Papers by Norman Wildberger
Geometriae Dedicata, 2007
One dimensional metrical geometry may be developed in either an affine or projective setting over... more One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this situation, such as the Triple quad formula, the Triple spread formula and the Spread polynomials, which are universal analogs of the Chebyshev polynomials of the first kind. Chromogeometry appears here, and the related metrical and algebraic properties of the projective line are brought to the fore.
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean ge... more By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are introduced here, and the main formulas extend those of rational trigonometry in the plane. This gives a unified, computational model of both spherical and hyperbolic geometries, allows the extension of many results of Euclidean geometry to the relativistic setting, and provides a new metrical approach to algebraic geometry.
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean ge... more By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are introduced here, and the main formulas extend those of rational trigonometry in the plane. This gives a unified, computational model of both spherical and hyperbolic geometries, allows the extension of many results of Euclidean geometry to the relativistic setting, and provides a new metrical approach to algebraic geometry.
Inventiones Mathematicae, 1989
The moment map of symplectic geometry is extended to associate to any unitary representation of a... more The moment map of symplectic geometry is extended to associate to any unitary representation of a nilpotent Lie group aG-invariant subset of the dual of the Lie algebra. We prove that this subset is the closed conex hull of the Kirillov orbit of the representation.
International Journal of Algebra and Computation
In [W], there is a graphic description of any irreducible, finite dimensional sl(3) module. This ... more In [W], there is a graphic description of any irreducible, finite dimensional sl(3) module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite dimensional U q (sl )-modules.
In \cite{W}, there is a graphic description of any irreducible, finite dimensional $\mathfrak{sl}... more In \cite{W}, there is a graphic description of any irreducible, finite dimensional $\mathfrak{sl}(3)$ module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite dimensional ${\mathcal U}\_q(\mathfrak{sl}(3))$-modules. In the present work, we generalize this construction to $\mathfrak{sl}(n)$. We show this is in fact a description of the reduced shape algebra, a quotient of the shape algebra of $\mathfrak{sl}(n)$. The basis used in \cite{W} is thus naturally parametrized with the so called quasi standard Young tableaux. To compute the matrix coefficients of the representation in this basis, it is possible to use Groebner basis for the ideal of reduced Pl\"{u}cker relations defining the reduced shape algebra.
This paper is a personal look at some issues in the representation theory of Lie groups having to... more This paper is a personal look at some issues in the representation theory of Lie groups having to do with the role of commutative hypergroups, bi-modules, and the construction of representations. We begin by considering Frobenius' original approach to the character theory of a finite group and extending it to the Lie group setting, and then introduce bi-modules as objects intermediate between characters and representations in the theory. A simplified way of understanding the formalism of geometric quantization, at least for compact Lie groups, is presented, which leads to a canonical bi-module of functions on an integral coadjoint orbit. Some meta-mathematical issues relating to the construction of representations are considered.
A new model for the irreducible representations of sl 3 is presented which is constructed over th... more A new model for the irreducible representations of sl 3 is presented which is constructed over the integers. This model utilizes the combinatorial geometry of certain polytopes in three dimensional space which we call diamonds. These are not Gelfand-Tsetlin polytopes, but share some of their properties. Matrix coefficients are directly computable in terms of maximal ladders of edges of given directions and type in the diamonds. We show that the generic diamond is the vector sum of dilates of the fundamental diamonds associated to quark and anti-quark triples, and is simultaneously both a classical and quantum object.
Advances in Applied Mathematics, 2003
This paper shows how to uniformly associate Lie algebras to the simply-laced Dynkin diagrams excl... more This paper shows how to uniformly associate Lie algebras to the simply-laced Dynkin diagrams excluding E 8 by constructing explicit combinatorial models of minuscule representations using only graph-theoretic ideas. This involves defining raising and lowering operators in a space of ideals of certain distributive lattices associated to sequences of vertices of the Dynkin diagram.
We show how to construct the simple exceptional Lie algebra of type G 2 by explicitly constructin... more We show how to construct the simple exceptional Lie algebra of type G 2 by explicitly constructing its 7 dimensional representation. Technically no knowledge of Lie theory is required. The structure constants have a combinatorial meaning involving convex subsets of a partially ordered multiset of six elements. These arise from playing the Numbers and Mutation games on a certain directed multigraph.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, a... more JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].
Uploads
Papers by Norman Wildberger