Lie Group - The Concept of A Lie Group, and Possibilities of Classification

The Concept of A Lie Group, and Possibilities of Classification

Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis. What must be understood is the nature of 'small' transformations, e.g., rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie algebra (Lie himself called them "infinitesimal groups"). It can be defined because Lie groups are manifolds, so have tangent spaces at each point.

The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" An, Bn, Cn and Dn, which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E8 is the largest of these.

Read more about this topic:  Lie Group

Famous quotes containing the words concept and/or lie:

    Modern man, if he dared to be articulate about his concept of heaven, would describe a vision which would look like the biggest department store in the world, showing new things and gadgets, and himself having plenty of money with which to buy them. He would wander around open-mouthed in this heaven of gadgets and commodities, provided only that there were ever more and newer things to buy, and perhaps that his neighbors were just a little less privileged than he.
    Erich Fromm (1900–1980)

    Probably nature itself gave man the ability to lie so that in difficult and tense moments he could protect his nest, just as do the vixen and wild duck.
    Anton Pavlovich Chekhov (1860–1904)