical category O in a quite elementary and explicit way. From this we deduce a conjecture of Bernstein, Frenkel, and Khovanov, 1999. As applications we propose a definition of a "Koszul" dual category for integral blocks of Harish-Chandra bimodules and for blocks outside the critical hyperplanes for the Kac-Moody category O.

Times Cited: 2

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Record 14 of 15

Author(s): Kleinsteuber, M (Kleinsteuber, Martin)

Title: A sort-Jacobi algorithm for semisimple Lie algebras

Source: LINEAR ALGEBRA AND ITS APPLICATIONS, 430 (1): 155-173 JAN 1 2009

Abstract: A structure preserving sort-Jacobi algorithm for computing eigenvalues or singular values is presented. It applies to an arbitrary semisimple Lie algebra on its (-1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra and generalizes well-known normal form problems such as e.g. the symmetric, Hermitian, skew-symmetric, symmetric and skew-symmetric R-Hamiltonian eigenvalue problem and the singular value decomposition. (C) 2008 Elsevier Inc. All rights reserved.

Times Cited: 0

DOI: 10.1016/j.laa.2008.07.004

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Record 15 of 15

Author(s): Zwicknagl, S (Zwicknagl, Sebastian)

Title: R-matrix Poisson algebras and their deformations

Source: ADVANCES IN MATHEMATICS, 220 (1): 1-58 JAN 15 2009

Abstract: We classify in this paper Poisson structures on modules over semisimple Lie algebras arising from classical r-matrices. We then study their quantizations and the relation to classical invariant theory. (c) 2008 Elsevier Inc. All tights reserved.

Times Cited: 0

DOI: 10.1016/j.aim.2008.08.006

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