Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν‎ and uses the usual convention that ν‎(0) = infinity. The chapter starts with a notation regarding the ring of integers of K and the natural map from it to the residue field, followed by a number of propositions regarding an anisotropic quadratic space. These include an anisotropic quadratic space with residual quadratic spaces, an unramified quadratic space of finite dimension, unramified finite-dimensional anisotropic quadratic forms over K, unramified anisotropic quadratic forms and a bilinear form, and a round quadratic space over K. The chapter concludes with a theorem that there exists an anisotropic quadratic form over K, unique up to isometry, and is non-singular.

The primary purpose of this chapter is the analysis of the group Wq(R) in the case of an arithmetic Dedekind domain R. The characterization of the elements of Br(R)2 as the Brauer classes of Clifford algebras of quadratic spaces over R of rank 4 and trivial Arf invariant is an important step along the way. The quadratic spaces which it provides make it possible to describe Ker Arf by use of the total signature. In the special case where R is the ring of integers in a number field, the quadratic group Qu(R) is closely related to the ideal class group, and this connection together with the description of Ker Arf implies that Wq(R) ≅ Cl(R)2 ⊕ G, where C1(R) is the ideal class group of R and G is a free Abelian group of rank with r the number of real embeddings of the number field. An additional focus is the comparison of the number theory of Wq(R) with that of W(R) and the structure of the quotient W(R)/Wq(R).

We apply G. Prasad's volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of SO(1, n).As a result we prove that for any even dimension n there exists a unique compact arithmetic hyperbolic n-orbifold of the smallest volume.We give a formula for the Euler-Poincaré characteristic of the orbifolds and present an explicit description of their fundamental groups as the stabilizers of certain lattices in quadratic spaces.We also study hyperbolic 4-manifolds defined arithmetically and obtain a number theoretical characterization of the smallest compact arithmetic 4-manifold.

The hidden universal Clifford algebra structure of M4(R) is ambiguous in the sense that this matrix algebra is the universal geometric algebra belonging to each of the real four-dimensional quadratic vector spaces R1,3 and R2,2. As a non-universal Clifford algebra, however, M4(R) is a geometric algebra of the unique five-dimensional quadratic vector space R2,3.

Small representations of integers by integral quadratic forms

The present article is motivated by the theorem of Cartan-Dieudonn\'e which states that every orthogonal transformation is a product of reflections. Its purpose is to determine, for each orthogonal transformation, the minimal number of factors in a decomposition into a product of reflections, and to propose an effective algorithm giving such a decomposition. With the orthogonal transformations $g$ of a quadratic space $(V,q)$, it associates couples $(S,\phi)$ where $S$ is a subspace of $V$, and $\phi$ an non-degenerate bilinear form on $S$ such that $\phi(y,y)=q(y)$ for every $y$ in $S$. In general, the minimal decompositions of $g$ into a product of reflections correspond to the bases of $S$ in which the matrix of $\phi$ is lower triangular. Therefore, we need an algorithm of triangularization of bilinear forms. Affine isometries are also taken into consideration.

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field [Formula: see text], p-ring spaces Vp(c) modulo c are introduced by defining a morphism ψ : f ↦ Vp(f) from the divisor lattice ℕ of positive integers to the lattice 𝒮 of subspaces of the direct product Vpof the p-elementary class group 𝒞/𝒞pand unit group U/Upof K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group Gal (N | ℚ) and sharing a common discriminant dNand conductor c over K. The number mp(d, c) of these extensions is given by a formula in terms of positions of p-ring spaces in 𝒮, whose complexity increases with the dimension of the vector space Vpover the finite field 𝔽p, called the modified p-class rank σpof K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with 0 ≤ σp≤ 1 only. Here, the results are extended to σp= 2, underpinned by concrete numerical examples.

We describe a new class of holonomy groups on pseudo-Riemannian manifolds. Namely, let $g$ be a nondegenerate bilinear form on a vector space $V$, and $L : V \to V$ a $g$-symmetric operator. Then the identity component of the centraliser of $L$ in $\mathrm{SO}(g)$ is a holonomy group for a suitable Levi-Civita connection.

Full field algebras, operads and tensor categories

Residually finite quantum group algebras

A Lie algebra is said to be metric if it admits a symmetric, invariant, and non-degenerate bilinear form. The harmonic oscillator algebra, which arises in the quantum mechanical description of a harmonic oscillator, is the smallest solvable non-abelian metric example. This algebra is the first step of a countable series of solvable Lie algebras that support invariant Lorentzian forms. Generalizing this situation, in this paper, we arrive at the oscillator Lie K -algebras as double extensions of metric spaces. The aim of this paper is to present some structural features, invariant metrics, and derivations of this class of algebras and to explore their possibilities of being extended to mixed metric Lie algebras.

Quadratic Dynamical Systems and Algebras

We show how both the contact structure and the metric structure of the thermodynamic phase space arise in a natural way from a generalized canonical probability distribution \ensuremath{\rho}. In particular, the metric form and the contact form are found to be derived from the microscopic entropy s=-ln\ensuremath{\rho}. Thus the first law and the second law of thermodynamics can be given the geometric interpretation that a thermodynamic system must possess both a contact and a compatible metric structure. We proceed to construct explicitly a new nondegenerate bilinear form on the thermodynamic phase space, whose restriction to state space yields the Weinhold-Ruppeiner metric, and whose restriction to Gibbs space can serve as an alternative to the metric proposed by Gilmore.

On orthogonal groups over the integers of a non-dyadic local field

Decompositions in categories of highest weight modules