Editing Universal enveloping algebra (section)

==Algebra of symbols==
The underlying vector space of <math>S(\mathfrak{g})</math> may be given a new algebra structure so that  <math>U(\mathfrak{g})</math> and <math>S(\mathfrak{g})</math> are isomorphic ''as associative algebras''. This leads to the concept of the '''algebra of symbols''' <math>\star(\mathfrak{g})</math>: the space of [[symmetric polynomial]]s, endowed with a product, the <math>\star</math>, that places the algebraic structure of the Lie algebra onto what is otherwise a standard associative algebra. That is, what the PBW theorem obscures (the commutation relations) the algebra of symbols restores into the spotlight.

The algebra is obtained by taking elements of <math>S(\mathfrak{g})</math> and replacing each generator <math>e_i</math> by an indeterminate, commuting variable <math>t_i</math> to obtain the space of symmetric polynomials <math>K[t_i]</math> over the field <math>K</math>. Indeed, the correspondence is trivial: one simply substitutes the symbol <math>t_i</math> for <math>e_i</math>. The resulting polynomial is called the '''symbol''' of the corresponding element of <math>S(\mathfrak{g})</math>. The inverse map is
:<math>w: \star(\mathfrak{g})\to U(\mathfrak{g})</math>
that replaces each symbol <math>t_i</math> by <math>e_i</math>. The algebraic structure is obtained by requiring that the product <math>\star</math> act as an isomorphism, that is, so that
:<math>w(p \star q) = w(p)\otimes w(q)</math>
for polynomials <math>p,q\in \star(\mathfrak{g}).</math>

The primary issue with this construction is that <math>w(p)\otimes w(q)</math> is not trivially, inherently a member of <math>U(\mathfrak{g})</math>, as written, and that one must first perform a tedious reshuffling of the basis elements (applying the [[structure constants]] as needed) to obtain an element of <math>U(\mathfrak{g})</math> in the properly ordered basis. An explicit expression for this product can be given: this is the '''Berezin formula'''.<ref>{{cite journal | last1 = Berezin | first1 = F.A. | author-link = Felix Berezin | year = 1967 | title = Some remarks about the associated envelope of a Lie algebra | journal = Funct. Anal. Appl. | volume = 1 | issue = 2| page = 91 | doi=10.1007/bf01076082| s2cid = 122356554 }}</ref> It follows essentially from the [[Baker–Campbell–Hausdorff formula]] for the product of two elements of a Lie group.

A closed form expression is given by<ref>Xavier Bekaert, "[http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/ModaveI/Xavier.pdf Universal enveloping algebras and some applications in physics]" (2005) ''Lecture, Modave Summer School in Mathematical Physics''.</ref>

:<math>p(t)\star q(t)= \left. \exp\left(t_i 
m^i \left(\frac{\partial}{\partial u}, \frac{\partial}{\partial v} \right)
\right) p(u)q(v)\right \vert_{u=v=t}</math>

where
:<math>m(A,B)=\log\left(e^Ae^B\right)-A-B</math>
and <math>m^i</math> is just <math>m</math> in the chosen basis.

The universal enveloping algebra of the [[Heisenberg algebra]] is the [[Weyl algebra]] (modulo the relation that the center be the unit); here, the <math>\star</math> product is called the [[Moyal product]].