Manuscript XIX-D@1 · XIX-D.tex · proposition · 1f47e33b

Proposition: A free closed observable need not remain closed after a differential deformation

Retarded memory, formal order, and gauge closure › Free closure is not interacting closure · lines 916–940 · proof included or linked

XIX-D@1#prop:free-interacting

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There is a graded algebra, a nilpotent free differential, and a nilpotent formal deformation for which a free closed element is not closed under the deformed differential.

Proof. Let

A = C [x, y] \otimes Λ (c), gh (x) = gh (y) = 0, gh (c) = 1.

Define a degree-one graded derivation $s_{κ}$ by

s_{κ} x = c, s_{κ} y = κ c, s_{κ} c = 0.

It squares to zero on generators and hence on the whole algebra. At $κ = 0$ , $s_{0} y = 0$ ; at positive formal coupling, $s_{κ} y = κ c \neq = 0$ .

Displayed formulas in this passage

unlabeled · line 925
$A = C [x, y] \otimes Λ (c), gh (x) = gh (y) = 0, gh (c) = 1.$
unlabeled · line 932
$s_{κ} x = c, s_{κ} y = κ c, s_{κ} c = 0.$

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section: Free closure is not interacting closure