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 | + | 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 | ||
− | 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.=== | ||
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− | + | 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 | |
− | + | ||
+ | <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 | + | 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 | ||
− | over | ||
− | while | ||
− | are matrices over | ||
− | A homomorphism | ||
− | 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> | |
− | |||
(the Berezinian); | (the Berezinian); | ||
− | 2) | + | 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091170/s09117014.png" />; |
− | 3) | + | 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. |
− | and | ||
− | they leave invariant an even, or odd, non-degenerate symmetric bilinear form. | ||
− | To every super-group | + | 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 | ||
− | of it there is related a super-manifold | ||
− | represented by a functor | ||
− | This super-manifold is a homogeneous space of | ||
====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 14:53, 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) |
Super-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Super-group&oldid=49457