# Questions tagged [algebraic-groups]

For questions about groups which have additional structure as algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. Consider using with the (group-theory) tag.

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Let $K$ be a field, for simplicity algebraically closed of characteristic $0$. Let $G$ be a reductive group over $K$. Def. A finite-dimensional $K$-algebra $R$ is an enveloping algebra for $G$, if $R^... 0answers 37 views ### Prove$A^\times$is an inner form of$GL_n$Question Let$A$be a central simple algebra over$F$where$F$is a field and$dim_FA=n^2$.$A^\times$can be seen as an algebraic group over$F$and let$G=A^\times$. Prove that$G$is an ... 0answers 31 views ### Prerequisites for Serre's Algebraic Groups and Class Fields What are the prerequisites for reading and understanding the book Algebraic Groups and Class Fields by Serre. Could you suggest some books to learn the prerequisites? Thanks 0answers 41 views ### Stabiliser in general position Consider the linear representation$V=$Sym$^3(\mathbb{C}^3)$of SL$_3(\mathbb{C})$. Now Theorem 7.2 of Popov-Vinberg asserts the following (Richardson [1972a], Luna ): For any action of a ... 1answer 34 views ### An ideal that contains the commutator of a solvable Lie algebra Let$\mathfrak g$be a real solvable Lie algebra and$\mathfrak n$be an ideal of$\mathfrak g$such that the commutator algebra$\mathfrak g'$is contained in$\mathfrak n$. Now, let$\mathfrak m$... 2answers 310 views ### Subgroups of$GL_n$containing upper triangular matrices EDIT: I rephrased the claim for clarity. Let$k$be a field (that we may assume to be algebraically closed, but I don't think it is necessary). Let$n\geq 1$and$T$denote the subgroup of$GL_n$... 1answer 68 views ### Show that$\mathfrak{sl}_n$is the Lie algebra of the algebraic group$SL_n$I am currently reading the book "Linear Algebraic Group" by Springer, more precisely in chapter 4 where Lie algebras of linear algebraic groups are introduced. I would like to prove that the Lie ... 1answer 17 views ### Expressing a cycle/set as odd or even. I have a cycle here that I have broken down into 2 disjoint cycles, them being (1,5,6) and (2,8). I'm wondering what is the process of telling whether the set is even or odd. Is it the number of ... 1answer 38 views ### Semisimple algebras of simple Lie algebras and their quotients Let$\mathfrak s$be a complex semisimple Lie algebra, then$\dim _\mathbb C\mathfrak s\geq 3$. But, however, is it possible for$\mathfrak s$to have a semisimple Lie subalgebra$\mathfrak h$such ... 1answer 110 views ### Kleiman's theorem on intersection theory I study the book 3264 & All That Intersection Theory in Algebraic Geometry by Eisenbud & Harris and I'm a little bit confused on on the proof of Kleiman’s theorem on pages 21: Theorem 1.7 (... 1answer 31 views ### Level set of characteristic polynomials closed in Zariski topology. Humphreys Proposition 18.2. I've been reading through Humphreys Linear Algebraic Groups, and this question concerns the proof of Proposition 18.2 (page 117). I believe the essence of my confusion is the following claim:$\...
Given two complex algebraic varieties $X,Y$ which are birational. Moreover, let $G$ be an algebraic group acting on $X$ and $Y$ and assume that the geometric quotients $X/G$ and $Y/G$ exist. What are ...