Solvable Lie algebra

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In mathematics, a Lie algebra g is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra is the subalgebra of g, denoted

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

that consists of all Lie brackets of pairs of elements of g. The derived series is the sequence of subalgebras

 \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 ...

If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is solvable.[1] The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory.

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 of a Lie algebra is called the radical.

Characterizations[edit]

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

  • (i) g is solvable.
  • (ii) ad(g), the adjoint representation of g, is solvable.
  • (iii) There is a finite sequence of ideals ai of g:
    \mathfrak{g} = \mathfrak{a}_0 \supset \mathfrak{a}_1 \supset ... \mathfrak{a}_r = 0, \quad \forall i [\mathfrak{a}_i, \mathfrak{a}_i] \subset \mathfrak{a}_{i+1}.
  • (iv) [g, g] is nilpotent.
  • (v) For g n-dimensional, there is a finite sequence of subalgebras ai of g:
    \mathfrak{g} = \mathfrak{a}_0 \supset \mathfrak{a}_1 \supset ... \mathfrak{a}_n = 0, \quad \forall i \operatorname{dim} \mathfrak{a}_{i}/\mathfrak{a}_{i + 1} = 1,
with each ai + 1 an ideal in ai.[2] A sequence of this type is called an elementary sequence.
  • (vi) There is a finite sequence of subalgebras gi of g,
    \mathfrak{g} = \mathfrak{g}_0 \supset \mathfrak{g}_1 \supset ... \mathfrak{g}_r = 0,
such that gi + 1 is an ideal in gi and gi/gi + 1 is abelian.[3]

Properties[edit]

Lie's Theorem states that if V is a finite-dimensional vector space over an algebraically closed field K of characteristic zero, and g is a solvable linear Lie algebra over a subfield k of K, and if π is a representation of g over V, then there exists a simultaneous eigenvector vV of the matrices π(X) for all elements Xg. More generally, the result holds if all eigenvalues of π(X) lie in K for all Xg.[5]

  • Every Lie subalgebra, quotient and extension of a solvable Lie algebra is solvable.
  • A solvable nonzero Lie algebra has a nonzero abelian ideal, the last nonzero term in the derived series.[6]
  • A homomorphic image of a solvable Lie algebra is solvable.[6]
  • If a is a solvable ideal in g and g/a is solvable, then g is solvable.[6]
  • If g is finite-dimensional, then there is a unique solvable ideal rg containing all solvable ideals in g. This ideal is the radical of g, denoted rad g.[6]
  • If a, bg are solvable ideals, then so is a + b.[1]
  • A solvable Lie algebra g has a unique largest nilpotent ideal n, the set of all Xg such that adX is nilpotent. If D is any derivation of g, then D(g) ⊂ n.[7]

Completely solvable Lie algebras[edit]

A Lie algebra g is called completely solvable or split solvable if it has an elementary sequence of ideals in g from 0 to g. 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.

  • (a) A solvable Lie algebra g is split solvable if and only if the eigenvalues of adX are in k for all X in g.[6]

Examples[edit]

X = \left(\begin{matrix}0 & \theta & x\\ -\theta & 0 & y\\ 0 & 0 & 0\end{matrix}\right), \quad \theta, x, y \in \mathbb{R}.
Then g is solvable, but not split solvable.[6] It is isomorphic with the Lie algebra of the group of translations and rotations in the plane.

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]

Notes[edit]

  1. ^ a b c Humphreys 1972
  2. ^ Knapp 2002 Proposition 1.23.
  3. ^ Fulton & Harris 1991
  4. ^ Knapp 2002 Proposition 1.46.
  5. ^ Knapp 2002 Theorem 1.25.
  6. ^ a b c d e f Knapp 2002
  7. ^ Knapp 2002 Proposition 1.40.

References[edit]

  • Fulton, W.; Harris, J. (1991). Representation theory. A first course. Graduate Texts in Mathematics 129. New York: Springer-Verlag. ISBN 978-0-387-97527-6. MR 1153249. 
  • Humphreys, James E. (1972). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics 9. New York: Springer-Verlag. ISBN 0-387-90053-5. 
  • Knapp, A. W. (2002). Lie groups beyond an introduction. Progress in Mathematics 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. ISBN 0-8176-4259-5. .