In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the...

Enveloping algebra in mathematics may refer to: Universal enveloping algebra, of a Lie algebra Associative enveloping algebra, of a general non-associative...

equivalent but larger, namely a group algebra or a universal enveloping algebra, then a group algebra or enveloping algebra can be "deformed", although the...

representations of its Lie algebra. In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with...

quantized enveloping algebra is a q-analog of a universal enveloping algebra. Given a Lie algebra g {\displaystyle {\mathfrak {g}}} , the quantum enveloping algebra...

operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum...

Q(a)=2L^{2}(a)-L(a^{2})\ .} The article on universal enveloping algebras describes the canonical construction of enveloping algebras, as well as the PBW-type theorems...

{g}}\mapsto U({\mathfrak {g}})} , called the universal enveloping algebra. To construct this: given a Lie algebra g {\displaystyle {\mathfrak {g}}} over F...

and a semisimple algebra over an algebraically closed field. The universal enveloping algebra of a Lie algebra is an associative algebra that can be used...

algebra can be given the structure of a Hopf algebra. See Tensor algebra for details. The symmetric algebra S(V) is the universal enveloping algebra of...

the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two coalgebra...

Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a...

the universal enveloping algebra of a free Lie algebra on a finite set is the shuffle algebra. This essentially follows because universal enveloping algebras...

It is a special case of the ★-product of the "algebra of symbols" of a universal enveloping algebra. The Moyal product is named after José Enrique Moyal...

comes about is given in the universal enveloping algebra article. Quantum deformations of the universal enveloping algebra lead to the notion of quantum...

about the given C*-algebra. This may also be called the universal enveloping von Neumann algebra, since it is given by a universal property; and (as always...

universal enveloping algebra. We continue the notation of the previous section: g {\displaystyle {\mathfrak {g}}} is a complex semisimple Lie algebra...

{sl}}_{2}\hookrightarrow {\mathfrak {g}}} , a W-algebra may be constructed from the universal enveloping algebra of the affine Lie algebra g ^ {\displaystyle {\hat {\mathfrak...

a representation of L is a Z2-graded representation of the universal enveloping algebra of L which respects the third equation above. A * Lie superalgebra...

theorem to hold). Just as for Lie algebras, the universal enveloping algebra of the Lie superalgebra can be given a Hopf algebra structure. Lie superalgebras...

creation modes. A basis of creation generators of the Virasoro algebra's universal enveloping algebra is the set L = { L − n 1 L − n 2 ⋯ L − n k } k ∈ N 0 < n...

algebra Symmetric algebra Tensor algebra Universal enveloping algebra Vertex operator algebra von Neumann algebra Weyl algebra Zinbiel algebra This is a list...

Lie algebras. The isomorphism maps the center Z ( U ( g ) ) {\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))} of the universal enveloping algebra U (...

the tensor algebra of any Lie algebra is a Poisson algebra. The universal enveloping algebra is obtained by modding out the Poisson algebra structure....

permutation. The shuffle algebra on a finite set is the graded dual of the universal enveloping algebra of the free Lie algebra on the set. Over the rational...

{\displaystyle {\mathfrak {g}}} over an algebraically closed field. The universal property of the universal enveloping algebra ensures that π {\displaystyle \pi...

{\displaystyle A_{n}} is also a quotient of the universal enveloping algebra of the Heisenberg algebra, the Lie algebra of the Heisenberg group, by setting the...

associative algebra on α generators, and the right hand side is the generating function for the universal enveloping algebra of the free Lie algebra on α generators...

commutator Lie algebra of its universal enveloping algebra. Ado's theorem states that every finite-dimensional Lie algebra over a field of characteristic...

Harish-Chandra (1951). "On some applications of the universal enveloping algebra of a semisimple Lie algebra". Trans. Amer. Math. Soc. 70 (1): 28–96. doi:10...