Central Extension of The Galilean Group
The Galilean group: Here, we will only look at its Lie algebra. It's easy to extend the results to the Lie group. The Lie algebra of L is spanned by H, Pi, Ci and Lij (antisymmetric tensor) subject to commutators, where
H is generator of time translations (Hamiltonian), Pi is generator of translations (momentum operator), Ci is generator of Galileian boosts and Lij stands for a generator of rotations (angular momentum operator).
We can now give it a central extension into the Lie algebra spanned by H', P'i, C'i, L'ij (antisymmetric tensor), M such that M commutes with everything (i.e. lies in the center, that's why it's called a central extension) and
Read more about this topic: Galilean Transformation
Famous quotes containing the words central, extension, galilean and/or group:
“The best laws cannot make a constitution work in spite of morals; morals can turn the worst laws to advantage. That is a commonplace truth, but one to which my studies are always bringing me back. It is the central point in my conception. I see it at the end of all my reflections.”
—Alexis de Tocqueville (1805–1859)
“The motive of science was the extension of man, on all sides, into Nature, till his hands should touch the stars, his eyes see through the earth, his ears understand the language of beast and bird, and the sense of the wind; and, through his sympathy, heaven and earth should talk with him. But that is not our science.”
—Ralph Waldo Emerson (1803–1882)
“The Galilean is not a favourite of mine. So far from owing him any thanks for his favour, I cannot avoid confessing that I owe a secret grudge to his carpentership.”
—Percy Bysshe Shelley (1792–1822)
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)