Zoran Skoda
symmetric ordering for Lie algebras

Given a Lie algebra LL over a field kk (for some facts it is enough to require just that LL is flat over the ground ring kk), there are many vector space isomorphisms between U(L)U(L) and S(L)S(L); there is also a more narrow class of those vector space isomorphisms which are tautological on kLk\oplus L. A strong form of PBW theorem says that in characteristics zero the unique kk-linear map e:S(L)U(L)e: S(L)\to U(L) such that

y 1y k1k! σS ky^ σ1y^ σky_1\cdots y_k \mapsto \frac{1}{k!}\sum_{\sigma\in S_k} \hat{y}_{\sigma 1} \cdots \hat{y}_{\sigma k}

for all y 1,y kLS(L)y_1\ldots, y_k\in L\subset S(L), where y^LU(L)\hat{y}\in L\subset U(L) is the same element as yy but understood as belonging to U(L)U(L), is the isomorphism of kk-coalgebras. This map is called the coexponential or the symmetrization map. It is also characterized by the property that for any element yLy\in L its nn-th power y ny^n maps to y^ n\hat{y}^n. In fact the coexponential map is the only such isomorphism of coalgebras which is tautological on kLk\oplus L and also functorial in LL. Other isomorphism of coalgebras tautological on kLk\oplus L are called generalized symmetrization maps.

Let LL be now nn-dimensional over LL where nn is finite. There is an embedding of U(L)U(L) into a semi-completed Weyl algebra A^ nS(L)S^(L *)\hat{A}_n\cong S(L)\sharp \hat{S}(L^*) given by a universal formula on generators. Effectively, the elements of U(L)U(L) are realized as “infinite order differential operators” (or first order if we use the automorphism x i i, jx jx_i\mapsto \partial^i, \partial_j\mapsto -x^j. Suppose x 1,,x nx_1,\ldots,x_n to be a basis of LL. Then the embedding is given on generators by x^ α λx λϕ α λ\hat{x}_\alpha \mapsto \sum_\lambda x_\lambda \phi^\lambda_\alpha where ϕ α λ= N=0 (1) NB NN!( N) α λ\phi^\lambda_\alpha = \sum_{N=0}^\infty \frac{(-1)^N B_N }{N!}(\mathbb{C}^N)^\lambda_\alpha, B NB_N are Bernoulli numbers and N\mathbb{C}^N is the NN-th power of the matrix \mathbb{C} which is a n×nn\times n-matrix with values in L *L^* with entries β α= γ=1 nC βγ α γ\mathbb{C}^\alpha_\beta = \sum_{\gamma = 1}^n C^\alpha_{\beta\gamma}\partial^\gamma and C βγ αC^\alpha_{\beta\gamma} are the structure constants determined by [x^ β,x^ γ]=C βγ αx α[\hat{x}_\beta, \hat{x}_\gamma] = C^\alpha_{\beta\gamma} x^\alpha.

If we compose the above embedding U(L)A^ nU(L)\hookrightarrow \hat{A}_n with the action of A^ n\hat{A}_n on the unit vector (“vacuum”) of the Fock module S(L)S(L) we obtain the inverse of the symmetrization map.

The symmetrization map e:S(L)U(L)e:S(L)\to U(L) transfers the noncommutative product from U(L)U(L) to S(L)S(L): if f,gS(L)f,g\in S(L) then fg=e 1(e(f) U(L)e(g)f\star g = e^{-1}(e(f)\cdot_{U(L)} e(g). The new product on S(L)S(L) is called the star product in symmetric ordering.

  • N. Bourbaki, Lie groups and algebras

  • N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, Journal of Algebra 309, n.1, pp.318–359 (2007) math.RT/0604096.

Revised on May 25, 2010 18:07:30 by Zoran Škoda (