Я вежливо поинтересовалась у администратора, работает ли сегодня касса.

Поинтересовалась means something like inquired / asked politely / took an interest in. It sounds a bit more formal or courteous than спросила, which is the basic verb asked.

So:

• спросила = asked
• поинтересовалась = inquired, asked in a polite or tactful way

In this sentence, вежливо and поинтересовалась work well together, because both suggest politeness.

That ending shows two things at once:

1. -л- marks the past tense.
2. -ась shows that the speaker is female.

So:

• поинтересовался = a man said this
• поинтересовалась = a woman said this

The -сь / -ся part is because the verb is поинтересоваться, which is a reflexive verb.

Yes, it can often be omitted in Russian if the subject is clear from the verb form or context.

Here, поинтересовалась already tells us the speaker is I in context and also female, so Russian often allows:

• Вежливо поинтересовалась у администратора, работает ли сегодня касса.

However, Я may still be included for clarity, emphasis, or because it sounds natural in the situation.

After verbs like спросить or поинтересоваться, Russian often uses у + genitive to mean from someone / of someone in the sense of asking them.

So:

• у администратора = from the administrator / of the administrator

This is a very common pattern:

• спросить у друга = ask a friend
• узнать у коллеги = find out from a colleague
• поинтересоваться у администратора = ask the administrator

Here администратора is in the genitive case because of the preposition у.

Because the preposition у requires the genitive.

Dictionary form:

• администратор

Genitive singular:

• администратора

So:

• у администратора = at/from the administrator

This is a fixed grammatical pattern, not something special about this one sentence.

Ли is a particle used to introduce an indirect yes/no question.

The clause means:

• whether the ticket office is open/operating today
• literally: is the cash desk working today or not?

In English, we use whether / if:

• I asked whether the ticket office was open today.

In Russian, ли does that job in indirect yes/no questions.

In Russian, ли usually comes right after the word that is being questioned or focused.

Here the question is basically:

• Работает ли сегодня касса?
• Is the ticket office operating today?

So ли follows работает, the finite verb.

This is the normal pattern for indirect yes/no questions:

• Я не знаю, придёт ли он. = I don’t know whether he will come.
• Она спросила, есть ли кофе. = She asked whether there was coffee.

Those versions are either unnatural or would change the emphasis.

The normal, neutral order is:

• работает ли сегодня касса

That means the question is simply whether it is operating.

If you move ли, you change what is being highlighted, and many placements sound awkward. For learners, the safest rule is:

• in indirect yes/no questions, put ли right after the verb or the element being questioned.

So in this sentence, работает ли is the natural choice.

It depends on context. Касса can mean:

• a cash desk / checkout
• a ticket office / ticket window
• the place where payments or ticket sales are handled

In a sentence like this, работает ли сегодня касса, it often means:

• whether the ticket office / payment desk is open today

Russian uses работать for places, services, and offices in the sense of being open / functioning.

So касса работает often means not literally the cash register is physically working, but the ticket/payment desk is open and operating.

Yes, literally работает means works / is working, but in Russian it is also very commonly used for businesses, offices, counters, services, websites, etc. to mean:

• is open
• is operating
• is functioning

Examples:

• Магазин работает до десяти. = The shop is open until ten.
• Банк сегодня не работает. = The bank isn’t open today.
• Касса работает? = Is the ticket office open?

So this is a very normal use of работать.

Russian word order is flexible, and сегодня is placed before касса simply as a natural adverb position.

• работает ли сегодня касса = whether the ticket office is open today

You could also hear:

• работает ли касса сегодня

Both are possible, but the version in your sentence sounds very natural. Putting сегодня earlier can make the time frame clear sooner.

Вежливо means politely. It modifies поинтересовалась:

• Я вежливо поинтересовалась... = I politely inquired...

It is not grammatically necessary, but it adds information about how the speaker asked. Since поинтересоваться already sounds fairly polite, вежливо reinforces that tone.

It contains an indirect question.

The whole sentence is a statement:

• I politely asked the administrator whether the ticket office was open today.

Inside it, the clause

• работает ли сегодня касса is an indirect question.

A direct question would be:

• Работает ли сегодня касса? or more colloquially:
• Сегодня касса работает?

So the sentence reports the question rather than asking it directly.

Поинтересоваться is the perfective partner of интересоваться in this meaning.

Here, perfective implies a completed single action:

• she asked once / made the inquiry

So:

• Я поинтересовалась... = I asked / inquired
• Я интересовалась... would usually suggest a more ongoing, repeated, or background interest, and would sound less appropriate here

With one completed act of asking, the perfective verb is the natural choice.

Only the past-tense verb form:

• Я вежливо поинтересовался у администратора, работает ли сегодня касса.

Everything else stays the same.

So:

• поинтересовалась = female speaker
• поинтересовался = male speaker

It is mostly neutral to slightly formal.

Why?

• поинтересовалась is more polished than спросила
• администратор sounds neutral/formal
• the structure with ли is standard and natural

In everyday speech, many people might say something simpler, such as:

• Я спросила у администратора, работает ли сегодня касса.

That is a bit more straightforward, but the original sentence is perfectly natural and polite.