Representation of a Lie superalgebra

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 ( c 1 A + c 2 B ) ⋅ X = c 1 A ⋅ X + c 2 B ⋅ X {\displaystyle (c_{1}A+c_{2}B)\cdot X=c_{1}A\cdot X+c_{2}B\cdot X} A ⋅ ( c 1 X + c 2 Y ) = c 1 A ⋅ X + c 2 A ⋅ Y {\displaystyle A\cdot (c_{1}X+c_{2}Y)=c_{1}A\cdot X+c_{2}A\cdot Y} ( − 1 ) A ⋅ X = ( − 1 ) A ( − 1 ) X {\displaystyle (-1)^{A\cdot X}=(-1)^{A}(-1)^{X}} [ A , B ] ⋅ X = A ⋅ ( B ⋅ X ) − ( − 1 ) A B B ⋅ ( A ⋅ X ) . {\displaystyle [A,B]\cdot X=A\cdot (B\cdot X)-(-1)^{AB}B\cdot (A\cdot X).} Equivalently, a representation of L is a Z2-graded representation of the universal enveloping algebra of L which respects the third equation above.

Source: Wikipedia — Representation of a Lie superalgebra (CC BY-SA 4.0)

Representation of a Lie superalgebra

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 ( c 1 A + c 2 B ) ⋅ X = c 1 A ⋅ X + c 2 B ⋅ X {\displaystyle (c_{1}A+c_{2}B)\cdot X=c_{1}A\cdot X+c_{2}B\cdot X} A ⋅ ( c 1 X + c 2 Y ) = c 1 A ⋅ X + c 2 A ⋅ Y {\displaystyle A\cdot (c_{1}X+c_{2}Y)=c_{1}A\cdot X+c_{2}A\cdot Y} ( − 1 ) A ⋅ X = ( − 1 ) A ( − 1 ) X {\displaystyle (-1)^{A\cdot X}=(-1)^{A}(-1)^{X}} [ A , B ] ⋅ X = A ⋅ ( B ⋅ X ) − ( − 1 ) A B B ⋅ ( A ⋅ X ) . {\displaystyle [A,B]\cdot X=A\cdot (B\cdot X)-(-1)^{AB}B\cdot (A\cdot X).} Equivalently, a representation of L is a Z2-graded representation of the universal enveloping algebra of L which respects the third equation above.

Source: Wikipedia "Representation of a Lie superalgebra" · CC BY-SA 4.0

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