nLab SL(2,H)



Group Theory

Spin geometry



By SL(2,)SL(2,\mathbb{H}) one denotes the special linear group of 2×22 \times 2 matrices with coefficients in the quaternions, where “special” refers to their Dieudonné determinant being unity:

SL(2,):={[a b c d]|adaca 1b=1} SL(2,\mathbb{H}) \;:=\; \left\{ \left. \left[ \array{ a & b \\ c & d } \right] \;\right\vert\; \left\Vert a d - a c a^{-1} b \right\Vert \;=\; 1 \right\}

(here q 2:=qq¯\left\Vert q\right\Vert^2 \;:=\; q \bar q \in \mathbb{R} is the norm(-square) on quaternions).


Relation to Sp(2)Sp(2)

Every quaternionic unitary matrix (hence in Sp(2)) happens to have unit Dieudonné determinant (Cohen-De Leo 99, Cor. 6.4). Therefore we have a subgroup inclusion

Sp(2)=U(2,)SL(2,). Sp(2) \;=\; U(2,\mathbb{H}) \;\subset\; SL(2,\mathbb{H}) \,.

Relation to Spin(5,1)Spin(5,1)

Under the conjugation action on 2×22 \times 2 Hermitian matrices with coefficients in the quaternions, SL(2,)SL(2,\mathbb{H}) is identified with Spin(5,1) and its canonical action on Minkowski spacetime 5,1\mathbb{R}^{5,1}.

For more on this see at spin representation, supersymmetry and division algebras and geometry of physics – supersymmetry.

exceptional spinors and real normed division algebras

AA\phantom{AA}spin groupnormed division algebra\,\, brane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})A\phantom{A} \mathbb{R} the real numberssuper 1-brane in 3d
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})A\phantom{A} \mathbb{C} the complex numberssuper 2-brane in 4d
6=5+16 = 5+1Spin(5,1)Spin(5,1) \simeq SL(2,H)A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1) {\simeq}SL(2,O)A\phantom{A} 𝕆\mathbb{O} the octonionsheterotic/type II string


Last revised on September 25, 2021 at 02:32:01. See the history of this page for a list of all contributions to it.