An Example Using Chevalley Restriction

Posted by Jason Polak on 24. January 2013 · Write a comment · Categories: algebraic-geometry, commutative-algebra · Tags: , ,

Here is a classic problem of geometric invariant theory: let G be a reductive linear algebraic group such as \mathrm{GL}_n and let \mathfrak{g} be its Lie algebra. Determine the invariant functions k[\mathfrak{g}]^G, where G acts on \mathfrak{g} via the adjoint action. This problem is motivated by the search for quotients: What is the quotient \mathfrak{g}/G? Here, the action of G on \mathfrak{g} is given by the adjoint action. More explicitly, an element g\in G acts via the differentiation of \mathrm{Int}_g, where \mathrm{Int}_g is conjugation by g on G.

For simplicity, we will stay in the realm of varieties over an algebraically closed field k of characteristic zero.

First, we should ask:

What should \mathfrak{g}/G even mean?

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Highlights in Linear Algebraic Groups 4: Lie Algebras III

Posted by Jason Polak on 04. September 2012 · Write a comment · Categories: algebraic-geometry, commutative-algebra · Tags: , , , ,

Last time in this series, we saw the definition of the Lie algebra of a linear algebraic group G over a arbitrary field k as the set of differentiations f:k[G]\to k; these are the k-linear maps satisfying f(ab) = \epsilon(a)f(b) + f(a)\epsilon(b) where \epsilon:k[G]\to k is the counit morphism corresponding to the identity in G.

In this post we will look at the geometric definition of the tangent space, which is natural when we consider the k-points of G as a subset of affine space. Furthermore, we shall see an example of the adjoint representation, and how morphisms of algebraic groups correspond to morphisms of Lie algebras in the explicit case of G embedded into \rm{GL}_n. This will allow us to write down an explicit formula for the adjoint representation.
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Highlights in Linear Algebraic Groups 3: Lie Algebras II

Posted by Jason Polak on 24. June 2012 · Write a comment · Categories: algebraic-geometry, group-theory · Tags: , , ,

In Linear Algebraic Groups 2, we defined the Lie algebra of an algebraic group G to be the Lie algebra of all left-invariant derivations D:k[G]\to k[G] where k[G] is the representing algebra of G. However, we were left trying to figure out exactly how a morphism \varphi:G\to H determines a morphism d\varphi:\mathcal{L}(G)\to\mathcal{L}(H). It turns out that the answer is slightly tricky, and thus the Lie algebra in terms of left-invariant derivations k[G]\to k[G] really can’t be the “real” or “most natural” definition!

Now, I could just give the formula for morphisms right away, but I think it would be a bit unmotivated. So before looking at the formula, let us first look at another way of defining the Lie algebra of an algebraic group. In fact, the next defintion is much more natural in that the functoriality of the Lie algebra construction will be clear, whereas in this case it was not.
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