On isoclasses of maximal subalgebras determined by automorphisms

AbstractWe study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field {\mathbb{K}} and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.

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Lower semimodular Lie algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500020496 ◽

1999 ◽

Vol 42 (3) ◽

pp. 521-540 ◽

Cited By ~ 1

Author(s):

V. R. Varea

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Maximal Subalgebras ◽

Modular Element ◽

Subalgebra Lattice ◽

The Relationship

This paper is concerned with the relationship between the properties of the subalgebra lattice ℒ(L) of a Lie algebra L and the structure of L. If the lattice ℒ(L) is lower semimodular, then the Lie algebra L is said to be lower semimodular. If a subalgebra S of L is a modular element in the lattice ℒ(L), then S is called a modular subalgebra of L. The easiest condition to ensure that L is lower semimodular is that dim A/B = 1 whenever B < A ≤ L and B is maximal in A (Lie algebras satisfying this condition are called sχ-algebras). Our aim is to characterize lower semimodular Lie algebras and sχ-algebras, over any field of characteristic greater than three. Also, we obtain results about the influence of two solvable modularmaximal subalgebras on the structure of the Lie algebra and some results on the structure of Lie algebras all of whose maximal subalgebras are modular.

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Soluble Lie algebras having finite-dimensional maximal subalgebras

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700043719 ◽

1974 ◽

Vol 11 (1) ◽

pp. 145-156 ◽

Cited By ~ 1

Author(s):

Ian N. Stewart

Keyword(s):

Lie Algebras ◽

Minimal Ideal ◽

Prime Characteristic ◽

Maximal Subalgebra ◽

Maximal Subalgebras ◽

Infinite Dimensional ◽

The Core ◽

Zero Characteristic ◽

Finite Dimensional ◽

Split Extensions

Infinite-dimensional soluble Lie algebras can possess maximal subalgebras which are finite-dimensional. We give a fairly complete description of such algebras: over a field of prime characteristic they do not exist; over a field of zero characteristic then, modulo the core of the aforesaid maximal subalgebra, they are split extensions of an abelian minimal ideal by the maximal subalgebra. If the field is algebraically closed, or if the maximal subalgebra is supersoluble, then all finite-dimensional maximal subalgebras are conjugate under the group of automorphisms generated by exponentials of inner derivations by elements of the Fitting radical. An example is given to indicate the differences encountered in the insoluble case, and the nonexistence of group-theoretic analogues is briefly discussed.

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