Computing faithful representations for nilpotent Lie algebras. (English) Zbl 1188.17007

Let \(L\) be a nilpotent Lie algebra of dimension \(n\) and class \(c\) over a field. The authors construct faithful representations of \(L\) searching for the smallest possible dimension, \(\mu(L)\). They use and compare three algorithms for this construction of the representations. The input in these algorithms is the structure constants and the output is the matrices representing a basis for \(L\). They find an upper bound for \(\mu(L)\). The second part of the paper considers to what extent \(\mu(L) \leq n+1\). It is known that this inequality does not always hold. A special class of filiform Lie algebras are introduced and some of their properties are derived. It is conjectured that for Lie algebras of dimension greater than 12 in this class, the above inequality fails.


17B30 Solvable, nilpotent (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)


Full Text: DOI arXiv


[1] Birkhoff, G., Representability of Lie algebras and Lie groups by matrices, Ann. of Math., 38, 526-532 (1937) · Zbl 0016.24402
[2] Burde, D., A refinement of Ado’s Theorem, Arch. Math., 70, 118-127 (1998) · Zbl 0904.17006
[3] Burde, D., Affine cohomology classes for filiform Lie algebras, (Contemp. Math., vol. 262 (2000)), 159-170 · Zbl 0967.17017
[5] de Graaf, W. A., Constructing faithful matrix representations of Lie algebras, (Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ISSAC’97 (1997), ACM Press: ACM Press New York), 54-59 · Zbl 0957.17001
[6] Jacobson, N., Lie Algebras (1979), Dover: Dover New York · JFM 61.1044.02
[7] Milnor, J., On fundamental groups of complete affinely flat manifolds, Adv. Math., 25, 178-187 (1977) · Zbl 0364.55001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.