Steve Shnider | Bar Ilan (original) (raw)
Papers by Steve Shnider
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013
Under a proper assignment of a metric and a connection, the (classical) dynamical trajectories ca... more Under a proper assignment of a metric and a connection, the (classical) dynamical trajectories can be identified as geodesics of the underlying manifold. We show how these geometric structures can be derived; specifically, we construct them explicitly for configuration and phase spaces of Hamiltonian systems. We demonstrate how the correspondence between geometry and dynamics can be applied to study the conserved quantities of a dynamical system. Lastly, we demonstrate how the mean-curvature of the energy level-sets in phase-space might be correlated with strongly chaotic behavior.
Notices of the American Mathematical Society, 2014
Vetus Testamentum, 2006
The theophany in Psalm xviii includes, together with the storm imagery, images of wings/flight an... more The theophany in Psalm xviii includes, together with the storm imagery, images of wings/flight and bows/arrows in a combination appearing nowhere else in the Hebrew Bible Hebrew (HB). On the other hand, in the iconography of the ancient Near East, these motifs are often part of a divine apparition, especially to a king in battle. One of the major examples is the winged disc, which in many cases contains the image of a god armed with a bow. We present a number of examples of the motifs of winged gods and bows from Egyptian and Neo-Assyrian sources, both iconographic and textual. In particular, the Neo-Assyrian parallels relate to the theme of the divine glory, kbd, Akk. melammu, and the divine empowerment of the king which assures his victory in battle. In the context of these examples, the theophany (vss. 8-18) and the battle scene (vss. 30, 33-43) can be understood as two perspectives on a single event involving God and the king. This approach leads us to suggest an emendation in the difficult verses, 35-36.
We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, ... more We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalisation of the Wirtinger inequality for the comass. Using a model for the classifying space BS^3 built inductively out of BS^1, we prove that the symmetric metrics of certain two-point homogeneous manifolds turn out not to be the systolically optimal metrics on those manifolds. We point out the unexpected role played by the exceptional Lie algebra E_7 in systolic geometry, via the calculation of Wirtinger constants. Using a technique of pullback with controlled systolic ratio, we calculate the optimal systolic ratio of the quaternionic projective plane, modulo the existence of a Joyce manifold with Spin(7) holonomy and unit middle-dimensional Betti number.
We study one and two parameter quantizations of the function algebra on a semisimple orbit in the... more We study one and two parameter quantizations of the function algebra on a semisimple orbit in the coadjoint representation of a simple Lie group subject to the condition that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group. We prove that the corresponding Poisson bracket must be the sum of the so-called R-matrix bracket and an invariant bracket. We classify such brackets for all semisimple orbits and show that they form a family of dimension equal to the rank equal to the second cohomology group of the orbit and then we quantize these brackets. A two parameter (or double) quantization corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on the orbit. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization o...
We analyze some of the main approaches in the literature to the method of `adequality' with w... more We analyze some of the main approaches in the literature to the method of `adequality' with which Fermat approached the problems of the calculus, as well as its source in the parisotes of Diophantus, and propose a novel reading thereof. Adequality is a crucial step in Fermat's method of finding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermat's collected works. We show that at least some of the manifestations of adequality amount to variational techniques exploiting a small, or infinitesimal, variation e. Fermat's treatment of geometric and physical applications suggests that an aspect of approximation is inherent in adequality, as well as an aspect of smallness on the part of e. We question the relevance to understanding Fermat of 19th century dictionary definitions of parisotes and adaequare, cited by Breger, and take issue with ...
We show that the method of stochastic reduction of linear superpositions can be applied to the pr... more We show that the method of stochastic reduction of linear superpositions can be applied to the process of disentanglement for the spin-0 state of two spin-1/2 particles. We describe the geometry of this process in the framework of the complex projective space
We examine prevailing philosophical and historical views about the origin of infinitesimal mathem... more We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leibniz, Euler, Cauchy and other giants of infinitesimal mathematics in a new light. We also detail several procedures of the historical infinitesimal calculus that were only clarified and formalized with the advent of modern infinitesimals. These procedures include Fermat's adequality; Leibniz's law of continuity and the transcendental law of homogeneity; Euler's principle of cancellation and infinite integers with the associated infinite products; Cauchy's infinitesimal-based definition of continuity and "Dirac" delta function. Such procedures were interpreted and formalized in Robinson's framework in terms of concepts like microcontinuity (S-continuity), the standard part principle, the transfer principle, and hyperfinite products. We evaluate the critiques of historical and modern infinitesimals by their foes from Berkeley and Cantor to Bishop and Connes. We analyze the issue of the consistency, as distinct from the issue of the rigor, of historical infinitesimals, and contrast the methodologies of Leibniz and Nieuwentijt in this connection. Contents 1. The ABC's of the history of infinitesimal mathematics 2 2. Adequality to Chimeras 5 2.1. Adequality 5 2.2. Archimedean axiom 6 2.3. Berkeley, George 7 2.4. Berkeley's logical criticism 8 2.
Recall that a Drinfel’d algebra (or a quasi-bialgebra in the original terminology of [1]) is an o... more Recall that a Drinfel’d algebra (or a quasi-bialgebra in the original terminology of [1]) is an object A = (V, · , ∆, Φ), where (V, · , ∆) is an associative, not necessarily coassociative, unital and counital k-bialgebra, Φ is an invertible element of V ⊗3, and the usual coassociativity property is replaced by the condition which we shall refer to as quasi-coassociativity: (1) (1 ⊗ ∆) ∆ · Φ = Φ · ( ∆ ⊗ 1)∆, where we use the dot · to indicate both the (associative) multiplication on V and the induced multiplication on V ⊗3. Moreover, the validity of the following “pentagon identity ” is required: (1 2 ⊗ ∆)(Φ) · ( ∆ ⊗ 1 2)(Φ) = (1 ⊗ Φ) · (1 ⊗ ∆ ⊗ 1)(Φ) · (Φ ⊗ 1), 1 ∈ V being the unit element and 1, the identity map on V. If ǫ: V → k (k being the ground field) is the counit of the coalgebra (V, ∆) then, by definition, (ǫ ⊗ 1) ∆ = (1⊗ǫ) ∆ = 1. We have a natural splitting V = V ⊕ k, V: = Ker(ǫ), given by the embedding k → V, k ∋ c ↦ → c · 1 ∈ V. For a (V, ·)-bimodule N, recall the follow...
Coherence phenomena appear in two different situations. In the context of category theory the ter... more Coherence phenomena appear in two different situations. In the context of category theory the term ‘coherence constraints’ refers to a set of diagrams whose commutativity implies the commutativity of a larger class of diagrams. In the context of algebra coherence constrains are a minimal set of generators for the second syzygy, that is, a set of equations which generate the full set of identities among the defining relations of an algebraic theory. A typical example of the first type is Mac Lane’s coherence theorem for monoidal categories [9, Theorem 3.1], an example of the second type is the result of [2] saying that pentagon identity for the ‘associator ’ Φ of a quasi-Hopf algebra implies the validity of a set of identities with higher instances of Φ. We show that both types of coherence are governed by a homological invariant of the operad for the underlying algebraic structure. We call this invariant the (space of) coherence constraints. In many cases these constraints can be ex...
The paper is devoted to the coherence problem for algebraic structures on a category. We describe... more The paper is devoted to the coherence problem for algebraic structures on a category. We describe coherence constraints in terms of the cohomology of the corresponding operad. Our approach enables us to introduce the concept of coherence even for structures which are not given by commutative diagrams. In the second part of the paper we discuss ‘quantizations ’ of various algebraic structures. We prove that there always exists the ‘canonical quantization ’ and show that the two prominent examples – Drinfel’d’s quasi-Hopf algebras and Gurevich’s generalized Lie algebras – are canonical quantizations of their ‘classical limits. ’ The second part (sections 6,7,8) can be read independently, though the abstract theory of the first part is necessary for the full understanding of the results.
Abstract. Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) ... more Abstract. Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) to study the comass of certain self-dual 4-forms on R 8. In particular, we prove that the Cayley 4-form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio (equation (1.4)) is SO(8)-conjugate to the Cayley 4-form. Contents
ABSTRACT. Coherence phenomena appear in two different situations. In the context of category theo... more ABSTRACT. Coherence phenomena appear in two different situations. In the context of category theory the term 'coherence constraints ' refers to a set of diagrams whose commutativity implies the commutativity of a larger class of diagrams. In the context of algebra coherence constrains are a minimal set of generators for the second syzygy, that is, a set of equations which generate the full set of identities among the defining relations of an algebraic theory. A typical example of the first type is Mac Lane's coherence theorem for monoidal categories [9, Theorem 3.1], an example of the second type is the result of [2] saying that pentagon identity for the 'associator ' $ of a quasi-Hopf algebra implies the validity of a set of identities with higher instances of $. We show that both types of coherence are governed by a homological invariant of the operad for the underlying algebraic structure. We call this invariant the (space of) coherence constraints. In ma...
A new approach to error analysis is introduced, based on the observation that many numerical proc... more A new approach to error analysis is introduced, based on the observation that many numerical procedures can be interpreted as computations of products in a suitable Lie group. The absence of an additive error law for such procedures is intimately related to the nonexistence of bi-invariant metrics on the relevant groups. Introducing the notion of an almost Inn(G) invariant metric (a left invariant, almost Inn(G) invariant metric can be constructed on any locally compact connected group having a countable basis for its identity neighborhoods), we show how error analysis can nevertheless be done for such procedures. We illustrate for what we call "scalar calculations without writing to memory"; the Horner algorithm for evaluation of a polynomial is such a calculation, and we give explicit error bounds for a floating point implementation of the Horner algorithm, and demonstrate their usefulness numerically. A left invariant, almost Inn(G) invariant metric on a group induces a metric on a homogeneous space of the group with useful properties for error analysis; treating R as a homogeneous space of the group of affine transformations of R we compute a new metric that unifies absolute and relative error.
In this paper we describe a multiparameter deformation of the function algebra of a semisimple co... more In this paper we describe a multiparameter deformation of the function algebra of a semisimple coadjoint orbit. In the first section we use the representation of the Lie algebra on a generalized Verma module to quantize the Kirillov bracket on the family of semisimple coadjoint orbits of a given orbit type. In the second section we extend this construction to define a deformation in the category of representations of the quantized enveloping algebra. In an earlier paper we used cohomological methods to prove the existence of a two parameter family quantizing a compatible pair of Poisson brackets on any symmetric coadjoint orbit. This paper gives a more explicit algebraic construction which includes more general orbit types and which we prove to be flat in all parameters. 1 Quantizing the Kirillov bracket Assume we have the following data: a simple Lie algebra, G, over the complex field, C, with corresponding Lie group, G, and a Cartan subalgebra, H, together with a system of simple positive roots. Let G = N − ⊕ H ⊕ N + , be the corresponding Cartan decomposition, where N + , N − are the nilpotent subalgebras which are, respectively, the sum of positive root spaces and the sum of the negative root spaces. Denote by G * and H * the C linear duals of G and H. Given λ ∈ G * , let O λ be the orbit of λ under the coadjoint action of G on G *. Finally, let S(G) be the symmetric algebra of G considered as polynomial functions on G * and I λ the ideal of polynomials vanishing on the orbit O λ. Then F λ = S(G)/I λ is the algebra of functions on O λ , with Poisson structure given by the standard Kostant-Kirillov-Souriau (KKS) bracket {f, g}(λ) = λ, [df λ , dg λ ]. Consider the G orbits of semisimple elements, that is, linear functions conjugate under the coadjoint action to ones which vanishes on N + ⊕ N −. In any such orbit we can pick as origin an element, λ, which is the trivial extension to G of a functional on H. The stabilizer of such an element will be a subalgebra generated by H and a subset of the simple roots. Such a subalgebra is called a Levi subalgebra. The set of all λ with stabilizer equal to a fixed Levi algebra L is parametrized by a subspace of H * minus its intersection with a family of coordinate hyperplanes.
SIAM Journal on Numerical Analysis, 1999
This paper introduces a new class of methods, which we call Möbius schemes, for the numerical sol... more This paper introduces a new class of methods, which we call Möbius schemes, for the numerical solution of matrix Riccati differential equations. The approach is based on viewing the Riccati equation in its natural geometric setting, as a flow on the Grassmanian of m-dimensional subspaces of an (n + m)-dimensional vector space. Since the Grassmanians are compact differentiable manifolds, and the coefficients of the equation are assumed continuous, there are no singularities or intrinsic instabilities in the associated flow. The presence of singularities and numerical instabilitites is an artefact of the coordinate system, but since Möbius schemes are based on the natural geometry, they are able to deal with numerical instability and pass accurately through the singularities. A number of examples are given to demonstrate these properties.
Perspectives on Science, 2013
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013
Under a proper assignment of a metric and a connection, the (classical) dynamical trajectories ca... more Under a proper assignment of a metric and a connection, the (classical) dynamical trajectories can be identified as geodesics of the underlying manifold. We show how these geometric structures can be derived; specifically, we construct them explicitly for configuration and phase spaces of Hamiltonian systems. We demonstrate how the correspondence between geometry and dynamics can be applied to study the conserved quantities of a dynamical system. Lastly, we demonstrate how the mean-curvature of the energy level-sets in phase-space might be correlated with strongly chaotic behavior.
Notices of the American Mathematical Society, 2014
Vetus Testamentum, 2006
The theophany in Psalm xviii includes, together with the storm imagery, images of wings/flight an... more The theophany in Psalm xviii includes, together with the storm imagery, images of wings/flight and bows/arrows in a combination appearing nowhere else in the Hebrew Bible Hebrew (HB). On the other hand, in the iconography of the ancient Near East, these motifs are often part of a divine apparition, especially to a king in battle. One of the major examples is the winged disc, which in many cases contains the image of a god armed with a bow. We present a number of examples of the motifs of winged gods and bows from Egyptian and Neo-Assyrian sources, both iconographic and textual. In particular, the Neo-Assyrian parallels relate to the theme of the divine glory, kbd, Akk. melammu, and the divine empowerment of the king which assures his victory in battle. In the context of these examples, the theophany (vss. 8-18) and the battle scene (vss. 30, 33-43) can be understood as two perspectives on a single event involving God and the king. This approach leads us to suggest an emendation in the difficult verses, 35-36.
We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, ... more We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalisation of the Wirtinger inequality for the comass. Using a model for the classifying space BS^3 built inductively out of BS^1, we prove that the symmetric metrics of certain two-point homogeneous manifolds turn out not to be the systolically optimal metrics on those manifolds. We point out the unexpected role played by the exceptional Lie algebra E_7 in systolic geometry, via the calculation of Wirtinger constants. Using a technique of pullback with controlled systolic ratio, we calculate the optimal systolic ratio of the quaternionic projective plane, modulo the existence of a Joyce manifold with Spin(7) holonomy and unit middle-dimensional Betti number.
We study one and two parameter quantizations of the function algebra on a semisimple orbit in the... more We study one and two parameter quantizations of the function algebra on a semisimple orbit in the coadjoint representation of a simple Lie group subject to the condition that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group. We prove that the corresponding Poisson bracket must be the sum of the so-called R-matrix bracket and an invariant bracket. We classify such brackets for all semisimple orbits and show that they form a family of dimension equal to the rank equal to the second cohomology group of the orbit and then we quantize these brackets. A two parameter (or double) quantization corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on the orbit. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization o...
We analyze some of the main approaches in the literature to the method of `adequality' with w... more We analyze some of the main approaches in the literature to the method of `adequality' with which Fermat approached the problems of the calculus, as well as its source in the parisotes of Diophantus, and propose a novel reading thereof. Adequality is a crucial step in Fermat's method of finding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermat's collected works. We show that at least some of the manifestations of adequality amount to variational techniques exploiting a small, or infinitesimal, variation e. Fermat's treatment of geometric and physical applications suggests that an aspect of approximation is inherent in adequality, as well as an aspect of smallness on the part of e. We question the relevance to understanding Fermat of 19th century dictionary definitions of parisotes and adaequare, cited by Breger, and take issue with ...
We show that the method of stochastic reduction of linear superpositions can be applied to the pr... more We show that the method of stochastic reduction of linear superpositions can be applied to the process of disentanglement for the spin-0 state of two spin-1/2 particles. We describe the geometry of this process in the framework of the complex projective space
We examine prevailing philosophical and historical views about the origin of infinitesimal mathem... more We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leibniz, Euler, Cauchy and other giants of infinitesimal mathematics in a new light. We also detail several procedures of the historical infinitesimal calculus that were only clarified and formalized with the advent of modern infinitesimals. These procedures include Fermat's adequality; Leibniz's law of continuity and the transcendental law of homogeneity; Euler's principle of cancellation and infinite integers with the associated infinite products; Cauchy's infinitesimal-based definition of continuity and "Dirac" delta function. Such procedures were interpreted and formalized in Robinson's framework in terms of concepts like microcontinuity (S-continuity), the standard part principle, the transfer principle, and hyperfinite products. We evaluate the critiques of historical and modern infinitesimals by their foes from Berkeley and Cantor to Bishop and Connes. We analyze the issue of the consistency, as distinct from the issue of the rigor, of historical infinitesimals, and contrast the methodologies of Leibniz and Nieuwentijt in this connection. Contents 1. The ABC's of the history of infinitesimal mathematics 2 2. Adequality to Chimeras 5 2.1. Adequality 5 2.2. Archimedean axiom 6 2.3. Berkeley, George 7 2.4. Berkeley's logical criticism 8 2.
Recall that a Drinfel’d algebra (or a quasi-bialgebra in the original terminology of [1]) is an o... more Recall that a Drinfel’d algebra (or a quasi-bialgebra in the original terminology of [1]) is an object A = (V, · , ∆, Φ), where (V, · , ∆) is an associative, not necessarily coassociative, unital and counital k-bialgebra, Φ is an invertible element of V ⊗3, and the usual coassociativity property is replaced by the condition which we shall refer to as quasi-coassociativity: (1) (1 ⊗ ∆) ∆ · Φ = Φ · ( ∆ ⊗ 1)∆, where we use the dot · to indicate both the (associative) multiplication on V and the induced multiplication on V ⊗3. Moreover, the validity of the following “pentagon identity ” is required: (1 2 ⊗ ∆)(Φ) · ( ∆ ⊗ 1 2)(Φ) = (1 ⊗ Φ) · (1 ⊗ ∆ ⊗ 1)(Φ) · (Φ ⊗ 1), 1 ∈ V being the unit element and 1, the identity map on V. If ǫ: V → k (k being the ground field) is the counit of the coalgebra (V, ∆) then, by definition, (ǫ ⊗ 1) ∆ = (1⊗ǫ) ∆ = 1. We have a natural splitting V = V ⊕ k, V: = Ker(ǫ), given by the embedding k → V, k ∋ c ↦ → c · 1 ∈ V. For a (V, ·)-bimodule N, recall the follow...
Coherence phenomena appear in two different situations. In the context of category theory the ter... more Coherence phenomena appear in two different situations. In the context of category theory the term ‘coherence constraints’ refers to a set of diagrams whose commutativity implies the commutativity of a larger class of diagrams. In the context of algebra coherence constrains are a minimal set of generators for the second syzygy, that is, a set of equations which generate the full set of identities among the defining relations of an algebraic theory. A typical example of the first type is Mac Lane’s coherence theorem for monoidal categories [9, Theorem 3.1], an example of the second type is the result of [2] saying that pentagon identity for the ‘associator ’ Φ of a quasi-Hopf algebra implies the validity of a set of identities with higher instances of Φ. We show that both types of coherence are governed by a homological invariant of the operad for the underlying algebraic structure. We call this invariant the (space of) coherence constraints. In many cases these constraints can be ex...
The paper is devoted to the coherence problem for algebraic structures on a category. We describe... more The paper is devoted to the coherence problem for algebraic structures on a category. We describe coherence constraints in terms of the cohomology of the corresponding operad. Our approach enables us to introduce the concept of coherence even for structures which are not given by commutative diagrams. In the second part of the paper we discuss ‘quantizations ’ of various algebraic structures. We prove that there always exists the ‘canonical quantization ’ and show that the two prominent examples – Drinfel’d’s quasi-Hopf algebras and Gurevich’s generalized Lie algebras – are canonical quantizations of their ‘classical limits. ’ The second part (sections 6,7,8) can be read independently, though the abstract theory of the first part is necessary for the full understanding of the results.
Abstract. Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) ... more Abstract. Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) to study the comass of certain self-dual 4-forms on R 8. In particular, we prove that the Cayley 4-form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio (equation (1.4)) is SO(8)-conjugate to the Cayley 4-form. Contents
ABSTRACT. Coherence phenomena appear in two different situations. In the context of category theo... more ABSTRACT. Coherence phenomena appear in two different situations. In the context of category theory the term 'coherence constraints ' refers to a set of diagrams whose commutativity implies the commutativity of a larger class of diagrams. In the context of algebra coherence constrains are a minimal set of generators for the second syzygy, that is, a set of equations which generate the full set of identities among the defining relations of an algebraic theory. A typical example of the first type is Mac Lane's coherence theorem for monoidal categories [9, Theorem 3.1], an example of the second type is the result of [2] saying that pentagon identity for the 'associator ' $ of a quasi-Hopf algebra implies the validity of a set of identities with higher instances of $. We show that both types of coherence are governed by a homological invariant of the operad for the underlying algebraic structure. We call this invariant the (space of) coherence constraints. In ma...
A new approach to error analysis is introduced, based on the observation that many numerical proc... more A new approach to error analysis is introduced, based on the observation that many numerical procedures can be interpreted as computations of products in a suitable Lie group. The absence of an additive error law for such procedures is intimately related to the nonexistence of bi-invariant metrics on the relevant groups. Introducing the notion of an almost Inn(G) invariant metric (a left invariant, almost Inn(G) invariant metric can be constructed on any locally compact connected group having a countable basis for its identity neighborhoods), we show how error analysis can nevertheless be done for such procedures. We illustrate for what we call "scalar calculations without writing to memory"; the Horner algorithm for evaluation of a polynomial is such a calculation, and we give explicit error bounds for a floating point implementation of the Horner algorithm, and demonstrate their usefulness numerically. A left invariant, almost Inn(G) invariant metric on a group induces a metric on a homogeneous space of the group with useful properties for error analysis; treating R as a homogeneous space of the group of affine transformations of R we compute a new metric that unifies absolute and relative error.
In this paper we describe a multiparameter deformation of the function algebra of a semisimple co... more In this paper we describe a multiparameter deformation of the function algebra of a semisimple coadjoint orbit. In the first section we use the representation of the Lie algebra on a generalized Verma module to quantize the Kirillov bracket on the family of semisimple coadjoint orbits of a given orbit type. In the second section we extend this construction to define a deformation in the category of representations of the quantized enveloping algebra. In an earlier paper we used cohomological methods to prove the existence of a two parameter family quantizing a compatible pair of Poisson brackets on any symmetric coadjoint orbit. This paper gives a more explicit algebraic construction which includes more general orbit types and which we prove to be flat in all parameters. 1 Quantizing the Kirillov bracket Assume we have the following data: a simple Lie algebra, G, over the complex field, C, with corresponding Lie group, G, and a Cartan subalgebra, H, together with a system of simple positive roots. Let G = N − ⊕ H ⊕ N + , be the corresponding Cartan decomposition, where N + , N − are the nilpotent subalgebras which are, respectively, the sum of positive root spaces and the sum of the negative root spaces. Denote by G * and H * the C linear duals of G and H. Given λ ∈ G * , let O λ be the orbit of λ under the coadjoint action of G on G *. Finally, let S(G) be the symmetric algebra of G considered as polynomial functions on G * and I λ the ideal of polynomials vanishing on the orbit O λ. Then F λ = S(G)/I λ is the algebra of functions on O λ , with Poisson structure given by the standard Kostant-Kirillov-Souriau (KKS) bracket {f, g}(λ) = λ, [df λ , dg λ ]. Consider the G orbits of semisimple elements, that is, linear functions conjugate under the coadjoint action to ones which vanishes on N + ⊕ N −. In any such orbit we can pick as origin an element, λ, which is the trivial extension to G of a functional on H. The stabilizer of such an element will be a subalgebra generated by H and a subset of the simple roots. Such a subalgebra is called a Levi subalgebra. The set of all λ with stabilizer equal to a fixed Levi algebra L is parametrized by a subspace of H * minus its intersection with a family of coordinate hyperplanes.
SIAM Journal on Numerical Analysis, 1999
This paper introduces a new class of methods, which we call Möbius schemes, for the numerical sol... more This paper introduces a new class of methods, which we call Möbius schemes, for the numerical solution of matrix Riccati differential equations. The approach is based on viewing the Riccati equation in its natural geometric setting, as a flow on the Grassmanian of m-dimensional subspaces of an (n + m)-dimensional vector space. Since the Grassmanians are compact differentiable manifolds, and the coefficients of the equation are assumed continuous, there are no singularities or intrinsic instabilities in the associated flow. The presence of singularities and numerical instabilitites is an artefact of the coordinate system, but since Möbius schemes are based on the natural geometry, they are able to deal with numerical instability and pass accurately through the singularities. A number of examples are given to demonstrate these properties.
Perspectives on Science, 2013