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wrote something at symbol of a differential operator
interesting description. I see that the entry symbol map treats something what can be interpreted as a very special case, namely of a Clifford algebra. On the other hand, when one has a filtered algebra whose associated graded is commutative, one can define a symbol map in much more generality, and the notion can be sheafified. The result is far more general than both notions (symbol map for Clifford algebras and symbol map for algebras and sheaves of differential operators) and widely used. Once I have a thought I will try to write something along those lines (or if somebody else is quicker than somebody else will).
There is a discussion of this article on MathOverflow: https://mathoverflow.net/questions/428375/the-principal-symbol-as-an-element-in-the-k-theory.
I removed redirect to principal symbol in the wake of creating a separate entry. Separating has several reasons, first of all it is not the same, and second, principal symbol is usually defined in the more general context of pseudodifferential operators; there are also versions in some filtered ring-theoretic setups.
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