"Mi nem ismerjük a zeneszerzőket."

why don't we say ismerrük and ismerje? what is the rule of different conjugatian? because it ends with r?

I'm not sure what you mean. We need first person plural definite conjugation (because we have a definite object here), which is -juk/jük. Now, the j assimilates after certain consonants in writing (& therefore in speech too), such as s, sz, z, zs, c, cs, dz & dzs if there's even a word ending with dzs.

but we say ismeri, why dont we say ismerje? is it irregular?

I'm not quite sure what you mean, ismer is completely regular.

The sentence has ismerjük, first person plural definite (we know it.)

Ismeri is third person singular definite (it knows it), ismerje is third person singular definite but with imperative mood (it should know it.)

so verb takes suffix according to have front or back vowels. if it has back vowels, it takes -ja , if it has front vowels, it takes -i. turkish is similar to hungarian in some extent

but in regular conjugation the stem takes -je, ja for third person?

For third person singular in present tense with indicative mood, indefinite conjugation has no ending (ismer, it knows something), definite conjugation has -ja or -i (ismeri, it knows it.) But here we have first person plural. You're talking about "him/her/it", while the sentence is about "us."

the suffix for 3rd singular is confusing my mind. it is sometimes -i sometimes -ja. keresi is conjugation for 3rd singular. also ismeri is like that. i thought there is a rule for it. you said some consonants for it. but there is no r consonant for it

Okay, I think I get what the problem is.

In Hungarian most suffixes have two or three forms. The form you should use depends on the so-called vowel harmony (Turkish has a similar system, you seem to be familiar with that language.) The so-called back (or deep) vowels are a, á, o, ó, u, ú. The so-called front (or high) vowels are e, é, i, í, ö, ő, ü, ű. In case of some suffixes there are two front vowel forms, an unrounded (e, é, i, í) & a rounded (ö, ő, ü, ű), but that isn't interesting in our case.

For 3rd person singular in present tense with indicative mood & definite conjugation there are two forms, -ja & -i. Here are some examples:

kap - kapja

mar - marja

lát - látja

tud - tudja

olvas - olvassa (here's the assimilation thing I talked about because I thought you were confused about that)

ismer - ismeri

üt - üti

süt - süti

The forms of most suffices differ usually only in the vowels, so this particular one might seem strange & can be confusing, I think.