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  "body": "\n## Definition\n\nA *Lie supergroup* is a real [[supermanifold|Z0SB9HQBAJ]] $G$ with three morphisms of supermanifolds\n$$m \\colon G \\times G \\to G, \\qquad i \\colon G \\to G, \\qquad e \\colon \\R^{0|0} \\to G,$$\ncalled multiplication, inverse and unit.\nThese must make the following diagrams of morphisms commute:\n\n1. associativity: $m \\circ (m \\times \\id) = m \\circ (\\id \\times m)$ as morphisms $G \\times G \\times G \\to G$;\n2. 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 $G$;\n3. 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.\n\n## Remarks\n\nExample: 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 $(0, 0)$.\nAssociativity holds since both bracketings give $(t + t' + t'' + \\theta\\theta' + \\theta\\theta'' + \\theta'\\theta'', \\theta + \\theta' + \\theta'')$.\nThe inverse works since $(t, \\theta)(-t, -\\theta) = (-\\theta\\theta, 0) = (0, 0)$, as $\\theta^2 = 0$.\n\nNon-example: on $\\R^{1|1}$ the law $(t, \\theta)(t', \\theta') = (tt', \\theta + \\theta')$ is associative with unit $(1, 0)$.\nBut it has no inverse morphism: on underlying points it is multiplication of real numbers, and $0$ has no inverse.\n\nThe underlying manifold of a Lie supergroup is an ordinary [[Lie group|7JBRGCVJRB]].\nThe left-invariant vector fields on $G$ form a Lie superalgebra, the Lie superalgebra of $G$.\nOrdinary Lie groups are the Lie supergroups with no odd coordinates.\n",
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