The content discusses the problem of normal ordering for enveloping algebras of direct sums of Lie algebras.
The key highlights are:
The authors consider a Lie k-algebra g that can be written as a direct sum of two of its subalgebras, g = g1 ⊕ g2. They introduce the associated linear maps ji and pi such that j1p1 + j2p2 = Idg.
The authors investigate the property that the map μstate, which allows multiplying members of U(g1) ⊗ U(g2) within U(g), is bijective.
The authors construct a normal form calculator by defining an action g ∗ (t ⊗ m) on T(g1) ⊗ U(g2) that is compatible with the quotient map ψ1 : T(g1) → U(g1).
They prove that this action g ∗ U extends to a Lie g-action on U(g1) ⊗ U(g2), and show that the map s : m → m ∗ (1 ⊗ 1) and μstate are mutually inverse, implying that μstate is bijective.
The authors discuss potential future work on quantized enveloping algebras and Lie superalgebras.
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