View source for Levi-Mal'tsev decomposition

The presentation of a finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l0582501.png" /> over a field of characteristic zero as a direct sum (as vector spaces) of its radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l0582502.png" /> (the maximal solvable ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l0582503.png" />) and a semi-simple Lie subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l0582504.png" />. It was obtained by E.E. Levi [[#References|[1]]] and A.I. Mal'tsev [[#References|[2]]]. The Levi–Mal'tsev theorem states that there always is such a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l0582505.png" />; moreover, the subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l0582506.png" /> is unique up to an automorphism of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l0582507.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l0582508.png" /> is the inner derivation of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l0582509.png" /> determined by an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825010.png" /> of the nil radical (the largest nilpotent ideal) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825012.png" /> is a connected and simply-connected real Lie group, then there are closed simply-connected analytic subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825016.png" /> is the maximal connected closed solvable normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825018.png" /> is a semi-simple subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825020.png" />, such that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825023.png" />, is an analytic isomorphism of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825024.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825025.png" />; in this case the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825026.png" /> is also called a Levi–Mal'tsev decomposition.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.E. Levi, ''Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.'' , '''40''' (1906) pp. 3–17</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Mal'tsev, "On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra" ''Dokl. Akad. Nauk SSSR'' , '''36''' : 2 (1942) pp. 42–45 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) {{MR|0148716}} {{MR|0143793}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) {{MR|0412321}} {{ZBL|0342.22001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) {{MR|0793377}} {{ZBL|0484.22018}} </TD></TR></table>



====Comments====
The existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825027.png" />, called the Levi factor (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825028.png" /> is a semi-simple subalgebra, respectively a semi-simple subgroup, also called a Levi subalgebra, respectively Levi subgroup), was established by Levi. The conjugacy of Levi factors was proved by Mal'tsev.

An analogue of the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825029.png" /> holds for an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825030.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825031.png" /> is the maximal unipotent normal subgroup and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058250/l05825032.png" /> is a maximal reductive subgroup (Mostow's theorem).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) {{MR|0682756}} {{ZBL|0319.17002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Hochschild, "The structure of Lie groups" , Holden-Day (1965) {{MR|0207883}} {{ZBL|0131.02702}} </TD></TR></table>