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''Lie super-group''
 
''Lie super-group''
  
A group object in the category of super-manifolds (cf. [[Super-manifold|Super-manifold]]). A super-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911701.png" /> is defined by a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911702.png" /> from the category of commutative superalgebras into the category of groups. Lie's theorems (cf. [[Lie theorem|Lie theorem]]) are transferred to super-groups, a fact that gives the correspondence between super-groups and finite-dimensional Lie superalgebras (cf. [[Superalgebra|Superalgebra]]).
+
A group object in the category of super-manifolds (cf. [[Super-manifold|Super-manifold]]). A super-group $  {\mathcal G} $
 +
is defined by a functor $  {\mathcal G} $
 +
from the category of commutative superalgebras into the category of groups. Lie's theorems (cf. [[Lie theorem|Lie theorem]]) are transferred to super-groups, a fact that gives the correspondence between super-groups and finite-dimensional Lie superalgebras (cf. [[Superalgebra|Superalgebra]]).
  
 
===Examples.===
 
===Examples.===
  
 +
1) The super-group  $  \mathop{\rm GL} _ {n\mid } m $
 +
is defined by the functor  $  C \mapsto  \mathop{\rm GL} _ {n\mid } m ( C ) $
 +
into groups of even invertible matrices from  $  M _ {n\mid } m ( C) $(
 +
see [[Super-space|Super-space]]), i.e. of matrices in the form
  
1) The super-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911703.png" /> is defined by the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911704.png" /> into groups of even invertible matrices from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911705.png" /> (see [[Super-space|Super-space]]), i.e. of matrices in the form
+
$$
 +
\left (  
 +
\begin{array}{cc}
 +
X  & Y  \\
 +
Z  & T  \\
 +
\end{array}
 +
\right ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911706.png" /></td> </tr></table>
+
where  $  X, T $
 +
are invertible matrices of orders  $  n, m $
 +
over  $  C _ {\overline{0}\; }  $,
 +
while  $  Y, Z $
 +
are matrices over  $  C _ {\overline{1}\; }  $.
 +
A homomorphism  $  \mathop{\rm GL} _ {n\mid } m ( C) \rightarrow C _ {\overline{0}\; }  ^  \star  $
 +
is defined by the formula
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911707.png" /> are invertible matrices of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911708.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911709.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117010.png" /> are matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117011.png" />. A homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117012.png" /> is defined by the formula
+
$$
 +
\mathop{\rm Ber}  \left (
 +
\begin{array}{cc}
 +
X  & Y  \\
 +
Z  & T  \\
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117013.png" /></td> </tr></table>
+
\right )  =   \mathop{\rm det} ( X- YT  ^ {-} 1 Z)  \mathop{\rm det}  T  ^ {-} 1
 +
$$
  
 
(the Berezinian);
 
(the Berezinian);
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117014.png" />;
+
2) $  \mathop{\rm SL} _ {n\mid } m = \mathop{\rm Ker}  \mathop{\rm Ber} $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117016.png" />; they leave invariant an even, or odd, non-degenerate symmetric bilinear form.
+
3) $  \mathop{\rm OSp} _ {n\mid } 2m \subset  \mathop{\rm GL} _ {n\mid } 2m $
 +
and $  \Pi _ {n} \subset  \mathop{\rm GL} _ {n\mid } m $;  
 +
they leave invariant an even, or odd, non-degenerate symmetric bilinear form.
  
To every super-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117017.png" /> and super-subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117018.png" /> of it there is related a super-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117019.png" />, represented by a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117020.png" />. This super-manifold is a homogeneous space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117021.png" />.
+
To every super-group $  {\mathcal G} $
 +
and super-subgroup $  {\mathcal H} $
 +
of it there is related a super-manifold $  {\mathcal G} / {\mathcal H} $,  
 +
represented by a functor $  C \mapsto {\mathcal G} ( C) / {\mathcal H} ( C) $.  
 +
This super-manifold is a homogeneous space of $  {\mathcal G} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Manin,  "Gauge fields and complex geometry" , Springer  (1988)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.A. Berezin,  "Introduction to superanalysis" , Reidel  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D.A. Leites (ed.) , ''Seminar on supermanifolds'' , Kluwer  (1990)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Manin,  "Gauge fields and complex geometry" , Springer  (1988)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.A. Berezin,  "Introduction to superanalysis" , Reidel  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D.A. Leites (ed.) , ''Seminar on supermanifolds'' , Kluwer  (1990)</TD></TR></table>

Revision as of 14:55, 7 June 2020


Lie super-group

A group object in the category of super-manifolds (cf. Super-manifold). A super-group $ {\mathcal G} $ is defined by a functor $ {\mathcal G} $ from the category of commutative superalgebras into the category of groups. Lie's theorems (cf. Lie theorem) are transferred to super-groups, a fact that gives the correspondence between super-groups and finite-dimensional Lie superalgebras (cf. Superalgebra).

Examples.

1) The super-group $ \mathop{\rm GL} _ {n\mid } m $ is defined by the functor $ C \mapsto \mathop{\rm GL} _ {n\mid } m ( C ) $ into groups of even invertible matrices from $ M _ {n\mid } m ( C) $( see Super-space), i.e. of matrices in the form

$$ \left ( \begin{array}{cc} X & Y \\ Z & T \\ \end{array} \right ) , $$

where $ X, T $ are invertible matrices of orders $ n, m $ over $ C _ {\overline{0}\; } $, while $ Y, Z $ are matrices over $ C _ {\overline{1}\; } $. A homomorphism $ \mathop{\rm GL} _ {n\mid } m ( C) \rightarrow C _ {\overline{0}\; } ^ \star $ is defined by the formula

$$ \mathop{\rm Ber} \left ( \begin{array}{cc} X & Y \\ Z & T \\ \end{array} \right ) = \mathop{\rm det} ( X- YT ^ {-} 1 Z) \mathop{\rm det} T ^ {-} 1 $$

(the Berezinian);

2) $ \mathop{\rm SL} _ {n\mid } m = \mathop{\rm Ker} \mathop{\rm Ber} $;

3) $ \mathop{\rm OSp} _ {n\mid } 2m \subset \mathop{\rm GL} _ {n\mid } 2m $ and $ \Pi _ {n} \subset \mathop{\rm GL} _ {n\mid } m $; they leave invariant an even, or odd, non-degenerate symmetric bilinear form.

To every super-group $ {\mathcal G} $ and super-subgroup $ {\mathcal H} $ of it there is related a super-manifold $ {\mathcal G} / {\mathcal H} $, represented by a functor $ C \mapsto {\mathcal G} ( C) / {\mathcal H} ( C) $. This super-manifold is a homogeneous space of $ {\mathcal G} $.

References

[1] Yu.I. Manin, "Gauge fields and complex geometry" , Springer (1988) (Translated from Russian)
[2] F.A. Berezin, "Introduction to superanalysis" , Reidel (1987) (Translated from Russian)
[3] D.A. Leites (ed.) , Seminar on supermanifolds , Kluwer (1990)
How to Cite This Entry:
Super-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Super-group&oldid=49457
This article was adapted from an original article by D.A. Leites (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article