In this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in positive characteristic settings. In the abstract case, we prove an independence result for compatible systems of Lie-irreducible representations, from which we deduce an independence result for compatible systems admitting what we call a Lie-irreducible decomposition. In the case of geometric compatible systems of Galois representations arising from certain classes of automorphic forms, we prove the existence of a Lie-irreducible decomposition. From this we deduce an independence result. We conclude with the case of compatible systems of Galois representations over global function fields, for which we prove the existence of a Lie-irreducible decomposition, and we deduce an independence result. From this we also deduce an independence result for compatible systems of lisse sheaves on normal varieties over finite fields.

在本文中，我们开发了一种通用方法来证明兼容表示系统中代数单项群的独立性，并将其应用于推导兼容系统在自守和正特征设置中的独立性结果。在抽象情况下，我们证明了李不可约表示的相容系统的独立结果，从中我们推导出了承认我们称之为李不可约分解的相容系统的独立结果。在由某些类自守形式产生的伽罗瓦表示的几何兼容系统的情况下，我们证明了李不可约分解的存在。由此我们推导出独立性结果。我们以全局函数域上伽罗瓦表示的兼容系统的情况作为结论，为此，我们证明了李不可约分解的存在，并推导出一个独立的结果。由此我们还推导出有限域上正常变体上的 Lisse 层相容系统的独立性结果。