Search results for "Lie group"

Multiplicative loops of 2-dimensional topological quasifields

2015

We determine the algebraic structure of the multiplicative loops for locally compact $2$-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a $1$-dimensional compact subgroup. In the last section we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension $4$ admitting an at least $7$-dimensional Lie group as collineation group.

Lie Algebras Generated by Extremal Elements

1999

We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals of L) over a field of characteristic distinct from 2. We prove that any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal number of extremal generators for the Lie algebras of type An, Bn (n>2), Cn (n>1), Dn (n>3), En (n=6,7,8), F4 and G2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.

Cohomology of Lie algebras

1995

This chapter is devoted to studying some concepts that will be extensively used in the last chapters, namely the cohomology of Lie algebras with values in a vector space, the Whitehead lemmas and Lie algebra extensions (which are related to second cohomology groups). The same three different cases of extensions of chapter 5 as well as the ℱ( M )-valued version of cohomology will be considered. In fact, the relation between Lie group and Lie algebra cohomology will be explored here, first with the simple example of central extensions of groups and algebras (governed by twococycles), and then in the higher order case, providing explicit formulae for obtaining Lie algebra cocycles from Lie gro…

Group-Theoretic analysis of the mixing angle in the electroweak gauge group

1996

In this paper the authors provide strong mathematical support for the idea that the experimentally measured magnitude 1 - M{sub W}{sup 2}/M{sub Z}{sup 2} associated with sin{sup 2}{theta}{sub w} in the standard model of electroweak interactions cannot be simultaneously identified with the squared quotient of the electric charge by the SU(2) charge, e{sup 2}/g{sup 2}. In fact, the natural, mathematical requirement that the Weinberg rotation between the gauge fields associated with the third component of the {open_quotes}weak isospin{close_quotes} (T{sub 3}) and the hypercharge (Y) proceeds from a global Lie-group homomorphism of the SU(2) {circle_times} U(1){sub y} gauge group in some locall…

Entropy, Lyapunov exponents, and rigidity of group actions

2018

This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teres\'opolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [Ergodic Theory Dynam. Systems…

A Comparison between Star Products on Regular Orbits of Compact Lie Groups

2001

In this paper an algebraic star product and differential one defined on a regular coadjoint orbit of a compact semisimple group are compared. It is proven that there is an injective algebra homomorphism between the algebra of polynomials with the algebraic star product and the algebra of differential functions with the differential star product structure.

Infinite lie groups of point transformations leaving invariant the linear equation which describes in the hodograph plane the isentropic one-dimensio…

1991

Abstract The group analysis of the hodograph equation which is equivalent to the non-linear system of one-dimensional isentropic gas dynamics reveals the existence of infinite groups of symmetry in correspondence with particular pressure laws. These turn out to be polytropes with selected indices, as is expected, as well as a new type of pressure. In all these cases the hodograph equation can be transformed, by a suitable change of variables, into the wave equationψ ζ = 0.

Basic theory of solvable Lie algebras and Lie groups

2020

Finite renormalization effects in the induceds¯dHvertex

1986

The finite renormalization contributions to the s-bard-italicH-italic vertex are examined in the standard model. They are explicitly shown to cancel each other among diagrams, so that the lower bound on the Higgs-boson mass M-italic/sub H-italic/>325 MeV is not affected by such effects.

Separation of representations with quadratic overgroups

2011

AbstractAny unitary irreducible representation π of a Lie group G defines a moment set Iπ, subset of the dual g⁎ of the Lie algebra of G. Unfortunately, Iπ does not characterize π. If G is exponential, there exists an overgroup G+ of G, built using real-analytic functions on g⁎, and extensions π+ of any generic representation π to G+ such that Iπ+ characterizes π.In this paper, we prove that, for many different classes of group G, G admits a quadratic overgroup: such an overgroup is built with the only use of linear and quadratic functions.