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''of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u0956701.png" /> over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u0956702.png" /> with a unit element''
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''of a Lie algebra $ \mathfrak{g} $ over a commutative ring $ \mathbb{k} $ with a unit element''
  
The associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u0956703.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u0956704.png" /> with a unit element, together with a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u0956705.png" /> for which the following properties hold: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u0956706.png" /> is a homomorphism of Lie algebras, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u0956707.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u0956708.png" />-linear and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u0956709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567010.png" />; 2) for every associative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567011.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567012.png" /> with a unit element and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567013.png" />-algebra mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567014.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567016.png" />, there exists a unique homomorphism of associative algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567017.png" />, mapping the unit to the unit, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567018.png" />. The universal enveloping algebra is unique up to an isomorphism and always exists: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567019.png" /> is the [[Tensor algebra|tensor algebra]] of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567020.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567022.png" /> is the two-sided ideal generated by all elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567025.png" /> is the canonical mapping, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567026.png" /> is the universal enveloping algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567027.png" />.
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The associative $ \mathbb{k} $-algebra $ U(\mathfrak{g}) $ with a unit element, together with a mapping $ \sigma: \mathfrak{g} \to U(\mathfrak{g}) $ for which the following properties hold:
 +
# $ \sigma $ is a homomorphism of Lie algebras, i.e., $ \sigma $ is a $ \mathbb{k} $-linear transformation and $ \sigma([x,y]) = \sigma(x) \sigma(y) - \sigma(y) \sigma(x) $ for all $ x,y \in \mathfrak{g} $.
 +
# For every associative $ \mathbb{k} $-algebra $ A $ with a unit element and every $ \mathbb{k} $-algebra homomorphism $ \alpha: \mathfrak{g} \to A $ such that $ \alpha([x,y]) = \alpha(x) \alpha(y) - \alpha(y) \alpha(x) $ for all $ x,y \in \mathfrak{g} $, there exists a unique homomorphism of associative algebras $ \beta: U(\mathfrak{g}) \to A $, mapping the unit to the unit, such that $ \alpha = \beta \circ \sigma $.
 +
The universal enveloping algebra is unique up to an isomorphism and always exists: If $ T(\mathfrak{g}) $ is the [[Tensor algebra|tensor algebra]] of the $ \mathbb{k} $-module $ \mathfrak{g} $, $ I $ is the two-sided ideal generated by all elements of the form $ [x,y] - (x \otimes y - y \otimes x) $ for $ x,y \in \mathfrak{g} $, and $ \sigma: \mathfrak{g} \to T(\mathfrak{g}) / I $ is the canonical map, then $ T(\mathfrak{g}) / I $ is the universal enveloping algebra of $ \mathfrak{g} $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567028.png" /> is Noetherian and the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567029.png" /> has finite order, then the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567030.png" /> is left and right Noetherian. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567031.png" /> is a free module over an integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567033.png" /> has no zero divisors. For any finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567034.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567035.png" /> the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567036.png" /> satisfies the Ore condition (cf. [[Imbedding of semi-groups|Imbedding of semi-groups]]) and so has a skew-field of fractions.
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If $ \mathbb{k} $ is Noetherian and the module $ \mathfrak{g} $ has finite order, then the algebra $ U(\mathfrak{g}) $ is left- and right-Noetherian. If $ \mathfrak{g} $ is a free module over an integral domain $ \mathbb{k} $, then $ U(\mathfrak{g}) $ has no zero divisors. For any finite-dimensional Lie algebra $ \mathfrak{g} $ over a field $ \mathbb{k} $, the algebra $ U(\mathfrak{g}) $ satisfies the Ore condition (cf. [[Imbedding of semi-groups|imbedding of semi-groups]]) and so has a skew-field of fractions.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567037.png" /> is any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567038.png" />-module, then every homomorphism of Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567039.png" /> extends to a homomorphism of associative algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567040.png" />. This establishes an isomorphism of the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567041.png" />-modules and the category of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567042.png" />-modules, whose existence forms the basis for the application of universal enveloping algebras in the theory of representations of Lie algebras (cf. [[#References|[3]]], [[#References|[4]]]).
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If $ V $ is any $ \mathbb{k} $-module, then every Lie-algebra homomorphism $ \mathfrak{g} \to \operatorname{End} V $ extends to a homomorphism of associative algebras $ U(\mathfrak{g}) \to \operatorname{End} V $. This establishes an isomorphism between the category of $ \mathfrak{g} $-modules and the category of left $ U(\mathfrak{g}) $-modules, whose existence forms the basis for the application of universal enveloping algebras in the theory of representations of Lie algebras ([[#References|[3]]], [[#References|[4]]]).
  
The universal enveloping algebra of the direct product of Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567043.png" /> is the tensor product of the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567044.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567045.png" /> is a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567048.png" /> are free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567049.png" />-modules, then the canonical homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567050.png" /> is an imbedding. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567051.png" /> is an extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567052.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567053.png" />. A universal enveloping algebra has a canonical filtration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567056.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567057.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567058.png" />-submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567059.png" /> generated by the products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567062.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567063.png" />. The graded algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567064.png" /> associated to this filtration is commutative and is generated by the image under the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567065.png" />; this mapping defines a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567066.png" /> of the [[Symmetric algebra|symmetric algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567067.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567068.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567069.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567070.png" />. By the Poincaré–Birkhoff–Witt theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567071.png" /> is an algebra isomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567072.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567073.png" />-module. The following is an equivalent formulation: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567074.png" /> is a basis of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567075.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567076.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567077.png" /> is a totally ordered set, then the family of monomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567080.png" />, forms a basis of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567081.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567082.png" /> (in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567083.png" /> is injective).
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The universal enveloping algebra of the direct product of Lie algebras $ \mathfrak{g}_{1},\ldots,\mathfrak{g}_{n} $ is the tensor product of the $ U(\mathfrak{g}_{i}) $’s. If $ \mathfrak{h} $ is a subalgebra of $ \mathfrak{g} $, where $ \mathfrak{h} $ and $ \mathfrak{g} / \mathfrak{h} $ are free $ \mathbb{k} $-modules, then the canonical homomorphism $ U(\mathfrak{h}) \to U(\mathfrak{g}) $ is an imbedding. If $ \mathbb{k}' $ is an extension of the field $ \mathbb{k} $, then $ U(\mathfrak{g} \otimes_{\mathbb{k}} \mathbb{k}') = U(\mathfrak{g}) \otimes_{\mathbb{k}} \mathbb{k}' $. A universal enveloping algebra has a canonical filtration $ {U_{0}}(\mathfrak{g}) \subseteq {U_{1}}(\mathfrak{g}) \subseteq \ldots $, where $ {U_{0}}(\mathfrak{g}) = \mathbb{k} \cdot 1 $ and $ {U_{n}}(\mathfrak{g}) $ for $ n \in \mathbf{N} $ is the $ \mathbb{k} $-submodule of $ U(\mathfrak{g}) $ generated by the products $ \sigma(x_{1}) \cdots \sigma(x_{m}) $, where $ m \in \mathbf{N}_{\leq n} $ and $ x_{i} \in \mathfrak{g} $ for all $ i \in \mathbf{N}_{\leq n} $. The graded algebra $ \operatorname{gr} U(\mathfrak{g}) $ associated to this filtration is commutative and is generated by the image under the natural homomorphism $ \mathfrak{g} \to \operatorname{gr} U(\mathfrak{g}) $; this mapping defines a homomorphism $ \delta $ of the [[Symmetric algebra|symmetric algebra]] $ S(\mathfrak{g}) $ of the $ \mathbb{k} $-module $ \mathfrak{g} $ onto $ \operatorname{gr} U(\mathfrak{g}) $. By the Poincaré-Birkhoff-Witt theorem, $ \delta: S(\mathfrak{g}) \to \operatorname{gr} U(\mathfrak{g}) $ is an algebra isomorphism if $ \mathfrak{g} $ is a free $ \mathbb{k} $-module. The following is an equivalent formulation: If $ (x_{i})_{i \in I} $ is an ordered basis of the $ \mathbb{k} $-module $ \mathfrak{g} $, where $ I $ is a totally ordered set, then the family of monomials $ \sigma(x_{i_{1}}) \cdots \sigma(x_{i_{n}}) $, for $ i_{1} \leq_{I} \ldots \leq_{I} i_{n} $ and $ n \in \mathbf{N}_{0} $, forms a basis of the $ \mathbb{k} $-module $ U(\mathfrak{g}) $ (in particular, $ \sigma $ is injective).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567084.png" /> be the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567085.png" />. Then for any finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567086.png" /> over a field of characteristic zero, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567087.png" /> consists of the subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567088.png" />-invariant elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567089.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567090.png" /> is semi-simple, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567091.png" /> is the algebra of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095670/u09567092.png" /> variables.
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Let $ Z(\mathfrak{g}) $ be the centre of $ U(\mathfrak{g}) $. Then for any finite-dimensional Lie algebra $ \mathfrak{g} $ over a field of characteristic zero, $ \operatorname{gr} Z(\mathfrak{g}) \subseteq \operatorname{gr} U(\mathfrak{g}) = S(\mathfrak{g}) $ consists of the subalgebra of $ G $-invariant elements of $ S(\mathfrak{g}) $. If $ \mathfrak{g} $ is semi-simple, then $ Z(\mathfrak{g}) $ is the algebra of polynomials in $ \operatorname{rank}(\mathfrak{g}) $ variables.
  
One of the important directions of research in universal enveloping algebra is the study of primitive ideals (cf. [[Primitive ideal|Primitive ideal]]; [[#References|[3]]]).
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One of the important directions of research in universal enveloping algebras is the study of [[Primitive ideal|primitive ideals]] ([[#References|[3]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Lie groups and Lie algebras" , ''Elements of mathematics'' , Hermann  (1975)  pp. Chapts. 1–3  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Groupes et algèbres de Lie" , Hermann  (1975)  pp. Chapts. VII-VIII</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Dixmier,  "Enveloping algebras" , North-Holland  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.M. Gel'fand,  "The centre of an infinitesimal group ring"  ''Mat. Sb.'' , '''26'''  (1950)  pp. 103–112  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR></table>
 
  
 +
<table>
 +
<TR><TD valign="top">[1]</TD><TD valign="top">
 +
N. Bourbaki, “Lie groups and Lie algebras”, ''Elements of mathematics'', Hermann (1975), pp. Chapts. 1–3. (Translated from French)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">
 +
N. Bourbaki, “Groupes et algèbres de Lie”, Hermann (1975), pp. Chapts. VII-VIII.</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">
 +
J. Dixmier, “Enveloping algebras”, North-Holland (1977). (Translated from French)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">
 +
A.A. Kirillov, “Elements of the theory of representations”, Springer (1976). (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top">
 +
I.M. Gel’fand, “The centre of an infinitesimal group ring”, ''Mat. Sb.'', '''26''' (1950), pp. 103–112. (In Russian)</TD></TR>
 +
<TR><TD valign="top">[6]</TD> <TD valign="top">
 +
J.-P. Serre, “Lie algebras and Lie groups”, Benjamin (1965). (Translated from French)</TD></TR>
 +
</table>
  
 +
====References====
  
====Comments====
+
<table>
 
+
<TR><TD valign="top">[a1]</TD><TD valign="top">
 
+
J.C. Jantzen, “Einhüllende Algebren halbeinfacher Lie-Algebren”, Springer (1983).</TD></TR>
====References====
+
</table>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.C. Jantzen,   "Einhüllende Algebren halbeinfacher Lie-Algebren" , Springer (1983)</TD></TR></table>
 

Revision as of 04:39, 24 April 2017

of a Lie algebra $ \mathfrak{g} $ over a commutative ring $ \mathbb{k} $ with a unit element

The associative $ \mathbb{k} $-algebra $ U(\mathfrak{g}) $ with a unit element, together with a mapping $ \sigma: \mathfrak{g} \to U(\mathfrak{g}) $ for which the following properties hold:

  1. $ \sigma $ is a homomorphism of Lie algebras, i.e., $ \sigma $ is a $ \mathbb{k} $-linear transformation and $ \sigma([x,y]) = \sigma(x) \sigma(y) - \sigma(y) \sigma(x) $ for all $ x,y \in \mathfrak{g} $.
  2. For every associative $ \mathbb{k} $-algebra $ A $ with a unit element and every $ \mathbb{k} $-algebra homomorphism $ \alpha: \mathfrak{g} \to A $ such that $ \alpha([x,y]) = \alpha(x) \alpha(y) - \alpha(y) \alpha(x) $ for all $ x,y \in \mathfrak{g} $, there exists a unique homomorphism of associative algebras $ \beta: U(\mathfrak{g}) \to A $, mapping the unit to the unit, such that $ \alpha = \beta \circ \sigma $.

The universal enveloping algebra is unique up to an isomorphism and always exists: If $ T(\mathfrak{g}) $ is the tensor algebra of the $ \mathbb{k} $-module $ \mathfrak{g} $, $ I $ is the two-sided ideal generated by all elements of the form $ [x,y] - (x \otimes y - y \otimes x) $ for $ x,y \in \mathfrak{g} $, and $ \sigma: \mathfrak{g} \to T(\mathfrak{g}) / I $ is the canonical map, then $ T(\mathfrak{g}) / I $ is the universal enveloping algebra of $ \mathfrak{g} $.

If $ \mathbb{k} $ is Noetherian and the module $ \mathfrak{g} $ has finite order, then the algebra $ U(\mathfrak{g}) $ is left- and right-Noetherian. If $ \mathfrak{g} $ is a free module over an integral domain $ \mathbb{k} $, then $ U(\mathfrak{g}) $ has no zero divisors. For any finite-dimensional Lie algebra $ \mathfrak{g} $ over a field $ \mathbb{k} $, the algebra $ U(\mathfrak{g}) $ satisfies the Ore condition (cf. imbedding of semi-groups) and so has a skew-field of fractions.

If $ V $ is any $ \mathbb{k} $-module, then every Lie-algebra homomorphism $ \mathfrak{g} \to \operatorname{End} V $ extends to a homomorphism of associative algebras $ U(\mathfrak{g}) \to \operatorname{End} V $. This establishes an isomorphism between the category of $ \mathfrak{g} $-modules and the category of left $ U(\mathfrak{g}) $-modules, whose existence forms the basis for the application of universal enveloping algebras in the theory of representations of Lie algebras ([3], [4]).

The universal enveloping algebra of the direct product of Lie algebras $ \mathfrak{g}_{1},\ldots,\mathfrak{g}_{n} $ is the tensor product of the $ U(\mathfrak{g}_{i}) $’s. If $ \mathfrak{h} $ is a subalgebra of $ \mathfrak{g} $, where $ \mathfrak{h} $ and $ \mathfrak{g} / \mathfrak{h} $ are free $ \mathbb{k} $-modules, then the canonical homomorphism $ U(\mathfrak{h}) \to U(\mathfrak{g}) $ is an imbedding. If $ \mathbb{k}' $ is an extension of the field $ \mathbb{k} $, then $ U(\mathfrak{g} \otimes_{\mathbb{k}} \mathbb{k}') = U(\mathfrak{g}) \otimes_{\mathbb{k}} \mathbb{k}' $. A universal enveloping algebra has a canonical filtration $ {U_{0}}(\mathfrak{g}) \subseteq {U_{1}}(\mathfrak{g}) \subseteq \ldots $, where $ {U_{0}}(\mathfrak{g}) = \mathbb{k} \cdot 1 $ and $ {U_{n}}(\mathfrak{g}) $ for $ n \in \mathbf{N} $ is the $ \mathbb{k} $-submodule of $ U(\mathfrak{g}) $ generated by the products $ \sigma(x_{1}) \cdots \sigma(x_{m}) $, where $ m \in \mathbf{N}_{\leq n} $ and $ x_{i} \in \mathfrak{g} $ for all $ i \in \mathbf{N}_{\leq n} $. The graded algebra $ \operatorname{gr} U(\mathfrak{g}) $ associated to this filtration is commutative and is generated by the image under the natural homomorphism $ \mathfrak{g} \to \operatorname{gr} U(\mathfrak{g}) $; this mapping defines a homomorphism $ \delta $ of the symmetric algebra $ S(\mathfrak{g}) $ of the $ \mathbb{k} $-module $ \mathfrak{g} $ onto $ \operatorname{gr} U(\mathfrak{g}) $. By the Poincaré-Birkhoff-Witt theorem, $ \delta: S(\mathfrak{g}) \to \operatorname{gr} U(\mathfrak{g}) $ is an algebra isomorphism if $ \mathfrak{g} $ is a free $ \mathbb{k} $-module. The following is an equivalent formulation: If $ (x_{i})_{i \in I} $ is an ordered basis of the $ \mathbb{k} $-module $ \mathfrak{g} $, where $ I $ is a totally ordered set, then the family of monomials $ \sigma(x_{i_{1}}) \cdots \sigma(x_{i_{n}}) $, for $ i_{1} \leq_{I} \ldots \leq_{I} i_{n} $ and $ n \in \mathbf{N}_{0} $, forms a basis of the $ \mathbb{k} $-module $ U(\mathfrak{g}) $ (in particular, $ \sigma $ is injective).

Let $ Z(\mathfrak{g}) $ be the centre of $ U(\mathfrak{g}) $. Then for any finite-dimensional Lie algebra $ \mathfrak{g} $ over a field of characteristic zero, $ \operatorname{gr} Z(\mathfrak{g}) \subseteq \operatorname{gr} U(\mathfrak{g}) = S(\mathfrak{g}) $ consists of the subalgebra of $ G $-invariant elements of $ S(\mathfrak{g}) $. If $ \mathfrak{g} $ is semi-simple, then $ Z(\mathfrak{g}) $ is the algebra of polynomials in $ \operatorname{rank}(\mathfrak{g}) $ variables.

One of the important directions of research in universal enveloping algebras is the study of primitive ideals ([3]).

References

[1] N. Bourbaki, “Lie groups and Lie algebras”, Elements of mathematics, Hermann (1975), pp. Chapts. 1–3. (Translated from French)
[2] N. Bourbaki, “Groupes et algèbres de Lie”, Hermann (1975), pp. Chapts. VII-VIII.
[3] J. Dixmier, “Enveloping algebras”, North-Holland (1977). (Translated from French)
[4] A.A. Kirillov, “Elements of the theory of representations”, Springer (1976). (Translated from Russian)
[5] I.M. Gel’fand, “The centre of an infinitesimal group ring”, Mat. Sb., 26 (1950), pp. 103–112. (In Russian)
[6] J.-P. Serre, “Lie algebras and Lie groups”, Benjamin (1965). (Translated from French)

References

[a1] J.C. Jantzen, “Einhüllende Algebren halbeinfacher Lie-Algebren”, Springer (1983).
How to Cite This Entry:
Universal enveloping algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universal_enveloping_algebra&oldid=18977
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article