Nonassociative rings and algebras Definition

Tits–Kantor–Koecher construction

Builds on Jordan pair Jordan triple system Jordan algebra Lie algebra

Definition

Let $F$ be a field of characteristic different from $2$ , and let $(V^+, V^-)$ be a Jordan pair over $F$ with triple products $\set{\cdot, \cdot, \cdot}$ . For $x \in V^+$ and $y \in V^-$ , let $D(x, y)$ be the linear map of $V^+ \oplus V^-$ given by $z \mapsto \set{x, y, z}$ on $V^+$ and $w \mapsto -\set{y, x, w}$ on $V^-$ . Let $\mathfrak{d}$ be the linear span of all $D(x, y)$ . The Tits–Kantor–Koecher algebra is the vector space $\mathrm{TKK}(V^+, V^-) = V^- \oplus \mathfrak{d} \oplus V^+$ with the bilinear bracket defined by:

$[x, x'] = 0$ for $x, x' \in V^+$ , and $[y, y'] = 0$ for $y, y' \in V^-$ ;
$[x, y] = D(x, y) = -[y, x]$ for $x \in V^+$ and $y \in V^-$ ;
$[D, v] = D(v) = -[v, D]$ for $D \in \mathfrak{d}$ and $v \in V^+ \cup V^-$ ;
$[D, D'] = DD' - D'D$ for $D, D' \in \mathfrak{d}$ .

For a Jordan algebra $J$ , one applies this to the pair $(J, J)$ with $\set{x, y, z} = (xy)z + (zy)x - (zx)y$ .

Remarks

The identity of a Jordan pair gives $[D(u, v), D(x, y)] = D(\set{u, v, x}, y) - D(x, \set{v, u, y})$ , so clause 4 stays in $\mathfrak{d}$ . With these clauses, $\mathrm{TKK}(V^+, V^-)$ is a Lie algebra, graded with $V^-$ , $\mathfrak{d}$ , $V^+$ in degrees $-1$ , $0$ , $1$ .

Example: $J = F$ with its usual product, so $\set{x, y, z} = xyz$ . Put $e = 1 \in V^+$ , $f = 2 \in V^-$ and $h = D(1, 2)$ . Then $h$ acts by $2$ on $V^+$ and by $-2$ on $V^-$ , so $[h, e] = 2e$ , $[h, f] = -2f$ and $[e, f] = h$ . These are the relations of $\mathfrak{sl}_2(F)$ .

Non-example: $\mathfrak{gl}_2(F)$ with the grading by $e_{12}$ , the diagonal matrices and $e_{21}$ is not the TKK algebra of $(F, F)$ . Its degree $0$ part is $2$ -dimensional, while $\mathfrak{d}$ is spanned by the single map $D(1, 1)$ .

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Jordan algebra concept
Jordan pair concept
Jordan triple system concept
Lie algebra concept
Tits–Kantor–Koecher construction definition

Tag	`EHMQC1ZBK9`
URI	`m2://EHMQC1ZBK9`
Kind	definition
Writing	W0 stub
MSC	17C50, 17B60, 17B70
Defines	Tits–Kantor–Koecher construction
Uses	Jordan pair, Jordan triple system, Jordan algebra, Lie algebra
Changed	3 November 2026

Tits–Kantor–Koecher construction

Definition

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