# What is the notation $\delta$ here?

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?

• Probably the kronecker delta $\delta_{i,j} := \begin{cases} 0, & \text{if } i \neq j , \\ 1, & \text{else.} \end{cases}$. – Viktor Glombik Apr 20 at 14:23
• Could you show that more implicitly? I’m starting to study Lie algebra at today’s morning. – Minh Apr 20 at 14:25
• 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 $e_{ij}$. – Dietrich Burde Apr 20 at 14:28
• 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. – Jair Taylor Apr 20 at 14:45