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Let $M_{p|q}(\mathbb{C}) = M_{p|q}(\mathbb{C})_0 \oplus M_{p|q}(\mathbb{C})_1$ be the super algebra of all $(p+q) \times (p+q)$ matrices.

Let $A, B \in M_{p|q}(\mathbb{C})$ (not necessarily homogeneous), If anybody can help me with finding the necessary and sufficient condition for $A$ and $B$ to super commute, that would be very helpful to me.

Super commutator is defined as $[X,Y] = XY - (-1)^{|X||Y|}XY \,\forall\, X,Y \in M_{p|q}(\mathbb{C})_0 \sqcup M_{p|q}(\mathbb{C})_1$ and extend this definition bilinearly to full $M_{p|q}(\mathbb{C})$. $|X| = \text{homogeneous degree of X}$

Thanks for your thoughts.

Have a good day.

GA316
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1 Answers1

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If $$A~=~A_0+A_1\qquad\text{and}\qquad B~=~B_0+B_1,\tag{1}$$ then the supercommutator $$[A,B]~=~[A,B]_0+[A,B]_1 \tag{2}$$ decomposes as $$[A,B]_0~=~[A_0,B_0]+[A_1,B_1]\qquad\text{and}\qquad [A,B]_1~=~[A_0,B_1]+[A_0,B_1].\tag{3}$$ In particular$$ A\text{ and }B\text{ supercommute}\quad\Leftrightarrow\quad [A,B]~=~0\quad\Leftrightarrow\quad [A,B]_0~=~0~~\wedge~~[A,B]_1 ~=~0,\tag{4}$$ cf. OP's question.

Qmechanic
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