Zoran Skoda
differential equation for realizations

Let x 1,,x nx_1,\ldots,x_n be commutative generators of a symmetric algebra S(V)S(V) where V=Span{x 1,,x n}V = Span\{x_1,\dots,x_n\}, and let x^ 1,,x^ n\hat{x}_1,\ldots,\hat{x}_n be the generators of nn-dimensional Lie algebra LL and also of he corresponding (noncommutative) generators of the enveloping algebra U(L)U(L).

Let A nA_n be the nn-th Weyl algebra, which is canonically isomorphic as a vector space to S(V)S(V *)S(V)\otimes S(V^*) as a vector space and A^ n\hat{A}_n its (semi-)completion by the degree of the differential operator. Hence in A^ n\hat{A}_n as a vector space is S(V)S^(V *)S(V)\otimes \hat{S}(V^*) where S^(V *)\hat{S}(V^*) is the space of formal power series in the dual coordinates 1,, n\partial^1,\ldots,\partial^n to x 1,,x nx_1,\ldots, x_n, which play role of the partial derivatives in the semi-completed Weyl algebra.

Suppose () ϕ:LLie(A n)()^\phi : L\to Lie(A_n) is a realization of the Lie algebra LL by differential operators (that is a homomorphism of Lie algebras into the Lie algebra of the associative algebra A nA_n) of the special form

x^ ix^ i ϕ= kx jϕ j i \hat{x}_i\mapsto \hat{x}_i^\phi = \sum_k x_j \phi^i_j

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

r(δ rϕ i k)ϕ j r(δ rϕ j k)ϕ i r= sC ij sϕ s k,i,j,k=1,n. \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 x^ i ϕ\hat{x}^\phi_i defines a homomorphism of Lie algebras () ϕ()^\phi if this differential system holds. In that case also ϕ(x^ i)( j)=ϕ i j\mathbf{\phi}(-\hat{x}_i)(\partial^j) = \phi^j_i extends to a homomorphism ϕ:LEnd(S^(V *))\mathbf{\phi}: L\to End(\hat{S}(V^*)) of Lie algebras. The latter is equivalent to a Hopf action ϕ˜:U(L)End(S^(V *))\tilde{\mathbf{\phi}} : U(L)\to End(\hat{S}(V^*)) which in turn defines a smash product A ϕ:=U(L)S(V *)^A_{\mathbf{\phi}} := U(L)\sharp \hat{S(V^*)}. If ϕ j i=δ j i+higherorderin\phi^i_j = \delta^i_j + higher order in \partial then there is an algebra isomorphism A ϕA nA_{\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.