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In the [[mathematics|mathematical]] field of [[representation theory]], a '''representation of a Lie superalgebra''' is an [[semigroup action|action]] of [[Lie superalgebra]] ''L'' on a [[graded vector space|'''Z'''<sub>2</sub>-graded vector space]] ''V'', such that if ''A'' and ''B'' are any two pure elements of ''L'' and ''X'' and ''Y'' are any two pure elements of ''V'', then
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:<math>(c_1 A+c_2 B)\cdot X=c_1 A\cdot X + c_2 B\cdot X\,</math>
 
:<math>A\cdot (c_1 X + c_2 Y)=c_1 A\cdot X + c_2 A\cdot Y\,</math>
 
:<math>(-1)^{A\cdot X}=(-1)^A(-1)^X\,</math>
 
:<math>[A,B]\cdot X=A\cdot (B\cdot X)-(-1)^{AB}B\cdot (A\cdot X).\,</math>
 
Equivalently, a representation of ''L'' is a '''Z'''<sub>2</sub>-graded representation of the [[universal enveloping algebra]] of ''L'' which respects the third equation above.
 
==Unitary representation of a star Lie superalgebra==
A <sup>*</sup> [[Lie superalgebra]] is a complex Lie superalgebra equipped with an [[Involution (mathematics)|involutive]] [[antilinear]] [[map]] <sup>*</sup> such that * respects the grading and
 
:[a,b]<sup>*</sup>=[b<sup>*</sup>,a<sup>*</sup>].
 
A [[unitary representation]] of such a Lie algebra is a '''Z'''<sub>2</sub> [[graded vector space|graded]] [[Hilbert space]] which is a representation of a Lie superalgebra as above together with the requirement that [[self-adjoint]] elements of the Lie superalgebra are represented by [[Hermitian]] transformations.
 
This is a major concept in the study of [[supersymmetry]] together with representation of a Lie superalgebra on an algebra. Say A is an [[star-algebra|*-algebra]] representation of the Lie superalgebra (together with the additional requirement that * respects the grading and L[a]<sup>*</sup>=-(-1)<sup>La</sup>L<sup>*</sup>[a<sup>*</sup>]) and '''H''' is the unitary rep and also, '''H''' is a [[unitary representation]] of A.
 
These three reps are all compatible if for pure elements a in A, |ψ> in '''H''' and L in the Lie superalgebra,
 
:L[a|ψ>)]=(L[a])|ψ>+(-1)<sup>La</sup>a(L[|ψ>]).
 
Sometimes, the Lie superalgebra is [[embedding|embedded]] within A in the sense that there is a homomorphism from the [[universal enveloping algebra]] of the Lie superalgebra to A. In that case, the equation above reduces to
 
:L[a]=La-(-1)<sup>La</sup>aL.
 
This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary [[Grassmann number]]s.
 
==See also==
* [[Graded vector space]]
* [[Lie algebra representation]]
* [[Representation theory of Hopf algebras]]
 
[[Category:Representation theory of Lie algebras]]
 
 
 
{{algebra-stub}}

Revision as of 13:40, 7 February 2014

Should conventional treatment falters try out alternative treatments. * After sex you may become aware of some white or clear discharge– this is liquefied sperm, and usually disappears within 24 hours of intercourse. You will need to take into consideration always currently being beneficial. Bear in mind, even a 10% reduction in body mass may help reestablish regular menstrual cycles and ovulation. Research has shown that smoking decreases the amount of eggs in a woman’s ovaries.

The end of the day can be tiring and weekends are generally reserved for chores that are not done during the week. " Well, if you need to understand how to get pregnant fast, then you'll definitely should be a lady with a program. One fantastic program called Pregnancy Miracle offers a report for potential moms-to-be. Simple research can help you to determine what these are. The two young people of the same age, the experience must be extraordinary.

This kind of meat is also a great source of vitamin B12 which is essential to maintain and develop the nervous system. You will find that reading some hints and tips for Sims game play will surely make your gaming experience a lot more fun as well as more interesting.   Choose the best vitamins, Fertil - Aid for Women: Female Fertility Supplement">herbs and supplements that work for you in order to achieve your goal. Living a healthier lifestyle can prevent you from having miscarriages and stillbirths. Alcohol can reduce fertility by as much as 50% and decrease sperm counts.

Just think about it for a moment, babies have an amazing effect on even the most hard-core men & women, a baby causes a staunch person go to all “gagga”. Day one of your menstrual cycle is the day you first see menstrual blood loss even if it is only a drop of blood. If you are to the point where you feel like you have tried everything, even procedures like IVF, and everything has failed, then it is time to start really taking care of your body. Plasma Packs – these look like juice boxes, but with plasma. No extra effort is needed for them to get pregnant.

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