Difference between revisions of "Super-group"
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''Lie super-group'' | ''Lie super-group'' | ||
− | A group object in the category of | + | A group object in the category of [[super-manifold]]s. A super-group $ {\mathcal G} $ |
+ | is defined by a functor $ {\mathcal G} $ | ||
+ | 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} $ | ||
+ | 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); | (the Berezinian); | ||
− | 2) | + | 2) $ \mathop{\rm SL} _ {n\mid m} = \mathop{\rm Ker} \mathop{\rm Ber} $; |
− | 3) | + | 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 | + | 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> |
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) |
Super-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Super-group&oldid=17496