Elementary Lie Theory
The one-parameter groups are the first instance of Lie theory. The compact case arises through Euler's formula in the complex plane. Other one-parameter groups occur in the split-complex number plane as the unit hyperbola
and in the dual number plane as the line In these cases the Lie algebra parameters have names: angle, hyperbolic angle, and slope. Using the appropriate "angle", and a radial vector, any one of these planes can be given a polar decomposition. Any one of these decompositions, or Lie algebra renderings, may be necessary for rendering the Lie subalgebra of a 2 × 2 real matrix.
There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere. Its Lie algebra is the subspace of quaternion vectors. Since the commutator ij − ji = 2k, the Lie bracket in this algebra is twice the cross product of ordinary vector analysis.
Another elementary 3-parameter example is given by the Heisenberg group and its Lie algebra. Standard treatments of Lie theory often begin with the Classical groups.
Read more about this topic: Lie Theory
Famous quotes containing the words elementary, lie and/or theory:
“If men as individuals surrender to the call of their elementary instincts, avoiding pain and seeking satisfaction only for their own selves, the result for them all taken together must be a state of insecurity, of fear, and of promiscuous misery.”
—Albert Einstein (1879–1955)
“... when one reflects on the books one never has written, and never may, though their schedules lie in the beautiful chirography which marks the inception of an unexpressed thought upon the pages of one’s notebook, one is aware, of any given idea, that the chances are against its ever being offered to one’s dearest readers.”
—Elizabeth Stuart Phelps (1844–1911)
“Don’t confuse hypothesis and theory. The former is a possible explanation; the latter, the correct one. The establishment of theory is the very purpose of science.”
—Martin H. Fischer (1879–1962)