Difference between revisions of "Super-group"

Revision as of 14:53, 7 June 2020

Lie super-group

A group object in the category of super-manifolds (cf. Super-manifold). A super-group is defined by a functor 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 is defined by the functor into groups of even invertible matrices from (see Super-space), i.e. of matrices in the form

where are invertible matrices of orders over , while are matrices over . A homomorphism is defined by the formula

(the Berezinian);

2) ;

3) and ; they leave invariant an even, or odd, non-degenerate symmetric bilinear form.

To every super-group and super-subgroup of it there is related a super-manifold , represented by a functor . This super-manifold is a homogeneous space of .

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=48908

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

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 ''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-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]]).
-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.===
-) 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
-$$
+) The super-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911703.png" /> is defined by the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911704.png" /> into groups of even invertible matrices from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911705.png" /> (see [[Super-space|Super-space]]), i.e. of matrices in the form
-\left (
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911706.png" /></td> </tr></table>
-where  $  X, T $
+where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911707.png" /> are invertible matrices of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911708.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s0911709.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117010.png" /> are matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117011.png" />. A homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117012.png" /> is defined by the formula
-are invertible matrices of orders  $  n, m $
-over  $  C _ {\overline{0}\; }  $,
-while  $  Y, Z $
-are matrices over  $  C _ {\overline{1}\; }  $.
-A homomorphism  $   \mathop{\rm GL} _ {n\mid } m ( C) \rightarrow C _ {\overline{0}\; }   ^  \star  $
-is defined by the formula
-$$
+<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>
- \mathop{\rm Ber}  \left (
 (the Berezinian);
-)  $   \mathop{\rm SL} _ {n\mid } m =  \mathop{\rm Ker}   \mathop{\rm Ber} $;
+) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117014.png" />;
-)  $   \mathop{\rm OSp} _ {n\mid } 2m \subset   \mathop{\rm GL} _ {n\mid } 2m $
+) <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.
-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} $
+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" />.
-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====
 <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>

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Difference between revisions of "Super-group"

Revision as of 14:53, 7 June 2020

Examples.

References