Continuous cochains on formal currents #

For formal currents R⸨X⸩ ⊗[R] L with their order topology, the continuity conditions of TauCeti.LieModule.Cohomology take the concrete form of the joint-continuity criterion: writing a_N for the currents of order at least N, a one-cochain is continuous when it vanishes on some a_N, and an alternating two-cochain c is continuous when each c u vanishes on some a_N and c vanishes on a_N × a_N for a single N. Since the bracket of currents is continuous, coboundaries of continuous one-cochains with trivial coefficients are continuous two-cocycles.

The residue cocycle (f ⊗ x, g ⊗ y) ↦ Res(f dg) Φ(x, y) of an invariant symmetric form Φ is continuous: every current pairs to zero with all currents of sufficiently high order, and the pairing vanishes on the positive currents a_0. Its class in continuous second cohomology is therefore defined, and it is nonzero as soon as Φ is nonzero; over a domain it has no torsion. Indeed no multiple t • c_Φ with t ≠ 0 is a coboundary of any one-cochain, continuous or not: a coboundary with trivial coefficients takes equal values on (X⁻¹ ⊗ x, X ⊗ y) and (1 ⊗ x, 1 ⊗ y), whose brackets agree, while t • c_Φ takes t Φ(x, y) and 0 there.

Conversely, when L is finite-dimensional with nondegenerate Killing form over a field of characteristic zero, for instance a semisimple Lie algebra, every continuous 2-cocycle c is c_Ψ + d₁₂ f for an invariant symmetric form Ψ and a continuous one-cochain f. On polynomial currents, the classification of TauCeti.Algebra.Lie.Loop.Cocycle writes c as c_Ψ + d₁₂ g for some one-cochain g. Joint continuity makes c vanish on pairs of currents of order at least some N₀ ≥ 1; since L is perfect, every mode x_n with n ≥ 2 N₀ is a sum of brackets ⁅y_{N₀}, z_{n - N₀}⁆, so g vanishes on these modes. Composing g with truncation at 2 N₀ therefore gives a continuous one-cochain f on formal currents extending g, and the continuous cochains c and c_Ψ + d₁₂ f agree on the dense polynomial currents, hence everywhere. Density alone would not suffice: it is the vanishing of g on high modes that makes g extend continuously. When Ψ spans the invariant forms, every continuous 2-cocycle is thus t • c_Ψ + d₁₂ f; for a simple Lie algebra over an algebraically closed field of characteristic zero, the continuous second cohomology of its formal loop algebra is one-dimensional, spanned by the residue class of the Killing form.

When Ψ is nonzero and spans the invariant forms, the classification extends to an arbitrary vector space Z of coefficients with trivial action, of any dimension: z ↦ [c_Ψ ⊗ z], the class of (f ⊗ x, g ⊗ y) ↦ Res(f dg) Ψ(x, y) • z, is a linear equivalence from Z onto the continuous second cohomology with coefficients in Z. Injectivity holds for torsion-free Z over a domain, since a coboundary takes equal values on (x_{-1}, y_1) and (x_0, y_0), while c_Ψ ⊗ z takes Ψ(x, y) • z and 0 there. For surjectivity, each functional φ turns a continuous cocycle c into a scalar one, which is t_φ • c_Ψ up to a coboundary. The scalars t_φ are the values φ z at a single vector z, read off from the values of c at (x_{-1}, y_1) and (x_0, y_0). Since formal currents are perfect, the difference c - c_Ψ ⊗ z is then the coboundary of a one-cochain with values in Z, which vanishes on the currents of order at least 2 N when c vanishes on pairs of currents of order at least N. No finite-dimensionality of Z is needed, as no basis of Z is used.

Main results #

TauCeti.CurrentAlgebra.mem_continuousTwoCochain_iff_orderSubmodule: the joint-continuity criterion for two-cochains on formal currents.
TauCeti.CurrentAlgebra.d₁₂_mem_continuousTwoCocycle_of_isTrivial: coboundaries of continuous one-cochains with trivial coefficients are continuous two-cocycles.
TauCeti.CurrentAlgebra.twoCochainOfBilinear_mem_continuousTwoCocycle: the residue cocycle is continuous.
TauCeti.CurrentAlgebra.residueClass: its class in continuous second cohomology.
TauCeti.CurrentAlgebra.residueClass_add, TauCeti.CurrentAlgebra.residueClass_smul: the residue class is linear in the form.
TauCeti.CurrentAlgebra.smul_residueClass_eq_zero_iff: a multiple of the residue class of a nonzero form over a domain vanishes only when the scalar does.
TauCeti.CurrentAlgebra.residueClass_inj: distinct invariant symmetric forms have distinct residue classes over every commutative ring.
TauCeti.CurrentAlgebra.exists_eq_twoCochainOfBilinear_add_d₁₂: every continuous 2-cocycle is the residue cocycle of an invariant symmetric form plus a continuous coboundary.
TauCeti.CurrentAlgebra.exists_eq_smul_twoCochainOfBilinear_add_d₁₂: when Ψ spans the invariant forms, every continuous 2-cocycle is a multiple of the residue cocycle of Ψ plus a continuous coboundary.
TauCeti.CurrentAlgebra.residueClassMap: the residue class z ↦ [c_Φ ⊗ z] with coefficients in a module Z with trivial action.
TauCeti.CurrentAlgebra.residueClassMap_injective: it is injective for a nonzero form over a domain and torsion-free Z.
TauCeti.CurrentAlgebra.exists_eq_twoCochainOfBilinearSMulRight_add_d₁₂: for a nonzero form Ψ spanning the invariant forms, every continuous 2-cocycle with values in a vector space Z is a residue cocycle c_Ψ ⊗ z plus a continuous coboundary.
TauCeti.CurrentAlgebra.residueClassEquiv: for a nonzero form Ψ spanning the invariant forms, the linear equivalence z ↦ [c_Ψ ⊗ z] from any vector space Z onto continuous second cohomology with coefficients in Z.
TauCeti.CurrentAlgebra.finrank_continuousSecondCohomology_eq_one: for a simple Lie algebra over an algebraically closed field of characteristic zero, continuous second cohomology of formal currents is one-dimensional.

References #

E. Frenkel, Langlands Correspondence for Loop Groups, §1.3, for the continuous cohomology of the formal loop algebra and its residue class.