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I'm reading the book "Introduction to Lie Algebra and Representation Theory - J. E. Humphrey", I have a question on an example on the page number $2$. That is

Example: For reference, we write down the multiplication table for $gl(n, F)$ relative to the standard basis consisting of the matrices $e_{ij}$ (having $1$ in the $(i, j)$ position and $0$ elsewhere). Since $e_{ij} e_{kl} = \delta_{jk} e_{il}$ it follows that: $$[e_{ij}, e_{kl}] = \delta_{jk} e_{il} - \delta_{li} e_{kj}.$$ Notice that the coefficients are all $1,-1$ or $0$; in particular, all of them lie in the prime field of $F$.

My question: What is the meaning of the notation $\delta$ here?

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    $\begingroup$ Probably the kronecker delta $\delta_{i,j} := \begin{cases} 0, & \text{if } i \neq j , \\ 1, & \text{else.} \end{cases}$. $\endgroup$ – Viktor Glombik Apr 20 at 14:23
  • $\begingroup$ Could you show that more implicitly? I’m starting to study Lie algebra at today’s morning. $\endgroup$ – Minh Apr 20 at 14:25
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    $\begingroup$ Minh, you should write down the $4$-dimensional Lie algebra $\mathfrak{gl}_2(F)$ explicitly yourself (not implicitly, but explicitly). The vector space has a basis $\{e_{11},e_{12},e_{21},e_{22}\}$. The bracket is given by $[A,B]=AB-BA$ for these matrices $e_{ij}$. $\endgroup$ – Dietrich Burde Apr 20 at 14:28
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    $\begingroup$ It's definitely the Kronecker delta. @Minh you will have to be more specific if you have another question, because your first question was answered. $\endgroup$ – Jair Taylor Apr 20 at 14:45

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