In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie, pronounced /liː/ ("lee"), not /laɪ/ ("lie") ) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... To meet Wikipedias quality standards and make it more accessible, this article may require cleanup. ... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ... The 1930s (years from 1930–1939) were described as an abrupt shift to more radical and conservative lifestyles, as countries were struggling to find a solution to the Great Depression, also known as the World Depression. ...

Definition and first properties

A Lie algebra is a type of algebra over a field; it is a vector space over some field F together with a binary operation [·, ·] In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... 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 same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

called the commutator or the Lie bracket, which satisfies the following axioms:

• Bilinearity:

for all scalars a, b in F and all elements x, y, z in

• Anticommutativity, or skew-symmetry:

for all elements x, y in When F is a field of characteristic two, one has to impose the stronger condition

[x,x] = 0

for all x in

• The Jacobi identity:

for all x, y, z in

For any associative algebra A with multiplication *, one can construct a Lie algebra L(A). As a vector space, L(A) is the same as A. The Lie bracket of two elements of L(A) is defined to be their commutator in A: In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... A mathematical operator (typically a binary operator, represented by *) is anticommutative if and only if it is true that x * y = −(y * x) for all x and y on the operators valid domain (e. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ...

[a,b] = a * b − b * a.

The associativity of the multiplication * in A implies the Jacobi identity of the commutator in L(A). In particular, the associative algebra of n × n matrices over a field F gives rise to the general linear Lie algebra The associative algebra A is called an enveloping algebra of the Lie algebra L(A). It is known that every Lie algebra can be embedded into one that arises from an associative algebra in this fashion. See universal enveloping algebra. In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ... In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). ...

Homomorphisms, subalgebras, and ideals

The Lie bracket is not an associative operation in general, meaning that [[x,y],z] need not equal [x,[y,z]]. Nonetheless, much of the terminology that was developed in the theory of associative rings or associative algebras is commonly applied to Lie algebras. A subspace of a Lie algebra that is closed under the Lie bracket is called a Lie subalgebra. If a subspace satisfies a stronger condition that In mathematics, associativity is a property that a binary operation can have. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ...

then I is called an ideal in the Lie algebra ^[1] A Lie algebra in which the commutator is not identically zero and which has no proper ideals is called simple. A homomorphism between two Lie algebras (over the same ground field) is a linear map that is compatible with the commutators:

for all elements x and y in As in the theory of associative rings, ideals are precisely the kernels of homomorphisms, given a Lie algebra and an ideal I in it, one constructs the factor algebra and the first isomorphism theorem holds for Lie algebras. Given two Lie algebras and their direct sum is the vector space consisting of the pairs with the operation In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

Categorical approach

A composition of two homomorphisms and is a homomorphism of the Lie algebras If a homomorphism is bijective, then it is invertible and is called an isomorphism, and these Lie algebras are called isomorphic. For many purposes, isomorphic Lie algebras are indistinguishable. The identity map on any Lie algebra is an isomorphism of the Lie algebra with itself. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

Examples

• Any vector space V endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called abelian, cf. below. Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket.

• The three-dimensional Euclidean space R³ with the Lie bracket given by the cross product of vectors becomes a three-dimensional Lie algebra.

• The Heisenberg algebra is a three-dimensional Lie algebra with generators x,y,z, whose commutation relations have the form

• Any Lie group G defines an associated real Lie algebra The definition in general is somewhat technical, but in the case of real matrix groups, it can be formulated via the exponential map, or the matrix exponent. The Lie algebra consists of those matrices X for which

for all real numbers t. The Lie bracket of is given by the commutator of matrices. As a concrete example, consider the special linear group SL(n,R), consisting of all n × n matrices with real entries and determinant 1. This is a matrix Lie group, and its Lie algebra consists of all n × n matrices with real entries and trace 0.

• The real vector space of all n × n skew-hermitian matrices is closed under the commutator and forms a real Lie algebra denoted u(n). This is the Lie algebra of the unitary group U(n).

• An important class of infinite-dimensional real Lie algebras arises in differential topology. The space of smooth vector fields on a differentiable manifold M forms a Lie algebra, where the Lie bracket is defined to be the commutator of vector fields. One way of expressing the Lie bracket is through the formalism of Lie derivatives, which identifies a vector field X with a first order partial differential operator L_X acting on smooth functions by letting L_X(f) be the directional derivative of the function f in the direction of X. The Lie bracket [X,Y] of two vector fields is the vector field defined through its action on functions by the formula:

This Lie algebra is related to the pseudogroup of diffeomorphisms of M.

• The commutation relations between the x, y, and z components of the angular momentum operator in quantum mechanics form a representation of a complex three-dimensional Lie algebra, which is the complexification of the Lie algebra so(3) of the three-dimensional rotation group:

Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... For the cross product in algebraic topology, see Künneth theorem. ... This article is about vectors that have a particular relation to the spatial coordinates. ... In mathematics, the term Heisenberg group, named after Werner Heisenberg, refers to the group of 3×3 upper triangular matrices of the form or its generalizations. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. ... There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A* is also its negative. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... 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 a Euclidean space. ... Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (such as quasigroup, for example). ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... This gyroscope remains upright while spinning due to its angular momentum. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... In mathematics, the complexification of a vector space V over the real number field is the corresponding vector space VC over the complex number field. ... In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ...

Structure theory and classification

Every finite-dimensional real or complex Lie algebra has a faithful representation by matrices (Ado's theorem). Lie's fundamental theorems describe a relation between Lie groups and Lie algebras. In particular, any Lie group gives rise to a canonically determined Lie algebra, and conversely, for any Lie algebra there is a corresponding connected Lie group (Lie's third theorem). This Lie group is not determined uniquely, however, any two connected Lie groups with the same Lie algebra are locally isomorphic, and in particular, have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) both give rise to the same Lie algebra, which is isomorphic to R³ with the cross-product, and SU(2) is a simply-connected twofold cover of SO(3). Real and complex Lie algebras can be classified to some extent, and this is often an important step toward the classification of Lie groups. In mathematics, Ados theorem states that every finite-dimensional Lie algebra L over a field K of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket. ... This article does not cite any references or sources. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ...

A Lie algebra is abelian if the Lie bracket vanishes, i.e. [x,y] = 0, for all x and y in . Abelian Lie algebras correspond to commutative (or abelian) connected Lie groups. A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra is nilpotent if the lower central series In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ...

becomes zero eventually. By Engel's theorem, a Lie algebra is nilpotent if and only if for every u in the adjoint endomorphism In mathematics, Engels theorem is one of the basic theorems in the theory of Lie algebras; it asserts that for a Lie algebra two concepts of nilpotency are identical. ... In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups. ...

is nilpotent. More generally still, a Lie algebra is said to be solvable if the derived series: In mathematics, a Lie algebra g is solvable if its derived series terminates in the zero subalgebra. ...

becomes zero eventually. Every Lie algebra has a unique maximal solvable ideal, called its radical. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras. The radical of a Lie algebra is a particular ideal of . ...

A Lie algebra is "simple" if it has no non-trivial ideals and is not abelian. A Lie algebra is called semisimple if its radical is zero. Equivalently, is semisimple if it does not contain any non-zero abelian ideals. In particular, a simple Lie algebra is semisimple. Conversely, it can be proven that any semisimple Lie algebra is the direct sum of its minimal ideals, which are canonically determined simple Lie algebras. In mathematics, a simple Lie group is a Lie group which is also a simple group. ... In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i. ...

In many ways, the classes of semisimple and solvable Lie algebras are at the opposite ends of the full spectrum of the Lie algebras. The Levi decomposition expresses an arbitrary Lie algebra as a semidirect product of its solvable radical and a semisimple Lie algebra, almost in a canonical way. Semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems. The classification of solvable Lie algebras is a 'wild' problem, and cannot be accomplished in general. In mathematics, Lie group decompositions are used to analyse the structure of Lie groups and associated objects, by showing how they are built up out of subgroups. ... In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, one of which is normal. ... In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ...

Cartan's criterion gives conditions for a Lie agebra to be nilpotent, solvable, or semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on defined by the formula Cartans criterion is an important mathematical theorem in the foundations of Lie algebra theory that gives conditions for a Lie agebra to be nilpotent, solvable, or semisimple. ... In mathematics, the Killing form, named for Wilhelm Killing (1847-1923), is a bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ...

where tr denotes the trace of a linear operator. A Lie algebra is nilpotent if and only if the Killing form is identically zero, and semisimple if and only if the Killing form is nondegenerate. A Lie algebra is solvable if and only if In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ...

The concept of semisimplicity for Lie algebras is closely related with the complete reducibility of their representations. When the ground field F has characteristic zero, semisimplicity of a Lie algebra over F is equivalent to the complete reducibility of all finite-dimensional representations of An early proof of this statement proceeded via connection with compact groups (Weyl's unitary trick), but later entirely algebraic proofs were found. In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ... In mathematics, a representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. ... In mathematics, the unitarian trick (occasionally unitarian trick) is a device in the representation theory of Lie groups, introduced by Hermann Weyl. ...

Relation to Lie groups

Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups. Given a Lie group, a Lie algebra can be associated to it either by endowing the tangent space to the identity with the differential of the adjoint map, or by considering the left-invariant vector fields as mentioned in the examples. This association is functorial, meaning that homomorphisms of Lie groups lift to homomorphisms of Lie algebras, and various properties are satisfied by this lifting: it commutes with composition, it maps Lie subgroups, kernels, quotients and cokernels of Lie groups to subalgebras, kernels, quotients and cokernels of Lie algebras, respectively. The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... For other uses, see identity (disambiguation). ... Suppose that φ : M → N is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. ... In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups. ... For functors in computer science, see the function object article. ...

The functor which takes each Lie group to its Lie algebra and each homomorphism to its differential is a full and faithful exact functor. This functor is not invertible; different Lie groups may have the same Lie algebra, for example SO(3) and SU(2) have isomorphic Lie algebras. Even worse, some Lie algebras need not have any associated Lie group. Nevertheless, when the Lie algebra is finite-dimensional, there is always at least one Lie group whose Lie algebra is the one under discussion, and a preferred Lie group can be chosen. Any finite-dimensional connected Lie group has a universal cover. This group can be constructed as the image of the Lie algebra under the exponential map. More generally, we have that the Lie algebra is homeomorphic to a neighborhood of the identity. But globally, if the Lie group is compact, the exponential will not be injective, and if the Lie group is not connected, simply connected or compact, the exponential map need not be surjective. In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint... There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ... This word should not be confused with homomorphism. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...

If the Lie algebra is infinite-dimensional, the issue is more subtle. In many instances, the exponential map is not even locally a homeomorphism (for example, in Diff(S¹), one may find diffeomorphisms arbitrarily close to the identity which are not in the image of exp). Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group. In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the related matter of the representation theory of Lie groups. Every representation of a Lie algebra lifts uniquely to a representation of the corresponding connected, simply connected Lie group, and conversely every representation of any Lie group induces a representation of the group's Lie algebra; the representations are in one to one correspondence. Therefore, knowing the representations of a Lie algebra settles the question of representations of the group. As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the center, once the classification of Lie algebras is known (solved by Cartan et al. in the semisimple case). In mathematics, the simple Lie groups were classified by Élie Cartan. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ...

Category theoretic definition

Using the language of category theory, a Lie algebra can be defined as an object A in the category of vector spaces together with a morphism [.,.]: A ⊗ A → A such that In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, especially category theory, the category K-Vect has all vector spaces over a fixed field K as objects and K-linear transformations as morphisms. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

where τ (a ⊗ b) := b ⊗ a and σ is the cyclic permutation braiding (id ⊗ τ_A,A) ° (τ_A,A ⊗ id). In diagrammatic form: A cyclic permutation is a permutation that shifts all elements of given ordered set by a fixed offset, with the elements shifted off the end inserted back at the beginning in the same order, i. ... Diagrammatic notation or Penrose graphical notation is a (usually handwritten) visual representation of tensors and tensor equations in physics, proposed by Roger Penrose. ...

Image File history File links latex compiled File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

See also

In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups. ... An anyonic Lie algebra is a U(1) graded vector space L over C equipped with a bilinear operator [.,.] and linear maps ε:L->C and Δ:L->L⊗L satisfying ε([X,Y])=ε(X)ε(Y) for pure graded elements X, Y and Z. See also Lie... Lie algebra cohomology is a cohomology theory for Lie algebras. ... In mathematics, a Lie bialgebra or Poisson-Hopf algebra, is the Lie algebra of a Poisson-Lie group. ... In mathematics a Lie coalgebra is a way of endowing a vector space with an algebraic structure in such a way that its dual vector space naturally carries the structure of a Lie algebra. ... In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. ... In mathematics, the Killing form, named for Wilhelm Killing (1847-1923), is a bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. ... There is a natural connection, first discovered by Eugene Wigner, between the properties of particles, the representation theory of Lie groups and Lie algebras, and the symmetries of the universe. ... A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz law. ... A quasi-Lie algebra is just like a Lie algebra, but with the axiom replaced by (anti-symmetry). ... In mathematics, a representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. ...

Notes

1. ^ Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.

References

• Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0
• Brian C. Hall Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, 2003. ISBN 0-387-40122-9
• Humphreys, James E. Introduction to Lie Algebras and Representation Theory, Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1978. ISBN 0-387-90053-5
• Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4
• Kac, Victor G. et al. Course notes for MIT 18.745: Introduction to Lie Algebras, http://www-math.mit.edu/~lesha/745lec/
• Varadarajan, V. S. Lie Groups, Lie Algebras, and Their Representations, 1st edition, Springer, 2004. ISBN 0-387-90969-9
• O'Connor, J. J. & Robertson, E.F. Biography of Sophus Lie, MacTutor History of Mathematics Archive, http://www-history.mcs.st-and.ac.uk/Biographies/Lie.html
• O'Connor, J. J. & Robertson, E.F. Biography of Wilhelm Killing, MacTutor History of Mathematics Archive, http://www-history.mcs.st-and.ac.uk/Biographies/Killing.html