Orbits of greatest dimension in semi-simple linear Lie groups

Infinite root systems, representations of graphs and invariant theory

“…[1]). There exists a point xeX~ which is smooth on X and on X 1 and for which the mapping ~b is smooth at the point (e, x), i.e.…”

“…To prove the existence of indecomposable representations of a graph we will use the following proposition which is a generalization of a statement from [1] (in this generality the statement will be applied in Sect. 2.7).…”

“…Tables P and Z in Sect. 1.1 statement (Lemma 2.4), which is a generalization of a lemma of Vinberg [1]: If G is an algebraic group operating on an algebraic variety X and A is an element from the Lie algebra of the stabilizer of a point x~X such that G. ~r=x, then rank A lrr~xl< rank ad A Ire<GI. Remarkably, this condition in our situation gives exactly the fundamental set M~A+ (Lemma 2.5).…”

“…An object of re(S, ~2) is a collection (U, q~) of finite-dimensional vector spaces U v, pes o, over IF, and linear maps ~bt: U,~)~ Us, ) for any edge leS 1…”

Left-symmetric algebras for 𝔤𝔩(𝔫)

“…[AVE,El,Li,S1,S2]. The tables in [S1] contain the coregular representations of SL(n) and the subclass of special modules may be extracted.…”