Examples in Particular Cases
If L is abelian (that is, the bracket is always 0), then U(L) is commutative; if a basis of the vector space L has been chosen, then U(L) can be identified with the polynomial algebra over K, with one variable per basis element.
If L is the Lie algebra corresponding to the Lie group G, U(L) can be identified with the algebra of left-invariant differential operators (of all orders) on G; with L lying inside it as the left-invariant vector fields as first-order differential operators.
To relate the above two cases: if L is a vector space V as abelian Lie algebra, the left-invariant differential operators are the constant coefficient operators, which are indeed a polynomial algebra in the partial derivatives of first order.
The center of U(L) is called Z(L) and consists of the left- and right- invariant differential operators; this in the case of G not commutative will often not be generated by first-order operators (see for example Casimir operator of a semi-simple Lie algebra).
Another characterisation in Lie group theory is of U(L) as the convolution algebra of distributions supported only at the identity element e of G.
The algebra of differential operators in n variables with polynomial coefficients may be obtained starting with the Lie algebra of the Heisenberg group. See Weyl algebra for this; one must take a quotient, so that the central elements of the Lie algebra act as prescribed scalars.
Read more about this topic: Universal Enveloping Algebra
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