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38
Matroids and geometric invariant theory of torus actions on flag spaces
, 2005
"... Abstract. Let F//T be a G.I.T. quotient of a flag manifold F by the natural action of the maximal torus T in SL(n, C). The construction of the quotient space depends upon the choice of a Tlinearized line bundle L of F. We study the case where L = Lλ is a very ample homogeneous line bundle determine ..."
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Abstract. Let F//T be a G.I.T. quotient of a flag manifold F by the natural action of the maximal torus T in SL(n, C). The construction of the quotient space depends upon the choice of a Tlinearized line bundle L of F. We study the case where L = Lλ is a very ample homogeneous line bundle determined by a dominant weight λ. We apply a theorem of Gel’fand, Goresky, MacPherson, and Serganova about matroids and their respective polytopes to study semistability of flags relative to a given Tlinearization of Lλ. The main theorem of this note is that regardless of the choice of linearization, the semistable flags are detected by invariant sections of Lλ; that is, for each semistable flag p ∈ F, there is a Tinvariant section s of Lλ such that s(p) ̸ = 0. Additionally we find that the closure of any Torbit in F is projectively normal for any projective embedding of F.
ON WITTEN MULTIPLE ZETAFUNCTIONS ASSOCIATED WITH SEMISIMPLE LIE ALGEBRAS II
"... Abstract. This is a continuation of our previous result, in which properties of multiple zetafunctions associated with simple Lie algebras of Ar type have been studied. In the present paper we consider more general situation, and discuss the Lie theoretic background structure of our theory. We show ..."
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Abstract. This is a continuation of our previous result, in which properties of multiple zetafunctions associated with simple Lie algebras of Ar type have been studied. In the present paper we consider more general situation, and discuss the Lie theoretic background structure of our theory. We show a recursive structure in the family of zetafunctions of sets of roots, which can be explained by the order relation among roots. We also point out that the recursive structure can be described in terms of Dynkin diagrams. Then we prove several analytic properties of zetafunctions associated with simple Lie algebras of Br, Cr, and Dr types. 1.
The structure of quantum Lie algebra for the classical series Bl
 Cl and Dl, J. Phys. A: Math. Gen
, 1998
"... Abstract. The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum ClebschGordan coeffic ..."
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Abstract. The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum ClebschGordan coefficients for adjoint ⊗ adjoint → adjoint. We present a practical method for the determination of these quantum ClebschGordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras Bl, Cl and Dl. In the quantum case also the structure constants of the Cartan subalgebra are nonzero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple qdeformations of the classical ones. 1.
Riemannian flag manifolds with homogeneous geodesics
 TRANS. AMER. MATH. SOC
, 2007
"... A geodesic in a Riemannian homogeneous manifold (M = G/K, g) is called a homogeneous geodesic if it is an orbit of a oneparameter subgroup of the Lie group G. We investigate Ginvariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when M = G/K is a flag manifo ..."
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A geodesic in a Riemannian homogeneous manifold (M = G/K, g) is called a homogeneous geodesic if it is an orbit of a oneparameter subgroup of the Lie group G. We investigate Ginvariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when M = G/K is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group G. We use an important invariant of a flag manifold M = G/K, itsTroot system, to give a simple necessary condition that M admits a nonstandard Ginvariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds M = G/K of a simple Lie group G, only the manifold Com(R2ℓ+2)=SO(2ℓ +1)/U(ℓ) of complex structures in R2ℓ+2, and the complex projective space CP 2ℓ−1 = Sp(ℓ)/U(1) · Sp(ℓ − 1) admit a nonnaturally reductive invariant metric with homogeneous geodesics. In all other cases the only Ginvariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra g of G). According to F. Podestà and G.Thorbergsson (2003), these manifolds are the only nonHermitian symmetric flag manifolds with coisotropic action of the stabilizer.
Generalized Chebyshev Polynomials Associated with Affine Weyl Groups
 Trans. Amer. Math. Soc
, 1988
"... ABSTRACT. We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in onetoone correspondence with affine Weyl groups. For each such figure in ndimensional Euclidean space, we construct a sequence of polynomi ..."
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ABSTRACT. We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in onetoone correspondence with affine Weyl groups. For each such figure in ndimensional Euclidean space, we construct a sequence of polynomials P/c: Rn — ► Rn so that the mapping P ^ is conjugate to stretching the figure by a factor A; and folding it back onto itself. If re = 1 and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials Pk are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials.
Infinite convolution products and refinable distributions on Lie groups
 Trans. Am. Math. Soc
, 2000
"... Abstract. Sufficient conditions for the convergence in distribution of an infinite convolution product µ1 ∗ µ2 ∗... of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions φ that satisfy Tφ = φ where T is a ..."
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Abstract. Sufficient conditions for the convergence in distribution of an infinite convolution product µ1 ∗ µ2 ∗... of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions φ that satisfy Tφ = φ where T is a refinement operator constructed from a measure µ and a dilation automorphism A. The existence of A implies G is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset K⊂Gsuch that for any open set U containing K, and for any distribution f on G with compact support, there exists an integer n(U,f) such that n ≥ n(U,f) implies supp(T n f) ⊂U. If µ is supported on an Ainvariant uniform subgroup Γ, then T is related, by an intertwining operator, to a transition operator W on C(Γ). Necessary and sufficient conditions for T n f to converge to φ ∈ L 2, and for the Γtranslates of φ to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of W to functions supported on Ω: = KK −1 ∩ Γ. 1.
Paving Hessenberg varieties by affines
, 2004
"... Abstract. Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety which arise in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all ..."
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Abstract. Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety which arise in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all classical types. This paving is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of this paving and their dimensions can be identified by a combinatorial condition on roots. We use this paving to prove these Hessenberg varieties have no odddimensional homology. 1.
Zeghib Actions of noncompact semisimple groups on Lorentz manifolds, Geom
 Funct. Anal
"... Abstract. The above title is the same, but with “semisimple ” instead of “simple, ” as that of a notice by Nadine Kowalsky. There, she announced many theorems on the subject of actions of simple Lie groups preserving a Lorentz structure. Unfortunately, she published proofs for essentially only half ..."
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Abstract. The above title is the same, but with “semisimple ” instead of “simple, ” as that of a notice by Nadine Kowalsky. There, she announced many theorems on the subject of actions of simple Lie groups preserving a Lorentz structure. Unfortunately, she published proofs for essentially only half of the announced results before her premature death. Here, using a different, geometric approach, we generalize her results to the semisimple case, and give proofs of all her announced results. 1.
Extended Lie algebraic stability analysis for switched systems with continuoustime and discretetime subsystems
 International Journal of Applied Mathematics and Computer Science 17(4): 447–454, DOI
, 2007
"... We analyze stability for switched systems which are composed of both continuoustime and discretetime subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadra ..."
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We analyze stability for switched systems which are composed of both continuoustime and discretetime subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunovlike function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.
The Toda Equations and the Geometry of Surfaces
"... The Toda equations are a particularly interesting example of a completely integrable Hamiltonian system and were initially studied in a dynamical context. Two particular forms of Toda equations, the "open" and "affine" equations, can be formulated for any simple Lie algebra and solutions to both for ..."
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The Toda equations are a particularly interesting example of a completely integrable Hamiltonian system and were initially studied in a dynamical context. Two particular forms of Toda equations, the "open" and "affine" equations, can be formulated for any simple Lie algebra and solutions to both forms arise naturally in a geometrical context from special types of harmonic maps into certain homogeneous spaces associated to the Lie algebra. Generally speaking, the open case is rather easier to deal with since the solutions correspond to holomorphic curves, but the affine case is in some ways more interesting since it has certain special features not enjoyed by the open case. All these aspects are discussed briefly below. The Toda equations are a remarkable phenomenon with many interesting facets. Since their discovery in the 1960s they have engendered an extensive and varied literature, much of which involves a detailed understanding of a number of different areas of mathematics. As a fi...