Difference between revisions of "Super-space"

Revision as of 14:53, 7 June 2020

A vector space over a field endowed with a -grading . The elements of the spaces and are said to be even and odd, respectively; for , the parity is defined to be . Each super-space has associated to it another super-space such that . The pair , where , , is called the dimension of the super-space . The field is usually considered as a super-space of dimension .

For two super-spaces and , the structure of a super-space on the spaces , , , etc., is defined naturally. In particular, a linear mapping is even if , and odd if . A homogeneous bilinear form is said to be symmetric if

and skew-symmetric if

All these concepts apply equally to -graded free modules over an arbitrary commutative superalgebra . The basis in is usually selected so that its first vectors are even and its last ones odd. Any endomorphism of the module is denoted in this basis by a block matrix

where , , such that if is even, then and consist of even elements and and consist of odd elements, whereas if is odd, then and consist of odd elements and and consist of even elements (in the former case the matrix is even, in the latter, odd).

References

[1]	F.A. Berezin, "Introduction to superanalysis" , Reidel (1987) (Translated from Russian)
[2]	D.A. Leites (ed.) , Seminar on super-manifolds , Kluwer (1990)

Comments

References

[a1]	F.A. Berezin, M.A. Shubin, "The Schrödinger equation" , Kluwer (1991) (Translated from Russian) (Supplement 3: D.A. Leites, Quantization and supermanifolds)

How to Cite This Entry:
Super-space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Super-space&oldid=49458

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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-<!--
+A [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911901.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911902.png" /> endowed with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911903.png" />-grading <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911904.png" />. The elements of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911906.png" /> are said to be even and odd, respectively; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911907.png" />, the parity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911908.png" /> is defined to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s0911909.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119010.png" />. Each super-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119011.png" /> has associated to it another super-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119013.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119014.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119017.png" />, is called the dimension of the super-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119018.png" />. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119019.png" /> is usually considered as a super-space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119020.png" />.
-s0911901.png
-$#A+1 = 51 n = 0
-$#C+1 = 51 : ~/encyclopedia/old_files/data/S091/S.0901190 Super\AAhspace
-Automatically converted into TeX, above some diagnostics.
-Please remove this comment and the {{TEX|auto}} line below,
-if TeX found to be correct.
--->
-{{TEX|auto}}
+For two super-spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119022.png" />, the structure of a super-space on the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119025.png" />, etc., is defined naturally. In particular, a linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119026.png" /> is even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119027.png" />, and odd if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119028.png" />. A homogeneous bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119029.png" /> is said to be symmetric if
-{{TEX|done}}
-A [[Vector space|vector space]]  $  V $
+<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/s091190/s09119030.png" /></td> </tr></table>
-over a field  $  k $
-endowed with a  $  \mathbf Z / 2 $-
-grading  $  V = V _ {\overline{0}\; }  \oplus V _ {\overline{1}\; }  $.
-The elements of the spaces  $  V _ {\overline{0}\; }  $
-and  $  V _ {\overline{1}\; }  $
-are said to be even and odd, respectively; for  $  x \in V _ {i} $,
-the parity  $  p( x) $
-is defined to be  $  i $
-$  ( i \in \mathbf Z / 2 = \{ \overline{0}\; , \overline{1}\; \} ) $.
-Each super-space  $  V $
-has associated to it another super-space  $  \Pi ( V) $
-such that  $  \Pi ( V) _ {i} = V _ {i+ \overline{1}\; }  $
-$  ( i \in \mathbf Z / 2 ) $.
-The pair  $  ( m, n) $,
-where  $  m =  \mathop{\rm dim}  V _ {\overline{0}\; }  $,
-$  n =  \mathop{\rm dim}  V _ {\overline{1}\; }  $,
-is called the dimension of the super-space  $  V $.
-The field  $  k $
-is usually considered as a super-space of dimension  $  ( 1, 0) $.
-For two super-spaces  $  V $
-and  $  W $,
-the structure of a super-space on the spaces  $  V \oplus W $,
-$   \mathop{\rm Hom} _ {k} ( V, W) $,
-$  V  ^  \star  $,
-etc., is defined naturally. In particular, a linear mapping  $  \phi :  V \rightarrow W $
-is even if  $  \phi ( V _ {i} ) \subset  W _ {i} $,
-and odd if  $  \phi ( V _ {i} ) \subset  W _ {i+ \overline{1}\; }  $.
-A homogeneous bilinear form  $  \beta :  V \otimes V \mapsto k $
-is said to be symmetric if
-$$
-\beta ( y, x)  =  (- 1) ^ {p( x) p( y)+ p( \beta )( p( x)+ p( y)) } \beta ( x, y),
-$$
 and skew-symmetric if
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119031.png" /></td> </tr></table>
-\beta ( y, x)  =  -(- 1) ^ {p( x) p( y)+ p( \beta )( p( x)+ p( y)) } \beta ( x, y).
-$$
-All these concepts apply equally to  $  \mathbf Z / 2 $-
+All these concepts apply equally to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119032.png" />-graded free modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119033.png" /> over an arbitrary commutative [[Superalgebra|superalgebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119034.png" />. The basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119035.png" /> is usually selected so that its first vectors are even and its last ones odd. Any endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119036.png" /> of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119037.png" /> is denoted in this basis by a block matrix
-graded free modules  $  V $
-over an arbitrary commutative [[Superalgebra|superalgebra]]  $  C $.
-The basis in  $  V $
-is usually selected so that its first vectors are even and its last ones odd. Any endomorphism  $  \phi $
-of the module  $  V $
-is denoted in this basis by a block matrix
-$$
+<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/s091190/s09119038.png" /></td> </tr></table>
-\alpha  =  \left (
-where  $  X \in M _ {n} ( C) $,
+where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119040.png" />, such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119041.png" /> is even, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119043.png" /> consist of even elements and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119045.png" /> consist of odd elements, whereas if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119046.png" /> is odd, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119048.png" /> consist of odd elements and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119050.png" /> consist of even elements (in the former case the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091190/s09119051.png" /> is even, in the latter, odd).
-$  T \in M _ {m} ( C) $,
-such that if  $  \phi $
-is even, then  $  X $
-and  $  T $
-consist of even elements and  $  Y $
-and  $  Z $
-consist of odd elements, whereas if  $  \phi $
-is odd, then  $  X $
-and  $  T $
-consist of odd elements and  $  Y $
-and  $  Z $
-consist of even elements (in the former case the matrix  $  \alpha $
-is even, in the latter, odd).
 ====References====
 <table><TR><TD valign="top">[1]</TD> <TD valign="top">  F.A. Berezin,   "Introduction to superanalysis" , Reidel  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.A. Leites (ed.) , ''Seminar on super-manifolds'' , Kluwer  (1990)</TD></TR></table>
 ====Comments====
 ====References====
 <table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.A. Berezin,   M.A. Shubin,   "The Schrödinger equation" , Kluwer  (1991)  (Translated from Russian)  (Supplement 3: D.A. Leites, Quantization and supermanifolds)</TD></TR></table>

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Difference between revisions of "Super-space"

Revision as of 14:53, 7 June 2020

References

Comments

References