Werner Heisenberg the lattice theory of crystals in 1921. While matrices were used in these cases, the algebra of matrices with their multiplication did not enter the picture as they did in the matrix formulation of quantum mechanics. In 1928, Albert Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics, The announcement of the Nobel Prize in Physics for 1932 was delayed until November 1933. It was at that time that it was announced Heisenberg had won the Prize for 1932 "for the creation of quantum mechanics, the application of which has, , led to the discovery of the
Heisenberg group operators on formula_103: for formula_109. So too do the multiplications by characters: for formula_111. These operators do not commute, and instead satisfy multiplication by a fixed unit modulus complex number. So the Heisenberg group formula_113 associated with "K" is a type of central extension of formula_114, via an exact sequence of groups: More general Heisenberg groups are described by 2-cocyles in the cohomology group formula_116. The existence of a duality between formula_117 and formula_101 gives rise to a canonical cocycle, but there are generally others. The Heisenberg group acts irreducibly on formula_103. Indeed, the continuous characters separate points so any