Oksana Boiko - Academia.edu (original) (raw)
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We establish the absence of zero divisors in the reduction algebra of a Lie algebra g with respec... more We establish the absence of zero divisors in the reduction algebra of a Lie algebra g with respect to its reductive Lie sub-algebra k. The class of reduction algebras include the Lie algebras (they arise when k is trivial) and the Gelfand–Kirillov conjecture extends naturally to the reduction algebras. We formulate the conjecture for the diagonal reduction algebras of sl type and verify it on a simplest example. 1 Preliminaries Let k be a reductive Lie subalgebra of a Lie algebra g; that is, the adjoint action of k on g is completely reducible (in particular, k is reductive). Fix a triangular decomposition of the Lie algebra k, k = n − + h + n +. (1.1) Denote by ∆ + and ∆ − the sets of positive and negative roots in the root system ∆ = ∆ + ∪ ∆ − of k. For each root α ∈ ∆ let h α = α ∨ ∈ h be the corresponding coroot vector. Denote by U(h) the ring of fractions of the commutative algebra U(h) relative to the set of denominators { h α + l | α ∈ ∆, l ∈ Z }. (1.2) The elements of this ring can also be regarded as rational functions on the vector space h *. The elements of U(h) ⊂ U(h) are then regarded as polynomial functions on h *. Let U(k) ⊂ ¯ A = U(g) be the rings of fractions of the algebras U(k) and A = U(g) relative to the set of denominators (1.2). These rings are well defined, because both U(k) and U(g) satisfy the Ore condition relative to (1.2); we give a short proof in the second part of Appendix. Define Z(g, k) to be the double coset space of ¯ A by its left ideal I + := ¯ An + , generated by elements of n + , and the right ideal I − := n − ¯ A, generated by elements of n − , Z(g, k) := 1 On leave of absence from P.N. Lebedev Physical Institute, Theoretical Department, Leninsky prospekt 53, 119991 Moscow, Russia 2 Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix–Marseille I, Aix–Marseille II et du Sud Toulon – Var; laboratoire affiliéà la FRUMAM (FR 2291)
We establish the absence of zero divisors in the reduction algebra of a Lie algebra g with respec... more We establish the absence of zero divisors in the reduction algebra of a Lie algebra g with respect to its reductive Lie sub-algebra k. The class of reduction algebras include the Lie algebras (they arise when k is trivial) and the Gelfand–Kirillov conjecture extends naturally to the reduction algebras. We formulate the conjecture for the diagonal reduction algebras of sl type and verify it on a simplest example. 1 Preliminaries Let k be a reductive Lie subalgebra of a Lie algebra g; that is, the adjoint action of k on g is completely reducible (in particular, k is reductive). Fix a triangular decomposition of the Lie algebra k, k = n − + h + n +. (1.1) Denote by ∆ + and ∆ − the sets of positive and negative roots in the root system ∆ = ∆ + ∪ ∆ − of k. For each root α ∈ ∆ let h α = α ∨ ∈ h be the corresponding coroot vector. Denote by U(h) the ring of fractions of the commutative algebra U(h) relative to the set of denominators { h α + l | α ∈ ∆, l ∈ Z }. (1.2) The elements of this ring can also be regarded as rational functions on the vector space h *. The elements of U(h) ⊂ U(h) are then regarded as polynomial functions on h *. Let U(k) ⊂ ¯ A = U(g) be the rings of fractions of the algebras U(k) and A = U(g) relative to the set of denominators (1.2). These rings are well defined, because both U(k) and U(g) satisfy the Ore condition relative to (1.2); we give a short proof in the second part of Appendix. Define Z(g, k) to be the double coset space of ¯ A by its left ideal I + := ¯ An + , generated by elements of n + , and the right ideal I − := n − ¯ A, generated by elements of n − , Z(g, k) := 1 On leave of absence from P.N. Lebedev Physical Institute, Theoretical Department, Leninsky prospekt 53, 119991 Moscow, Russia 2 Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix–Marseille I, Aix–Marseille II et du Sud Toulon – Var; laboratoire affiliéà la FRUMAM (FR 2291)