Search results for "Adjoint representation of a Lie algebra"

Lie Algebras Generated by Extremal Elements

1999

We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals of L) over a field of characteristic distinct from 2. We prove that any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal number of extremal generators for the Lie algebras of type An, Bn (n>2), Cn (n>1), Dn (n>3), En (n=6,7,8), F4 and G2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.

Actions of complex Lie groups on analytic ?-algebras

1987

On a reduced analytic .ℂ-algebraR there are faithful analytic actions of complex Lie groups of arbitrarily high dimension if and only ifR has Krull dimension ≥2.

Simple and semisimple Lie algebras and codimension growth

1999

Irreducible finitary Lie algebras over fields of positive characteristic

2000

A Lie subalgebra L of [gfr ][lfr ][ ](V) is said to be finitary if it consists of elements of finite rank. We study the situation when L acts irreducibly on the infinite-dimensional vector space V and show: if Char [ ] > 7, then L has a unique minimal ideal I. Moreover I is simple and L/I is solvable.

Non-integrality of the PI-exponent of special Lie algebras

2013

If L is a special Lie algebra over a field of characteristic zero, its sequence of codimensions is exponentially bounded. The PI-exponent measures the exponential rate of growth of such sequence and here we give a first example of a special Lie algebra whose (upper and lower) PI-exponent is non-integer.

On the Codimension Growth of Finite-Dimensional Lie Algebras

1999

Abstract We study the exponential growth of the codimensions cn(L) of a finite-dimensional Lie algebra L over a field of characteristic zero. We show that if the solvable radical of L is nilpotent then lim n → ∞ c n ( L ) exists and is an integer.

LEFT INVARIANT COMPLEX STRUCTURES ON NILPOTENT SIMPLY CONNECTED INDECOMPOSABLE 6-DIMENSIONAL REAL LIE GROUPS

2007

Integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras have been computed in a previous paper, along with normal forms for representatives of the various equivalence classes under the action of the automorphism group. Here we go to the connected simply connected Lie group G0 associated to such a Lie algebra 𝔤. For each normal form J of integrable complex structures on 𝔤, we consider the left invariant complex manifold G = (G0, J) associated to G0 and J. We explicitly compute a global holomorphic chart for G and we write down the multiplication in that chart.

Some criteria for detecting capable Lie algebras

2013

Abstract In virtue of a recent bound obtained in [P. Niroomand, F.G. Russo, A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra 39 (2011) 1293–1297], we classify all capable nilpotent Lie algebras of finite dimension possessing a derived subalgebra of dimension one. Indirectly, we find also a criterion for detecting noncapable Lie algebras. The final part contains a construction, which shows that there exist capable Lie algebras of arbitrary big corank (in the sense of Berkovich–Zhou).

A restriction on the schur multiplier of nilpotent lie algebras

2011

An improvement of a bound of Yankosky (2003) is presented in this paper, thanks to a restriction which has been recently obtained by the authors on the Schur multiplier M(L) of a finite dimensional nilpotent Lie algebra L. It is also described the structure of all nilpotent Lie algebras such that the bound is attained. An important role is played by the presence of a derived subalgebra of maximal dimension. This allows precision on the size of M(L). Among other results, applications to the non-abelian tensor square L ⊗ L are illustrated.

Nilpotent Lie algebras with 2-dimensional commutator ideals

2011

Abstract We classify all (finitely dimensional) nilpotent Lie k -algebras h with 2-dimensional commutator ideals h ′ , extending a known result to the case where h ′ is non-central and k is an arbitrary field. It turns out that, while the structure of h depends on the field k if h ′ is central, it is independent of k if h ′ is non-central and is uniquely determined by the dimension of h . In the case where k is algebraically or real closed, we also list all nilpotent Lie k -algebras h with 2-dimensional central commutator ideals h ′ and dim k h ⩽ 11 .