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Here’s a practical reason to look at derivations, from one algebraist’s perspective, broken down in a few steps.

  1. Suppose you would rather look at the group of automorphisms of the algebra, instead of the derivations. There are good reasons to do that, even if what you are really interested is the algebra itself. Practically, you can sometimes translate a statement you want to prove about the algebra into a corresponding statement about the automorphism group and then apply different techniques to get your desired proof. Philosophically, this is an algebraic analogue of Klein’s Erlangen program, and that analogy is meaningful thanks to Galois descent for example (see the discussion in section III.1 of Serre’s book [2]).

  2. But the group of automorphisms can be technically more challenging. For example, the octonions are an 8-dimensional algebra but their automorphism group is some 14-dimensional Lie group (manifold with a group structure) so maybe it’s hard to work directly with it.

  3. So you may choose to simplify the problem by linearizing it, i.e., replacing the manifold with its tangent space. When you do that, you replace the automorphism group with its tangent space at the identity, which is the Lie algebra of derivations of the algebra. It is technically easier to work with, because for example it is easier to tell if two subspaces of a vector space are equal than if two submanifolds of a manifold are equal.

You don’t lose too much when you make this switch from looking at the automorphism group to the Lie algebra of derivations (the tangent space) because there are theorems saying that such-and-such a property holds for the identity component of the group if and only if some other property holds for the Lie algebra, see for example section II.7 in Borel’s book [2][1]. And historically this has been a very productive way of approaching the problem, since it led to things like the Killing-CartanKilling–Cartan classification of simple Lie groups.

That’s the idea. Here is a technical detail that may or may not be relevant to your situation. The relationship between results for the derivations and results for the automorphism group is tightest over fields of characteristic zero. However, by the 1970s, say, it was clear that many of the theorems about automorphism groups or derivation algebras should be true over fields of all or almost all characteristics. But the technicalities involved in carefully proving theorems about the automorphism groups over fields of non-zero characteristic (along the lines of Grothendieck’s SGA3) are a substantial step up in difficulty and step down in power from the situation in characteristic zero. So it was very natural for researchers at that time to focus their attention on the derivations instead of on the automorphism group and simply exclude a handful of characteristics where the derivation algebras don’t tell the whole story.

References

[1] Borel, Armand, Linear algebraic groups, 2nd enlarged edition. Springer. (1991).

[2] Serre, Jean-Pierre, Galois cohomology. Transl. from the French by Patrick Ion, Berlin: Springer. x, 210 p. (1997). ZBL0902.12004.

Here’s a practical reason to look at derivations, from one algebraist’s perspective, broken down in a few steps.

  1. Suppose you would rather look at the group of automorphisms of the algebra, instead of the derivations. There are good reasons to do that, even if what you are really interested is the algebra itself. Practically, you can sometimes translate a statement you want to prove about the algebra into a corresponding statement about the automorphism group and then apply different techniques to get your desired proof. Philosophically, this is an algebraic analogue of Klein’s Erlangen program, and that analogy is meaningful thanks to Galois descent for example (see the discussion in section III.1 of Serre’s book [2]).

  2. But the group of automorphisms can be technically more challenging. For example, the octonions are an 8-dimensional algebra but their automorphism group is some 14-dimensional Lie group (manifold with a group structure) so maybe it’s hard to work directly with it.

  3. So you may choose to simplify the problem by linearizing it, i.e., replacing the manifold with its tangent space. When you do that, you replace the automorphism group with its tangent space at the identity, which is the Lie algebra of derivations of the algebra. It is technically easier to work with, because for example it is easier to tell if two subspaces of a vector space are equal than if two submanifolds of a manifold are equal.

You don’t lose too much when you make this switch from looking at the automorphism group to the Lie algebra of derivations (the tangent space) because there are theorems saying that such-and-such a property holds for the identity component of the group if and only if some other property holds for the Lie algebra, see for example section II.7 in Borel’s book [2]. And historically this has been a very productive way of approaching the problem, since it led to things like the Killing-Cartan classification of simple Lie groups.

That’s the idea. Here is a technical detail that may or may not be relevant to your situation. The relationship between results for the derivations and results for the automorphism group is tightest over fields of characteristic zero. However, by the 1970s, say, it was clear that many of the theorems about automorphism groups or derivation algebras should be true over fields of all or almost all characteristics. But the technicalities involved in carefully proving theorems about the automorphism groups over fields of non-zero characteristic (along the lines of Grothendieck’s SGA3) are a substantial step up in difficulty and step down in power from the situation in characteristic zero. So it was very natural for researchers at that time to focus their attention on the derivations instead of on the automorphism group and simply exclude a handful of characteristics where the derivation algebras don’t tell the whole story.

References

[1] Borel, Armand, Linear algebraic groups, 2nd enlarged edition. Springer. (1991).

[2] Serre, Jean-Pierre, Galois cohomology. Transl. from the French by Patrick Ion, Berlin: Springer. x, 210 p. (1997). ZBL0902.12004.

Here’s a practical reason to look at derivations, from one algebraist’s perspective, broken down in a few steps.

  1. Suppose you would rather look at the group of automorphisms of the algebra, instead of the derivations. There are good reasons to do that, even if what you are really interested is the algebra itself. Practically, you can sometimes translate a statement you want to prove about the algebra into a corresponding statement about the automorphism group and then apply different techniques to get your desired proof. Philosophically, this is an algebraic analogue of Klein’s Erlangen program, and that analogy is meaningful thanks to Galois descent for example (see the discussion in section III.1 of Serre’s book [2]).

  2. But the group of automorphisms can be technically more challenging. For example, the octonions are an 8-dimensional algebra but their automorphism group is some 14-dimensional Lie group (manifold with a group structure) so maybe it’s hard to work directly with it.

  3. So you may choose to simplify the problem by linearizing it, i.e., replacing the manifold with its tangent space. When you do that, you replace the automorphism group with its tangent space at the identity, which is the Lie algebra of derivations of the algebra. It is technically easier to work with, because for example it is easier to tell if two subspaces of a vector space are equal than if two submanifolds of a manifold are equal.

You don’t lose too much when you make this switch from looking at the automorphism group to the Lie algebra of derivations (the tangent space) because there are theorems saying that such-and-such a property holds for the identity component of the group if and only if some other property holds for the Lie algebra, see for example section II.7 in Borel’s book [1]. And historically this has been a very productive way of approaching the problem, since it led to things like the Killing–Cartan classification of simple Lie groups.

That’s the idea. Here is a technical detail that may or may not be relevant to your situation. The relationship between results for the derivations and results for the automorphism group is tightest over fields of characteristic zero. However, by the 1970s, say, it was clear that many of the theorems about automorphism groups or derivation algebras should be true over fields of all or almost all characteristics. But the technicalities involved in carefully proving theorems about the automorphism groups over fields of non-zero characteristic (along the lines of Grothendieck’s SGA3) are a substantial step up in difficulty and step down in power from the situation in characteristic zero. So it was very natural for researchers at that time to focus their attention on the derivations instead of on the automorphism group and simply exclude a handful of characteristics where the derivation algebras don’t tell the whole story.

References

[1] Borel, Armand, Linear algebraic groups, 2nd enlarged edition. Springer. (1991).

[2] Serre, Jean-Pierre, Galois cohomology. Transl. from the French by Patrick Ion, Berlin: Springer. x, 210 p. (1997). ZBL0902.12004.

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Here’s a practical reason to look at derivations, from one algebraist’s perspective, broken down in a few steps.

  1. Suppose you would rather look at the group of automorphisms of the algebra, instead of the derivations. There are good reasons to do that, even if what you are really interested is the algebra itself. Practically, you can sometimes translate a statement you want to prove about the algebra into a corresponding statement about the automorphism group and then apply different techniques to get your desired proof. Philosophically, this is an algebraic analogue of Klein’s Erlangen program, and that analogy is meaningful thanks to Galois descent for example (see the discussion in section III.1 of Serre’s book [2]).

  2. But the group of automorphisms can be technically more challenging. For example, the octonions are an 8-dimensional algebra but their automorphism group is some 14-dimensional Lie group (manifold with a group structure) so maybe it’s hard to work directly with it.

  3. So you may choose to simplify the problem by linearizing it, i.e., replacing the manifold with its tangent space. When you do that, you replace the automorphism group with its tangent space at the identity, which is the Lie algebra of derivations of the algebra. It is technically easier to work with, because for example it is easier to tell if two subspaces of a vector space are equal than if two submanifolds of a manifold are equal.

You don’t lose too much when you make this switch from looking at the automorphism group to the Lie algebra of derivations (the tangent space) because there are theorems saying that such-and-such a property holds for the identity component of the group if and only if some other property holds for the Lie algebra, see for example section II.7 in Borel’s book [2]. And historically this has been a very productive way of approaching the problem, since it led to things like the Killing-Cartan classification of simple Lie groups.

That’s the idea. Here is a technical detail that may or may not be relevant to your situation. The relationship between results for the derivations and results for the automorphism group is tightest over fields of characteristic zero. However, by the 1970s, say, it was clear that many of the theorems about automorphism groups or derivation algebras should be true over fields of all or almost all characteristics. But the technicalities involved in carefully proving theorems about the automorphism groups over fields of non-zero characteristic (along the lines of Grothendieck’s SGA3) are a substantial step up in difficulty and step down in power from the situation in characteristic zero. So it was very natural for researchers at that time to focus their attention on the derivations instead of on the automorphism group and simply exclude a handful of characteristics where the derivation algebras don’t tell the whole story.

References

[1] Borel, Armand, Linear algebraic groups, 2nd enlarged edition. Springer. (1991).

[2] Serre, Jean-Pierre, Galois cohomology. Transl. from the French by Patrick Ion, Berlin: Springer. x, 210 p. (1997). ZBL0902.12004.