| Universal Enveloping Algebra |
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| CATEGORIES ABOUT UNIVERSAL ENVELOPING ALGEBRA | |
| lie algebras | |
| ring theory | |
| hopf algebras | |
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To understand the basic idea of this construction, first note that any associative algebra ''A'' over the Field ''K'' becomes a Lie algebra over ''K'' with the bracket : {Link without Title} = ''ab'' − ''ba''. That is, from an associative product, one can construct a Lie bracket by simply taking the Commutator with respect to that associative product. We denote this Lie algebra by ''AL''. Construction of the universal enveloping algebra attempts to reverse this process: to a given Lie algebra ''L'' over ''K'' we find the "most general" unital associative ''K''-algebra ''A'' such that the Lie algebra ''AL'' contains ''L''; this algebra ''A'' is ''U''(''L''). The important constraint is to preserve the representation theory: the Representations of ''L'' correspond in a one-to-one manner to the Module s over ''U''(''L''). In a typical context where ''L'' is acting by '' Infinitesimal Transformation s'', the elements of ''U''(''L'') act like Differential Operator s, of all orders. UNIVERSAL PROPERTY Let ''L'' be any Lie algebra over ''K''. Given a unital associative ''K''-algebra ''U'' and a Lie algebra homomorphism h (notation as above) we say that ''U'' is the universal enveloping algebra of ''L'' if it satisfies the following Universal Property : for any unital associative ''K''-algebra ''A'' and Lie algebra homomorphism f there exists a ''unique'' unital algebra homomorphism g such that f DIRECT CONSTRUCTION For general reasons having to do with universal properties, we can say that ''if'' a Lie algebra has a universal enveloping algebra, then this enveloping algebra is uniquely determined by ''L'' (up to a unique algebra isomorphism). By the following construction, which suggests itself on general grounds (for instance, as part of a pair of Adjoint Functors ), we establish that indeed every Lie algebra does have a universal enveloping algebra. Starting with the Tensor Algebra ''T''(''L'') on the Vector Space underlying ''L'', we take ''U''(''L'') to be the quotient of ''T''(''L'') made by imposing the relations a for all ''a'' and ''b'' in (the image in ''T''(''L'') of) ''L'', where the "." on the LHS denotes the associative multiplication in ''T''(''L''), and the bracket on the RHS now means the given Lie algebra product, in ''L''. Formally, we define U where |
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