Global analysis, analysis on manifolds Definition

Lie supergroup

Definition

A Lie supergroup is a real supermanifold $G$ with three morphisms of supermanifolds $m \colon G \times G \to G, \qquad i \colon G \to G, \qquad e \colon \R^{0|0} \to G,$ called multiplication, inverse and unit. These must make the following diagrams of morphisms commute:

associativity: $m \circ (m \times \id) = m \circ (\id \times m)$ as morphisms $G \times G \times G \to G$ ;
unit: $m \circ (e \times \id) = \id = m \circ (\id \times e)$ , after identifying $\R^{0|0} \times G$ and $G \times \R^{0|0}$ with $G$ ;
inverse: $m \circ (i \times \id) \circ \Delta = e \circ \pi = m \circ (\id \times i) \circ \Delta$ , where $\Delta \colon G \to G \times G$ is the diagonal and $\pi \colon G \to \R^{0|0}$ is the unique morphism to the point.

Remarks

Example: on $\R^{1|1}$ , with even coordinate $t$ and odd coordinate $\theta$ , define $m$ by $(t, \theta)(t', \theta') = (t + t' + \theta\theta', \theta + \theta')$ , inverse $(t, \theta) \mapsto (-t, -\theta)$ and unit $(0, 0)$ . Associativity holds since both bracketings give $(t + t' + t'' + \theta\theta' + \theta\theta'' + \theta'\theta'', \theta + \theta' + \theta'')$ . The inverse works since $(t, \theta)(-t, -\theta) = (-\theta\theta, 0) = (0, 0)$ , as $\theta^2 = 0$ .

Non-example: on $\R^{1|1}$ the law $(t, \theta)(t', \theta') = (tt', \theta + \theta')$ is associative with unit $(1, 0)$ . But it has no inverse morphism: on underlying points it is multiplication of real numbers, and $0$ has no inverse.

The underlying manifold of a Lie supergroup is an ordinary Lie group. The left-invariant vector fields on $G$ form a Lie superalgebra, the Lie superalgebra of $G$ . Ordinary Lie groups are the Lie supergroups with no odd coordinates.

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Lie group concept
Lie supergroup definition

Tag	`5X6G3Q5CSQ`
URI	`m2://5X6G3Q5CSQ`
Kind	definition
Writing	W0 stub
MSC	58A50, 22E99
Defines	Lie supergroup
Uses	Lie group

Lie supergroup

Definition

Remarks

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