Solvable Lie algebra

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In mathematics, a Lie algebra \mathfrak{g} is solvable if its derived series terminates in the zero subalgebra. That is, writing

[\mathfrak{g},\mathfrak{g}]

for the derived Lie algebra of \mathfrak{g}, generated by the set of values

[x,y]

for x and y in \mathfrak{g}, the derived series

 \mathfrak{g} \geq [\mathfrak{g},\mathfrak{g}] \geq [[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]] \geq [[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]],[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]]  \geq ...

becomes constant eventually at 0.

Any nilpotent Lie algebra is solvable, a fortiori, but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition.

A maximal solvable subalgebra is called a Borel subalgebra. The largest solvable ideal is called the radical.

Properties[edit]

Let \mathfrak{g} be a finite-dimensional Lie algebra over a field of characteristic 0. The following are equivalent.

  • (i) \mathfrak{g} is solvable.
  • (ii) \operatorname{ad}(\mathfrak{g}), the adjoint representation of \mathfrak{g}, is solvable.
  • (iii) There is a finite sequence of ideals \mathfrak{a}_i of \mathfrak{g} such that:
    \mathfrak{g} = \mathfrak{a}_0 \supset \mathfrak{a}_1 \supset ... \mathfrak{a}_r = 0 where [\mathfrak{a}_i, \mathfrak{a}_i] \subset \mathfrak{a}_{i+1} for all i.
  • (iv) [\mathfrak{g}, \mathfrak{g}] is nilpotent.

Lie's Theorem states that if V is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and \mathfrak{g} is a solvable linear Lie algebra over V, then there exists a basis of V relative to which the matrices of all elements of \mathfrak{g} are upper triangular.

Completely solvable Lie algebras[edit]

A Lie algebra \mathfrak{g} is called completely solvable if it has a finite chain of ideals from 0 to \mathfrak{g} such that each has codimension 1 in the next. A finite-dimensional nilpotent Lie algebra is completely solvable, and a completely solvable Lie algebra is solvable. Over an algebraically closed field and solvable Lie algebra is completely solvable, but the 3-dimensional real Lie algebra of the group of Euclidean isometries of the plane is solvable but not completely solvable.

Example[edit]

Solvable Lie groups[edit]

The terminology arises from the solvable groups of abstract group theory. There are several possible definitions of solvable Lie group. For a Lie group G, there is

  • termination of the usual derived series, in other words taking G as an abstract group;
  • termination of the closures of the derived series;
  • having a solvable Lie algebra.

To have equivalence one needs to assume G connected. For connected Lie groups, these definitions are the same, and the derived series of Lie algebras are the Lie algebra of the derived series of (closed) subgroups.

See also[edit]

External links[edit]

References[edit]

  • Humphreys, James E. Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1972. ISBN 0-387-90053-5