On sl(2)-equivariant quantizations

Abstract

By computing certain cohomology of Vect(M ) of smooth vector fields we prove that on 1-dimensional manifolds M there is no quantization map intertwining the action of non-projective embeddings of the Lie algebra sl(2) into the Lie algebra Vect(M ). Contrariwise, for projective embeddings sl(2)-equivariant quantization exists.

The space of m-ary differential operators as a module over the Lie algebra of vector fields

“…the symbol may not exit, as already pointed out in [3] for the unary case. Throughout this paper, sl( 2) is realized as in (1.2).…”

“…Let us put the direct sum ⊕F λ counted R(i, m)-times. The symbol map is as follows: Theorem 6.1 For all δ ∈ {1, 3 2 , 2, . .…”

“…, k. We can ask whether there exists an appropriate principal symbol for which the equivariant quantization maps turn into Vect(M )-equivariant ones. Theorem 8.1 For δ ∈ {1, 3 2 , . .…”

“…Theorem 10.2 (i) For δ = 1, all modules D 1 λ;µ are isomorphic provided they have the same shift δ. (ii) For δ = 1, 3 2 , 2, all modules D 2 λ;µ are isomorphic provided they have the same shift δ; however, the modules…”

“…(i) For δ = 1, a sl(2)-equivariant map S δ → D λ;µ exists only for λ = 0. (ii) For δ =3 2 , a sl(2)-equivariant map S δ → D λ;µ exists only for λ = 0 or λ = − 1 2 1 j , where j = 1, . .…”