In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). This construction passes from the non-associative structure L to a (more familiar, and possibly easier to handle) unital associative algebra which captures the important properties of L. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. ...

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 In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ...

[a,b] = 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 A_L. For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if and only...

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 A_L 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 modules over U(L). In a typical context where L is acting by infinitesimal transformations, the elements of U(L) act like differential operators, of all orders. In mathematics, if φ: G→H is a homomorphism of Lie groups, and g and h are the Lie algebras of G and H respectively, then the induced map φ* on tangent spaces is a homomorphism of Lie algebras, i. ... In abstract algebra, a module is a generalization of a vector space. ... In mathematics, an infinitesimal transformation is a limiting form of small transformation. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

Universal property

Let L be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism

h: L → U_L,

(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 In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

f: L → A_L

there exists a unique unital algebra homomorphism

g: U → A

such that

f = gh.

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. The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

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 In mathematics, the tensor algebra is an abstract algebra construction of a unital associative algebra T(V) from a vector space V. In a sense, T(V) is the most general algebra containing V. If we take basis vectors for V, those become non-commuting variables in T(V), subject... The fundamental concept in linear algebra is that of a vector space or linear space. ...

a.b − b.a = [a,b]

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. In mathematics, LHS is informal shorthand for the left-hand side of an equation. ... In mathematics, LHS is informal shorthand for the left-hand side of an equation. ...

Formally, we define

U(L) = T(L)/I

where

I = ([a,b] − a.b + b.a | a, b in L)

is the (two-sided) ideal in T(L) generated by all elements of the form [a,b] − a.b + b.a for a,b in L. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

The natural map L → T(L) gives rise to a map h : L → U(L), and this is the Lie algebra homomorphism used in the universal property given above.

The analogous construction for Lie superalgebras is straightforward. In mathematics, a Lie superalgebra is a kind of generalisation of a Lie algebra. ...

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. In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... The fundamental concept in linear algebra is that of a vector space or linear space. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

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. In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ...

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. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. ...

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 not be generated by first-order operators (see for example Casimir operator). The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ... In mathematics, a Casimir invariant of a Lie algebra is a member of the center of the universal enveloping algebra of the 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. This article is about the mathematical concept of convolution. ... This page deals with mathematical distributions. ... The word support has several specialized meanings: In mathematics, see support (mathematics). ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

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. In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form Elements a,b,c can be taken from some (arbitrary) commutative ring. ... In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X...

Further description of structure

The fundamental Poincaré-Birkhoff-Witt theorem gives a precise description of U(L); the most important consequence is that L can be viewed as a subspace of U(L). More precisely: the canonical map h : L → U(L) is always injective. Furthermore, U(L) is generated as a unital associative algebra by L. In the theory of Lie algebras, the Poincaré-Birkhoff-Witt theorem is a fundamental result characterizing the universal enveloping algebra of a Lie algebra. ... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

L acts on itself by the Lie algebra adjoint representation, and this action can be extended to a representation of L on U(L): L acts as an algebra of derivations on T(L), and this action respects the imposed relations, so it actually acts on U(L). (This is the purely infinitesimal way of looking at the invariant differential operators mentioned above.) The adjoint representation of a Lie group G is the linearized version of the action of G on itself by conjugation. ... In mathematics, if φ: G→H is a homomorphism of Lie groups, and g and h are the Lie algebras of G and H respectively, then the induced map φ* on tangent spaces is a homomorphism of Lie algebras, i. ... There are several meanings of derivation: A derivation in abstract algebra is a linear map that satisfies Leibniz law. ...

Under this representation, the elements of U(L) invariant under the action of L (i.e. such that any element of L acting on them gives zero) are called invariant elements. They are generated by the Casimir invariants. In mathematics, a Casimir invariant of a Lie algebra is a member of the center of the universal enveloping algebra of the Lie algebra. ...

As mentioned above, the construction of universal enveloping algebras is part of a pair of adjoint functors. U is a functor from the category of Lie algebras over K to the category of unital associative K-algebras. This functor is left adjoint to the functor which maps an algebra A to the Lie algebra A_L. It should be noted that the universal enveloping algebra construction is not exactly inverse to the formation of A_L: if we start with an associative algebra A, then U(A_L) is not equal to A; it is much bigger. The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... In category theory, a functor is a special type of mapping between categories. ...

The facts about representation theory mentioned earlier can be made precise as follows: the abelian category of all representations of L is isomorphic to the abelian category of all left modules over U(L). In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ... In mathematics, if φ: G→H is a homomorphism of Lie groups, and g and h are the Lie algebras of G and H respectively, then the induced map φ* on tangent spaces is a homomorphism of Lie algebras, i. ... In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i. ...

The construction of the group algebra for a given group is in many ways analogous to constructing the universal enveloping algebra for a given Lie algebra. Both constructions are universal and translate representation theory into module theory. Furthermore, both group algebras and universal enveloping algebras carry natural comultiplications which turn them into Hopf algebras. In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, coalgebras are structures that are in a certain sense dual to the unital associative algebras. ... In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes . (Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ...