differential equation for realizations

Let $x_1,\ldots,x_n$ be commutative generators of a symmetric algebra $S(V)$ where $V = Span\{x_1,\dots,x_n\}$, and let $\hat{x}_1,\ldots,\hat{x}_n$ be the generators of $n$-dimensional Lie algebra $L$ and also of he corresponding (noncommutative) generators of the enveloping algebra $U(L)$.

Let $A_n$ be the $n$-th Weyl algebra, which is canonically isomorphic as a vector space to $S(V)\otimes S(V^*)$ as a vector space and $\hat{A}_n$ its (semi-)completion by the degree of the differential operator. Hence in $\hat{A}_n$ as a vector space is $S(V)\otimes \hat{S}(V^*)$ where $\hat{S}(V^*)$ is the space of formal power series in the dual coordinates $\partial^1,\ldots,\partial^n$ to $x_1,\ldots, x_n$, which play role of the partial derivatives in the semi-completed Weyl algebra.

Suppose $()^\phi : L\to Lie(A_n)$ is a realization of the Lie algebra $L$ by differential operators (that is a homomorphism of Lie algebras into the Lie algebra of the associative algebra $A_n$) of the special form

$\hat{x}_i\mapsto \hat{x}_i^\phi = \sum_k x_j \phi^i_j$

where $\phi^i_j \in \hat{S}(V^*)$ are formal power series in $\partial^1,\ldots,\partial^n$. Denote $\delta_j = \frac{\partial}{\partial(\partial^j)}$. Then the following system of $n^3$ formal differential equations holds

$\sum_r (\delta_r \phi^k_i)\phi^r_j - (\delta_r \phi^k_j) \phi^r_i = \sum_s C^s_{ij} \phi^k_s, \,\,\,i,j,k = 1,\ldots n.$

Conversely, the formula above for $\hat{x}^\phi_i$ defines a homomorphism of Lie algebras $()^\phi$ if this differential system holds. In that case also $\mathbf{\phi}(-\hat{x}_i)(\partial^j) = \phi^j_i$ extends to a homomorphism $\mathbf{\phi}: L\to End(\hat{S}(V^*))$ of Lie algebras. The latter is equivalent to a Hopf action $\tilde{\mathbf{\phi}} : U(L)\to End(\hat{S}(V^*))$ which in turn defines a smash product $A_{\mathbf{\phi}} := U(L)\sharp \hat{S(V^*)}$. If $\phi^i_j = \delta^i_j + higher order in \partial$ then there is an algebra isomorphism $A_{\mathbf{\phi}}\cong A_n$.

Created on June 24, 2011 at 10:59:39. See the history of this page for a list of all contributions to it.