Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 1st 2012
   • (edited Mar 1st 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a bit to [[Heisenberg Lie algebra]]. Mostly, I wrote a section _[Relation to Poisson algebra](http://ncatlab.org/nlab/show/Heisenberg+Lie+algebra#RelationToPoissonAlgebra)_ with a discussion of how the Heisenberg algebra naturally sits inside the Lie algebra underlying the Poisson algebra.

   added a bit to Heisenberg Lie algebra.

   Mostly, I wrote a section Relation to Poisson algebra with a discussion of how the Heisenberg algebra naturally sits inside the Lie algebra underlying the Poisson algebra.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 1st 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexHm, right after saving this, the Lab went down...

   Hm, right after saving this, the Lab went down…

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeMar 1st 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexBut, there seems no entry [[Heisenberg algebra]].

   But, there seems no entry Heisenberg algebra.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 1st 2012
   • (edited Mar 1st 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexDarn. Sorry. I meant to point to _[[Heisenberg Lie algebra]]_. [edit: I have made _[[Heisenberg algebra]]_ a redirect now. ]

   Darn. Sorry. I meant to point to Heisenberg Lie algebra.

   [edit: I have made Heisenberg algebra a redirect now. ]

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeMar 1st 2012
   • (edited Mar 1st 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexOh, now I see I even contributed to it (version [1](http://ncatlab.org/nlab/revision/Heisenberg+Lie+algebra/1))... :) It is growing nicely :)

   Oh, now I see I even contributed to it (version 1)… :) It is growing nicely :)

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 1st 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, thanks!

   Okay, thanks!

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeMar 1st 2012
   • (edited Mar 1st 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI added a paragraph on the relation of [[Heisenberg Lie algebra]] to certain construction of associative algebras, so called [[Heisenberg double]], which generalized the [[Weyl algebra]]. In fact there are several associative algebras related to Heisenberg algebra. One can look simply at its universal enveloping algebra. Then one can extend it by a central element and/or with the "number operator". One obtains variants called Heisenbeg-Weyl algebra, Weyl algebra, CCR-algebra and alike, which for various authors mean the same or a different thing, depending if the central element is 1 or not, if the number operator $N$ is a separate generator or not and so on. I do not know how will $n$Lab solve this terminologically difficult point.

   I added a paragraph on the relation of Heisenberg Lie algebra to certain construction of associative algebras, so called Heisenberg double, which generalized the Weyl algebra. In fact there are several associative algebras related to Heisenberg algebra. One can look simply at its universal enveloping algebra. Then one can extend it by a central element and/or with the “number operator”. One obtains variants called Heisenbeg-Weyl algebra, Weyl algebra, CCR-algebra and alike, which for various authors mean the same or a different thing, depending if the central element is 1 or not, if the number operator $N$ is a separate generator or not and so on. I do not know how will $n$Lab solve this terminologically difficult point.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMar 1st 2012
   • (edited Mar 2nd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. I have split that up into two new subsections: _Relation to Weyl algebra_ and _Relation to Heisenberg double_. I am not sure if I understood your "if another central ement is added" correctly: let's see: the Weyl algebra is the quotient of the universal enveloping algebra of the Heisenberg Lie algebra obtained by identifying the central elements with multiples of the identity (hence by removing a central element). Agreed?

   Thanks. I have split that up into two new subsections: Relation to Weyl algebra and Relation to Heisenberg double.

   I am not sure if I understood your “if another central ement is added” correctly:

   let’s see: the Weyl algebra is the quotient of the universal enveloping algebra of the Heisenberg Lie algebra obtained by identifying the central elements with multiples of the identity (hence by removing a central element).

   Agreed?

9. • CommentRowNumber9.
   • CommentAuthorzskoda
   • CommentTimeMar 2nd 2012
   • (edited Mar 2nd 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI agree; Weyl algebra is smaller. [[Heisenberg double]] generalizes Weyl algebra. CCR algebra (canonical commutation relation algebra, or boson oscillator algebra) is essentially the same as the Weyl algebra, possibly in infinite dimension, but the noncentral number operator is taken as central, however $N-a^\dagger a$ is central, so the question is if it is set to zero or not.

   I agree; Weyl algebra is smaller. Heisenberg double generalizes Weyl algebra. CCR algebra (canonical commutation relation algebra, or boson oscillator algebra) is essentially the same as the Weyl algebra, possibly in infinite dimension, but the noncentral number operator is taken as central, however $N-a^\dagger a$ is central, so the question is if it is set to zero or not.