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Binomial operation's Jacobi Identity 📂Abstract Algebra

Binomial operation's Jacobi Identity

Definition

Set SS and binary operations ∗:S×S→S\ast : S \times S \to S, commutative binary operations +:S×S→S+ : S \times S \to S, are considered. The following form of equation is called the Jacobi identity.

a∗(b∗c)+c∗(a∗b)+b∗(c∗a)=0,a,b,c∈S a \ast (b \ast c) + c \ast (a \ast b) + b \ast (c \ast a) = 0,\quad a,b,c \in S

If the above equation holds, ∗\ast satisfies the Jacobi identity.

Description

It refers to an equation where rotating variables and summing them results in 00.

  • The vector cross product ×\times satisfies the Jacobi identity.

    x×(y×z)+z×(x×y)+y×(z×x)=0 \mathbf{x} \times (\mathbf{y} \times \mathbf{z}) + \mathbf{z} \times (\mathbf{x} \times \mathbf{y}) + \mathbf{y} \times (\mathbf{z} \times \mathbf{x}) = 0

  • In a ring (R,+,⋅)(R, + , \cdot), the commutator [a,b]=ab−ba[a, b] = ab - ba satisfies the Jacobi identity.

    a∗(b∗c)+c∗(a∗b)+b∗(c∗a)=a∗(bc−cb)+c∗(ab−ba)+b∗(ca−ac)=((abc−acb)−(bca−cba))+((cab−cba)−(abc−bac))+((bca−bac)−(cab−acb))=0 \begin{align*} &a \ast (b \ast c) + c \ast (a \ast b) + b \ast (c \ast a) \\ &= a \ast (bc- cb) + c \ast (ab - ba) + b \ast (ca - ac) \\ &= (({\color{red}\cancel{\color{black}abc}} - {\color{blue}\cancel{\color{black}acb}}) - ({\color{green}\cancel{\color{black}bca}} - {\color{black}\cancel{\color{black}cba}})) + (({\color{magenta}\cancel{\color{black}cab}} - {\color{black}\cancel{\color{black}cba}}) - ({\color{red}\cancel{\color{black}abc}} - {\color{orange}\cancel{\color{black}bac}})) \\ & \quad + (({\color{green}\cancel{\color{black}bca}} - {\color{orange}\cancel{\color{black}bac}}) - ({\color{magenta}\cancel{\color{black}cab}} - {\color{blue}\cancel{\color{black}acb}})) \\ &= 0 \end{align*}

    • The commutator [A,B]=AB−BA[A, B] = AB - BA of two matrices A,B∈Mn×nA, B \in M_{n \times n} satisfies the Jacobi identity.

    • The Lie bracket of vector fields [X,Y]=XY−YX[X, Y] = XY - YX satisfies the Jacobi identity.