Simple Lie group in the context of Linear algebraic group

👉 Simple Lie group in the context of Linear algebraic group

In mathematics, a linear algebraic group is a subgroup of the group of invertible $n\times n$ matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation $M^{T}M=I_{n}$ where $M^{T}$ is the transpose of $M$ .

Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and Kolchin (1948). In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today.

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Simple Lie group in the context of Georgi–Glashow model

In particle physics, the Georgi–Glashow model is a particular Grand Unified Theory (GUT) proposed by Howard Georgi and Sheldon Glashow in 1974. In this model, the Standard Model gauge groups SU(3) × SU(2) × U(1) are combined into a single simple gauge group SU(5). The unified group SU(5) is then thought to be spontaneously broken into the Standard Model subgroup below a very high energy scale called the grand unification scale.

Since the Georgi–Glashow model combines leptons and quarks into single irreducible representations, there exist interactions which do not conserve baryon number, although they still conserve the quantum number B – L associated with the symmetry of the common representation. This yields a mechanism for proton decay, and the rate of proton decay can be predicted from the dynamics of the model. However, proton decay has not yet been observed experimentally, and the resulting lower limit on the lifetime of the proton contradicts the predictions of this model. Nevertheless, the elegance of the model has led particle physicists to use it as the foundation for more complex models which yield longer proton lifetimes, particularly SO(10) in basic and SUSY variants.

View the full Wikipedia page for Georgi–Glashow model

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Simple Lie group in the context of Linear algebraic group

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👉 Simple Lie group in the context of Linear algebraic group

Simple Lie group in the context of Georgi–Glashow model