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Difference between revisions of "Super-group"

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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  $  {\mathcal G} $
+
A group object in the category of [[super-manifold]]s. A super-group  $  {\mathcal G} $
 
is defined by a functor  $  {\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]]).
+
from the category of commutative [[superalgebra]]s 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.
  
 
===Examples.===
 
===Examples.===
  
1) The super-group  $  \mathop{\rm GL} _ {n\mid } m $
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1) The super-group  $  \mathop{\rm GL} _ {n\mid m} $
is defined by the functor  $  C \mapsto  \mathop{\rm GL} _ {n\mid } m ( C ) $
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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) $(
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into groups of even invertible matrices from  $  M _ {n\mid m} ( C) $(
see [[Super-space|Super-space]]), i.e. of matrices in the form
+
see [[Super-space]]), i.e. of matrices in the form
  
 
$$  
 
$$  
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while  $  Y, Z $
 
while  $  Y, Z $
 
are matrices over  $  C _ {\overline{1}\; }  $.  
 
are matrices over  $  C _ {\overline{1}\; }  $.  
A homomorphism  $  \mathop{\rm GL} _ {n\mid } m ( C) \rightarrow C _ {\overline{0}\; }  ^  \star  $
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A homomorphism  $  \mathop{\rm GL} _ {n\mid m} ( C) \rightarrow C _ {\overline{0}\; }  ^  \star  $
 
is defined by the formula
 
is defined by the formula
  
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\end{array}
 
\end{array}
  
\right )  =  \mathop{\rm det} ( X- YT  ^ {-} 1 Z)  \mathop{\rm det}  T  ^ {-} 1
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\right )  =  \mathop{\rm det} ( X- YT  ^ {-1} Z)  \mathop{\rm det}  T  ^ {-1}
 
$$
 
$$
  
 
(the Berezinian);
 
(the Berezinian);
  
2)  $  \mathop{\rm SL} _ {n\mid } m =  \mathop{\rm Ker}  \mathop{\rm Ber} $;
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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 $
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3)  $  \mathop{\rm OSp} _ {n\mid 2m} \subset  \mathop{\rm GL} _ {n\mid 2m} $
and  $  \Pi _ {n} \subset  \mathop{\rm GL} _ {n\mid } m $;  
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and  $  \Pi _ {n} \subset  \mathop{\rm GL} _ {n\mid m} $;  
 
they leave invariant an even, or odd, non-degenerate symmetric bilinear form.
 
they leave invariant an even, or odd, non-degenerate symmetric bilinear form.
  
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====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>

Latest revision as of 19:04, 18 July 2020


Lie super-group

A group object in the category of super-manifolds. 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.

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=49615
This article was adapted from an original article by D.A. Leites (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article