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In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra ''L'' on a Z2-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 : : : : Equivalently, a representation of ''L'' is a Z2-graded representation of the universal enveloping algebra of ''L'' which respects the third equation above. ==Unitary representation of a star Lie superalgebra== A * Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map * such that * respects the grading and :() *=(). A unitary representation of such a Lie algebra is a Z2 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 *-algebra representation of the Lie superalgebra (together with the additional requirement that * respects the grading and L() *=-(-1)LaL *()) 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()=(L())|ψ>+(-1)Laa(L()). Sometimes, the Lie superalgebra is 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()=La-(-1)LaaL. This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary Grassmann numbers. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Representation of a Lie superalgebra」の詳細全文を読む スポンサード リンク
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