Let be a commutative ring. Let be a http://planetmath.org/node/5420module over http://planetmath.org/node/5420freely generated by two sets and , where is an index set, and a further element . Define a product by bilinear extension by setting
The module together with this product is called a Heisenberg algebra. The element is called the central element.
It is easy to see that the product also fulfills the Jacobi identity, so a Heisenberg algebra is actually a Lie algebra of rank (opposed to the rank of as free module, which is ) with one-dimensional center generated by .
Heisenberg algebras arise in quantum mechanics with and typically , but also in the theory of vertex with .
|Date of creation||2013-03-22 15:24:57|
|Last modified on||2013-03-22 15:24:57|
|Last modified by||GrafZahl (9234)|