What is Derivations: Definition and 106 Discussions

In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D : A → A that satisfies Leibniz's law:




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{\displaystyle D(ab)=aD(b)+D(a)b.}
More generally, if M is an A-bimodule, a K-linear map D : A → M that satisfies the Leibniz law is also called a derivation. The collection of all K-derivations of A to itself is denoted by DerK(A). The collection of K-derivations of A into an A-module M is denoted by DerK(A, M).
Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on Rn. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold. It follows that the adjoint representation of a Lie algebra is a derivation on that algebra. The Pincherle derivative is an example of a derivation in abstract algebra. If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K. An algebra A equipped with a distinguished derivation d forms a differential algebra, and is itself a significant object of study in areas such as differential Galois theory.

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  1. H

    I About derivations of lie algebra

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  2. S

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  3. Q

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  5. K

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  6. E

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  8. A

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  9. R

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  10. JacobPhys

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  11. Math Amateur

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  13. wirefree

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  14. Wrichik Basu

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  15. N

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  16. S

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  17. D

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  18. D

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  19. DavideGenoa

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  20. W

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  21. mertcan

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  22. M

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  24. A

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  25. S

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  33. S

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  39. C

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  40. topsquark

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  41. B

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  42. Z

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  43. B

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  44. C

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  45. G

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  46. D

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  47. B

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  48. D

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  49. N

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