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| In [[mathematics]] a '''Lie ring''' is a structure related to [[Lie algebra]]s that can arise as a generalisation of Lie algebras, or through the study of the [[lower central series]] of [[Group (mathematics)|groups]].
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| Lie rings need not be [[Lie group]]s under addition. Any [[Lie algebra]] is an example of a Lie ring. Any associative ring can be made into a Lie ring by defining a bracket operator <math>[x,y] = xy - yx</math>. Conversely to any Lie algebra there is a corresponding ring, called the [[universal enveloping algebra]].
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| Lie rings are used in the study of finite [[p-group]]s through the [[Lazard correspondence]]. The lower central factors of a ''p''-group are finite abelian ''p''-groups, so modules over '''Z'''/''p'''''Z'''. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the [[commutator]] of two coset representatives. The Lie ring structure is enriched with another module homomorphism, then ''p''th power map, making the associated Lie ring a so-called restricted Lie ring.
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| Lie rings are also useful in the definition of a [[p-adic analytic group]]s and their endomorphisms by studying Lie algebras over rings of integers such as the [[p-adic integers]]. The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and the reducing modulo ''p'' to get a Lie algebra over a finite field.
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| == Formal definition ==
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| A '''Lie ring''' is defined as a [[nonassociative ring]] with multiplication that is [[anticommutative]] and satisfies the [[Jacobi identity]]. More specifically we can define a Lie ring <math>L</math> to be an [[abelian group]] with an operation <math>[\cdot,\cdot]</math> that has the following properties:
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| * Bilinearity:
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| ::<math> [x + y, z] = [x, z] + [y, z], \quad [z, x + y] = [z, x] + [z, y] </math>
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| :for all ''x'', ''y'', ''z'' ∈ ''L''.
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| * The ''Jacobi identity'':
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| :: <math> [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 \quad </math>
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| :for all ''x'', ''y'', ''z'' in ''L''.
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| * For all ''x'' in ''L'':
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| ::<math> [x,x]=0 \quad </math>
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| == Examples ==
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| * Any [[Lie algebra]] over a general [[Ring (mathematics)|ring]] instead of a [[Field (mathematics)|field]] is an example of a Lie ring. Lie rings are ''not'' [[Lie group]]s under addition, despite the name.
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| * Any associative ring can be made into a Lie ring by defining a bracket operator <math>[x,y] = xy - yx</math>.
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| * For an example of a Lie ring arising from the study of [[Group (mathematics)|groups]], let <math>G</math> be a group with <math>(x,y) = x^{-1}y^{-1}xy</math> the commutator operation, and let <math>G = G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots \supseteq G_n \supseteq \cdots</math> be a [[central series]] in <math>G</math> — that is the commutator subgroup <math>(G_i,G_j)</math> is contained in <math>G_{i+j}</math> for any <math>i,j</math>. Then
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| :: <math>L = \bigoplus G_i/G_{i+1}</math>
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| :is a Lie ring with addition supplied by the group operation (which will be commutative in each homogeneous part), and the bracket operation given by
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| :: <math>[xG_i, yG_j] = (x,y)G_{i+j}\ </math>
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| :extended linearly. Note that the centrality of the series ensures the commutator <math>(x,y)</math> gives the bracket operation the appropriate Lie theoretic properties.
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| [[Category:Lie algebras]]
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| [[Category:Non-associative algebra]]
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