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Some questions on semisimple linear algebraic groups

Discussion in 'Mathematics' started by Anna Abasheva, Oct 8, 2018.

  1. Let $G$ be a connected semisimple linear algebraic group of finite type over a field $K$. I have several, I suppose rather base, questions concerning the theory of such groups.

    1. Does there always exist an embedding of $G$ into $GL_n$ for some $n$? As I understand the embedding exists in the context of Lie groups over $\mathbb R$ but I ask about an algebraic embedding. (EDIT) Yes, it exists for an arbitrary linear group of finite type.

    2. Suppose we fix an embedding $\iota\colon G\hookrightarrow GL_n$. In $GL_n$ we can define eigenvalues of the elements of $G$. Let us choose $g\in G$ with eigenvalues $\lambda_1,...\lambda_n$ and we define $K_{g,\iota}$ as $K(\lambda_1,...\lambda_n)$. Does $K_{g,\iota}$ depend on $\iota$?

    3. Suppose again we have an embedding $\iota\colon G\hookrightarrow GL_n$. Then we have the determinant map $\Delta\colon GL_n\to \mathbb G_m$. Does the map $\Delta\circ\iota$ depend on the choice of $\iota$?

    4. Let $G\hookrightarrow GL_n\cong GL(V)\subset \textrm{End}(V)$ where $\textrm{End}(V)$ is a ring algebraic variety of endomorphisms of $V$. Let us call $R_\iota$ the minimal ring variety containing $G$. Does $R_\iota$ depend on the embedding?

    If these objects are independent of the embedding can they be described in a natural way? And in fact the only question that can't be generalized to arbitrary linear groups is (2), can something be said for questions (1), (3) and (4) for arbitrary linear groups?

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