Baer correspondence up to isoclinism: Difference between revisions

Latest revision as of 06:56, 14 February 2012

Definition

The Baer correspondence up to isoclinism is a correspondence defined as follows:

Equivalence classes under isoclinism of groups of nilpotency class at most two $\leftrightarrow$ Equivalence classes under isoclinism of Lie rings of nilpotency class at most two

The correspondence is as follows: A Lie ring $L$ is identified with a group $G$ via a pair of isomorphisms:

An isomorphism $ζ$ between the additive group of $L / Z (L)$ and the inner automorphism group $G / Z (G)$ , and
An isomorphism $ϕ$ between the additive group of $[L, L]$ and the derived subgroup $[G, G]$

such that for $x, y \in L$ , with images $\bar{x}, \bar{y}$ mod $Z (L)$ , we have:

$ϕ ([x, y]) = [ζ (\bar{x}), ζ (\bar{y})]$

where the bracket on the left is the Lie bracket and the bracket on the right is the group commutator, well defined because the group commutator in $G$ of two elements depends only on their cosets mod $Z (G)$ .

@@ Line 4: / Line 4: @@
 Equivalence classes under [[isoclinism]] of groups of nilpotency class at most two <math>\leftrightarrow</math> Equivalence classes under [[isoclinism of Lie rings|isoclinism]] of Lie rings of nilpotency class at most two
+The correspondence is as follows: A Lie ring <math>L</math> is identified with a group <math>G</math> via a pair of isomorphisms:
+* An isomorphism <math>\zeta</math> between the additive group of <math>L/Z(L)</math> and the [[inner automorphism group]] <math>G/Z(G)</math>, and
+* An isomorphism <math>\phi</math> between the additive group of <math>[L,L]</math> and the [[derived subgroup]] <math>[G,G]</math>
+such that for <math>x,y \in L</math>, with images <math>\overline{x},\overline{y}</math> mod <math>Z(L)</math>, we have:
+<math>\phi([x,y]) = [\zeta(\overline{x}),\zeta(\overline{y})]</math>
+where the bracket on the left is the Lie bracket and the bracket on the right is the group commutator, well defined because the group commutator in <math>G</math> of two elements depends only on their cosets mod <math>Z(G)</math>.