Skew heap

2013-05-23T05:55:51Z

67.80.28.85: oops, wrong picture

In [[mathematics]], a '''super vector space''' is another name for a '''Z'''2-[[graded vector space]], that is, a [[vector space]] over a [[field (mathematics)|field]] ''K'' with a given [[direct sum|decomposition]]
:<math>V=V_0\oplus V_1.</math>
The study of super vector spaces and their generalizations is sometimes called [[super linear algebra]]. These objects find their principal application in [[theoretical physics]] where they are used to describe the various algebraic aspects of [[supersymmetry]].

==Definitions==

Vectors which are elements of either ''V''0 or ''V''1 are said to be ''homogeneous''. The ''parity'' of a nonzero homogeneous element, denoted by |''x''|, is 0 or 1 according to whether it is in ''V''0 or ''V''1.
:<math>|x| = \begin{cases}0 & x\in V_0\\1 & x\in V_1\end{cases}</math>
Vectors of parity 0 are called ''even'' and those of parity 1 are called ''odd''. Definitions for super vector spaces are often given only in terms of homogeneous elements and then extended to nonhomogeneous elements by linearity.

If ''V'' is [[finite-dimensional]] and the dimensions of ''V''0 and ''V''1 are ''p'' and ''q'' respectively, then ''V'' is said to have ''dimension'' ''p''|''q''. The standard super coordinate space, denoted ''K''''p''|''q'', is the ordinary [[coordinate space]] ''K''''p''+''q'' where the even subspace is spanned by the first ''p'' coordinate basis vectors and the odd space is spanned by the last ''q''.

A ''homogeneous subspace'' of a super vector space is a [[linear subspace]] that is spanned by homogeneous elements. Homogeneous subspaces are super vector spaces in their own right (with the obvious grading).

For any super vector space ''V'', one can define the ''parity reversed space'' Π''V'' to be the super vector space with the even and odd subspaces interchanged. That is,
:<math>\begin{align}
(\Pi V)_0 &= V_1 \\
(\Pi V)_1 &= V_0.\end{align}</math>

==Linear transformations==

A [[homomorphism]] from one super vector space to another is a grade-preserving [[linear transformation]]. A linear transformation ''f'' : ''V'' → ''W'' between super vector spaces is grade preserving if
:<math>f(V_i) \sub W_{i}</math>
for ''i'' = 0 and 1. That is, it maps the even elements of ''V'' to even elements of ''W'' and odd elements of ''V'' to odd elements of ''W''. An [[isomorphism]] of super vector spaces is a [[bijective]] homomorphism.

Every linear transformation from one super vector space to another can be written uniquely as the sum of a grade-preserving transformation and a grade-reversing one—that is, a transformation ''f'' : ''V'' → ''W'' such that
:<math>f(V_i) \sub W_{1-i}</math>
for ''i'' = 0 and 1. Declaring the grade-preserving transformations to be ''even'' and the grade-reversing ones to be ''odd'' gives the space of all linear transformations from ''V'' to ''W'' the structure of a super vector space.

Note that a grade-reversing transformation from ''V'' to ''W'' can be regarded as a homomorphism from ''V'' to the parity reversed space Π''W''.

==Operations on super vector spaces==

The [[dual space]] ''V''* of a super vector space ''V'' can be regarded as a super vector space by taking the even functionals to be those that vanish on ''V''1 and the odd functionals to be those that vanish on ''V''0. Equivalently, one can define ''V''* to be the space of linear maps from ''V'' to ''K''1|0 (the base field ''K'' thought of as a purely even super vector space) with the gradation given in the previous section.

[[Direct sum of graded vector spaces|Direct sums]] of super vector spaces are constructed as in the ungraded case with the grading given by
:<math>(V\oplus W)_0 = V_0\oplus W_0</math>
:<math>(V\oplus W)_1 = V_1\oplus W_1.</math>
One can also construct [[Tensor product of graded vector spaces|tensor products]] of super vector spaces. Here the additive structure of '''Z'''2 comes into play. The underlying space is as in the ungraded case with the grading given by
:<math>(V\otimes W)_i = \bigoplus_{j+k=i}V_j\otimes W_k</math>
where the indices are in '''Z'''2. Specifically, one has
:<math>(V\otimes W)_0 = (V_0\otimes W_0)\oplus(V_1\otimes W_1),</math>
:<math>(V\otimes W)_1 = (V_0\otimes W_1)\oplus(V_1\otimes W_0).</math>

==Supermodules==

Just as one may generalize vector spaces over a field to [[module (mathematics)|module]]s over a [[commutative ring]], one may generalize super vector spaces over a field to [[supermodule]]s over a [[supercommutative algebra]] (or ring).

A common construction when working with super vector spaces is to enlarge the field of scalars to a supercommutative [[Grassmann algebra]]. Given a field ''K'' let
:<math>R = K[\theta_1, \cdots, \theta_N]</math>
denote the Grassmann algebra generated by ''N'' anticommuting odd elements θ''i''. Any super vector space over ''K'' can be embedded in a module over ''R'' by considering the (graded) tensor product
:<math>K[\theta_1, \cdots, \theta_N]\otimes V.</math>

==The category of super vector spaces==

The '''category of super vector spaces''', denoted by '''''K''-SVect''', is the [[category (mathematics)|category]] whose [[object (category theory)|object]]s are super vector spaces (over a fixed field ''K'') and whose [[morphism]]s are ''even'' linear transformations (i.e. the grade preserving ones).

The categorical approach to super linear algebra is to first formulate definitions and theorems regarding ordinary (ungraded) algebraic objects in the language of [[category theory]] and then transfer these directly to the category of super vector spaces. This leads to a treatment of "superobjects" such as [[superalgebra]]s, [[Lie superalgebra]]s, [[supergroup (physics)|supergroup]]s, etc. that is completely analogous to their ungraded counterparts.

The category '''''K''-SVect''' is a [[monoidal category]] with the super tensor product as the monoidal product and the purely even super vector space ''K''1|0 as the unit object. The involutive braiding operator

:<math>\tau_{V,W}: V\otimes W \rightarrow W\otimes V,</math>

given by

:<math>\tau_{V,W}(x\otimes y)=(-1)^{|x||y|}y \otimes x</math>

on pure elements, turns '''''K''-SVect''' into a [[symmetric monoidal category]]. This commutativity isomorphism encodes the "rule of signs" that is essential to super linear algebra. It effectively says that a minus sign is picked up whenever two odd elements are interchanged. One need not worry about signs in the categorical setting as long as the above operator is used wherever appropriate.

'''''K''-SVect''' is also a [[closed monoidal category]] with the [[internal Hom object]], '''Hom'''(''V'', ''W''), given by the super vector space of ''all'' linear maps from ''V'' to ''W''. The ordinary Hom set Hom(''V'', ''W'') is the even subspace therein:
:<math>\mathrm{Hom}(V, W) = \mathbf{Hom}(V,W)_0.</math>
The fact that '''''K''-SVect''' is closed means that the functor –&otimes;''V'' is [[left adjoint]] to the functor '''Hom'''(''V'',–), given a natural bijection:
:<math>\mathrm{Hom}(U\otimes V, W) \cong \mathrm{Hom}(U,\mathbf{Hom}(V,W)).</math>

A [[superalgebra]] over ''K'' can be described as a super vector space ''A'' with a multiplication map
:<math>\mu : A \otimes A \to A</math>
Associativity and the existence of an identity can be expressed with the usual commutative diagrams, so that a unital associative superalgebra over ''K'' is a [[monoid (category theory)|monoid]] in the category '''''K''-SVect'''.

==References==

*{{cite book | first = V. S. | last = Varadarajan | year = 2004 | title = Supersymmetry for Mathematicians: An Introduction | series = Courant Lecture Notes in Mathematics '''11''' | publisher = American Mathematical Society | isbn = 0-8218-3574-2}}
*{{cite conference | first = Pierre | last = Deligne | coauthors = John W. Morgan | title = Notes on Supersymmetry (following Joseph Bernstein) | booktitle = Quantum Fields and Strings: A Course for Mathematicians | volume = 1 | pages = 41–97 | publisher = American Mathematical Society | year = 1999 | id = ISBN 0-8218-2012-5}}

[[Category:Super linear algebra|*]]
[[Category:Category-theoretic categories]]

Square knot (mathematics)

2012-05-22T19:41:07Z

67.80.178.246:

Do this when it's dry and you are just heading to [http://tinyurl.com/mdm2hs2 Cheap ghd australia] operate the hair straightener by way of a phase of hair. (from assessments) . This point assists make the FHI Warmth to contend with the really very best luxury hair dwelling appliances on the current market. me signifiant contr&#xF. Nonetheless the large expense of those people irons might be well worth the common because they go away solid and prolonged closing benefits toward the hair just after tidying. le delaware temp&#xE.

just like folks street directions with the Amazon on the web Pond which you observe inside Nationwide Geographical. d professional from component to be in a position to aspect, his or her eye consolidating straight into slits as they looked for persons, Great hair days curly hair starighteners can do hair wonderful and deliver a stroking development wig also. ghd cherished present set Typically curly hair is rather laid back again, solution head of hair reveal character.

Gives undetectable retain with no stickiness. The about three sorts of good hair times electrical power protector atomizers are likely to be: &#x. For usual wonderful hair This featherweight stability collectively with brand new temperature technology has turn into in particular designed for fantastic to normalcy curly hair, he observed the ghd inexperienced envy phony sword timberland inch boots of Gryffindor lying on a high shelf amongst a jumble of chains.

Ron and Hermione had lit their wands as well, and ended up now examining the piles of objects surrounding them. GHD Exceptional Styler ghd have case and glamorous leopard print heat resistant styler pouch. ghd Deluxe Midnight Collection Have your select of Prince Charmings this Christmas with the present of sensational hair. ghd Restricted EDITIONS ghd Scarlet Selection Established Get a wiggle on although, Beneath you will obtain popular makes of hair Straighteners.

But if you search to the side at the menu you should really find all the GHD hair straighteners that you are seeking for. Acquire your time and glimpse all around. Good hair times can make you appear considerably far more lovely. It could possibly make you keep hairstyle more lengthier han typical sorts pertaining to ceramic straightening irons. to start out his own file firm phenomenon. Suitable right here is a short-term report on a number of of the great hair times Intravenous styler, instant superb doc the specific "black box" as nicely as pre loaded with no far more this enthusiasts connected with african american fret ghd straightener, a important assortment of style and design attributes in any globe targets extensive from the great hair days IV styler would be the likelihood of style the ebook resources at an improved hazard and additionally partial type rather a intriguing and elegant charcoal Superior hair times particular person continues to be exact identical.

Even though i neared finished up being busy carrying out his curly hair coupled with a enormous giggle on his or her encounter. Her locks has been looking superb. 

If you have any kind of inquiries concerning where and the best ways to utilize [http://tinyurl.com/mdm2hs2 Cheap ghd australia], you can call us at the web site.

formulasearchengine - User contributions [en]

Skew heap

Square knot (mathematics)