Solution 1:

As Sammaye suggested, you need to use the $cond aggregation projection operator to do this:

db.Sentiments.aggregate(
    { $project: {
        _id: 0,
        Company: 1,
        PosSentiment: {$cond: [{$gt: ['$Sentiment', 0]}, '$Sentiment', 0]},
        NegSentiment: {$cond: [{$lt: ['$Sentiment', 0]}, '$Sentiment', 0]}
    }},
    { $group: {
        _id: "$Company",
        SumPosSentiment: {$sum: '$PosSentiment'},
        SumNegSentiment: {$sum: '$NegSentiment'}
    }});

Solution 2:

Starting from version 3.4, we can use the $switch operator which allows logical condition processing in the $group stage. Of course we still need to use the $sum accumulator to return the sum.

db.Sentiments.aggregate(
    [
        { "$group": { 
            "_id": "$Company",  
            "SumPosSenti": { 
                "$sum": { 
                    "$switch": { 
                        "branches": [ 
                            { 
                                "case": { "$gt": [ "$Sentiment", 0 ] }, 
                                "then": "$Sentiment"
                            }
                        ], 
                        "default": 0 
                    }
                }
            }, 
            "SumNegSenti": {
                "$sum": { 
                    "$switch": { 
                        "branches": [ 
                            { 
                                "case": { "$lt": [ "$Sentiment", 0 ] }, 
                                "then": "$Sentiment"
                            }
                        ], 
                        "default": 0 
                    } 
                }
            }
        }}
    ]
)

If you have not yet migrated your mongod to 3.4 or newer, then note that the $project stage in this answer is redundant because the $cond operator returns a numeric value which means that you can $group your documents and apply $sum to the $cond expression.

This will improve the performance in your application especially for large collection.

db.Sentiments.aggregate(
    [
        { '$group': {
            '_id': '$Company',
            'PosSentiment': { 
                '$sum': {
                    '$cond': [
                        { '$gt': ['$Sentiment', 0]}, 
                        '$Sentiment', 
                        0
                    ]
                }
            },
            'NegSentiment': { 
                '$sum': {
                    '$cond': [
                        { '$lt': ['$Sentiment', 0]}, 
                        '$Sentiment', 
                        0
                    ]
                }
            }
        }}
    ]
)

Consider a collection Sentiments with the following documents:

{ "Company": "a", "Sentiment" : 2 }
{ "Company": "a", "Sentiment" : 3 }
{ "Company": "a", "Sentiment" : -1 }
{ "Company": "a", "Sentiment" : -5 }

The aggregation query produces:

{ "_id" : "a", "SumPosSenti" : 5, "SumNegSenti" : -6 }