Minimal faithful representations of reductive Lie algebras

Abstract

We prove an explicit formula for the invariant µ(g) for finitedimensional semisimple, and reductive Lie algebras g over C.Here µ(g) is the minimal dimension of a faithful linear representation of g. The result can be used to study Dynkin's classification of maximal reductive subalgebras of semisimple Lie algebras. Mathematics Subject Classification (2000). 17B20, 17B10.

A lower bound for faithful representations of nilpotent Lie algebras

“…The value of µ(n) has been obtained only for very few families of Lie algebras n (see, for instance [Be,B,BM1,CRo,Ro,S]).…”

Faithful Lie algebra modules and quotients of the universal enveloping algebra

“…Proof. The claim is clear for simple and abelian Lie algebras, see [5]. Since the µinvariant is subadditive, it also follows for reductive Lie algebras.…”

“…Proof. If n is abelian, then µ(n) ≥ ⌈2 √ n − 1⌉ ≥ 2 = c + 1 by proposition 2.4 of [5]. Assume now that n is not abelian.…”

“…Also, the only abelian Lie algebras satisfying the condition are the ones of dimension n ≤ 4. On the other hand, any reductive Lie algebra g satisfying µ(g) = dim(g) must be either semisimple or abelian: if g = s ⊕ C ℓ+1 with ℓ ≥ 0 and a non-trivial semisimple Lie algebra s, then µ(s ⊕ C) = µ(s), see [5], and…”

The minimal dimensions of faithful representations for Heisenberg Lie superalgebras

“…In 1998 Burde concluded that µ(h m ) = m + 2 for Heisenberg Lie algebra h m of dimension 2m + 1 [3]. In 2008 Burde and Moens established an explicit formula of µ(g) for semi-simple and reductive Lie algebras [4]. In 2009 Cagliero and Rojas obtained a formula µ(h m,p ) for the current Heisenberg Lie algebra h m,p [5].…”