После тренировки мне нельзя поднимать тяжёлую штангу.

Because после always takes the genitive case.

• Dictionary form: тренировка
• Genitive singular: тренировки

So:

• после тренировки = after the workout / after training

This is a very common pattern:

• после урока = after the lesson
• после работы = after work
• после тренировки = after the workout

So the ending changes because the preposition после requires it.

Because нельзя is used in an impersonal construction, and the person affected is put in the dative case.

So:

• я = I
• мне = to me / for me

In Russian, phrases like мне нельзя, мне можно, мне нужно, мне холодно are very common. English often uses I in these situations, but Russian often uses to me / for me grammatically.

So:

• мне нельзя literally feels like it is not allowed for me or I must not

That is why мне, not я, is used.

Нельзя means something like:

• it is not allowed
• one must not
• it is not advisable
• it is forbidden / not okay

In this sentence, it most naturally means:

• I shouldn’t lift a heavy barbell after training
• I’m not supposed to lift a heavy barbell after training

It does not necessarily mean simple physical inability. For that, Russian more often uses не могу.

Compare:

• Мне нельзя поднимать тяжёлую штангу. = I must not / I shouldn’t lift a heavy barbell.
• Я не могу поднять тяжёлую штангу. = I can’t lift a heavy barbell.

So нельзя is about prohibition, rule, medical advice, or what is considered improper.

Russian usually does not use a present-tense copula like English is / am / are in sentences like this.

So where English says:

• It is not allowed for me
• I am not allowed

Russian simply says:

• мне нельзя

Here нельзя itself works as the predicate. This is normal Russian style, not an incomplete sentence.

This is a question of aspect.

• поднимать = imperfective
• поднять = perfective

In this sentence, поднимать is used because the meaning is general:
after training, I shouldn’t be lifting a heavy barbell.

The imperfective often fits:

• general rules
• repeated actions
• ongoing activity
• broad statements about what one should or should not do

Compare:

• Мне нельзя поднимать тяжёлую штангу. = I shouldn’t lift / be lifting a heavy barbell.
• Мне нельзя поднять эту штангу. = I can’t lift this barbell up / it’s impossible for me to lift it (more like a single completed action, depending on context)

So the imperfective here sounds more natural for a general restriction.

Because тяжёлая штанга is the direct object of поднимать, so it goes into the accusative case.

Dictionary forms:

• тяжёлая = heavy
• штанга = barbell

Accusative singular feminine:

• тяжёлую
• штангу

So:

• поднимать тяжёлую штангу = to lift a heavy barbell

This happens because штанга is a feminine noun, and the adjective has to agree with it in:

• gender
• number
• case

That is why both words change together.

In this sentence, штанга means barbell.

That is the most likely meaning here, especially with тяжёлая and the context of training.

However, in other contexts, штанга can also mean a rod, bar, or pole-like piece of equipment. Russian words are often a bit broader than their most common English translation.

But here you should understand it as:

• heavy barbell

Yes, you could, but the meaning changes.

• Мне нельзя поднимать тяжёлую штангу. = I’m not allowed to / I shouldn’t / I mustn’t lift a heavy barbell.
• Я не могу поднимать тяжёлую штангу. = I can’t lift a heavy barbell / I’m unable to do it.

So:

• нельзя = prohibition, rule, medical restriction, advice
• не могу = inability

If the speaker means a doctor told them not to do it after training, мне нельзя is much better.

Yes. Russian word order is flexible, although not random.

The given sentence:

• После тренировки мне нельзя поднимать тяжёлую штангу.

is very natural. It starts with the time phrase after training, then gives the person affected, then the restriction.

You could also say:

• Мне нельзя после тренировки поднимать тяжёлую штангу.

This is also natural and means the same thing.

The difference is mostly emphasis and information flow, not basic meaning.

Russian often places the most important or contextual information earlier depending on what the speaker wants to highlight.

Most naturally, it means after a workout / after the training session.

Here тренировки is genitive singular from тренировка.

If you wanted after workouts in a more general plural sense, you would usually say:

• после тренировок

So:

• после тренировки = after the workout / after a training session
• после тренировок = after workouts

Context can make the singular sound general too, but grammatically this form is singular.

By itself, it most naturally sounds like a rule, restriction, or recommendation, often something medical or practical.

For example:

• a doctor advised it
• a coach prohibited it
• it is unsafe after exercise
• the speaker is under some restriction

It does not sound like a mere preference such as I don’t want to.

If you wanted a personal preference, Russian would use something else, for example:

• Я не хочу поднимать тяжёлую штангу после тренировки. = I don’t want to lift a heavy barbell after training.

So мне нельзя suggests an external or objective restriction, not just a choice.