Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Lie algebras were introduced to study the concept of infinitesimal transformations. In Mathematics, an infinitesimal transformation is a limiting form of small transformation. The term "Lie algebra" (after Sophus Lie, pronounced /ˈliː/ ("lee"), not /ˈlaɪ/ ("lie") ) was introduced by Hermann Weyl in the 1930s. Marius Sophus Lie (liː as "Lee" ( 17 December 1842 - 18 February 1899) was a Norwegian -born Mathematician. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. The 1930s were described as an abrupt shift to more radical and conservative lifestyles as countries were struggling to find a solution to the Great Depression. In older texts, the name "infinitesimal group" is used.

1 Definition and first properties
- 1.1 Homomorphisms, subalgebras, and ideals
- 1.2 Categorical approach
2 Examples
3 Structure theory and classification
4 Relation to Lie groups
5 Category theoretic definition
6 See also
7 Notes
8 References

Definition and first properties

A Lie algebra is a type of algebra over a field; it is a vector space $\mathfrak{g}$ over some field F together with a binary operation [·, ·]

$[\cdot,\cdot]: \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$

called the Lie bracket, which satisfies the following axioms:

Bilinearity:

$[a x + b y, z] = a [x, z] + b [y, z], \quad [z, a x + b y] = a[z, x] + b [z, y]$

for all scalars a, b in F and all elements x, y, z in $\mathfrak{g}.$

Anticommutativity, or skew-symmetry:

$[x,y]=-[y,x]\,$

for all elements x, y in $\mathfrak{g}.$ When F is a field of characteristic two, one has to impose the stronger condition:

$[x,x]=0\$

for all x in $\mathfrak{g}.$

The Jacobi identity:

$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 \quad$

for all x, y, z in $\mathfrak{g}.$

For any associative algebra A with multiplication *, one can construct a Lie algebra L(A). In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with 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 In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Mathematics, a bilinear map is a function of two arguments that is linear in each In mathematics anticommutativity refers to the property of an operation being anticommutative, i In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's 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 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:

$[a,b]=a*b-b*a.\$

The associativity of the multiplication * in A implies the Jacobi identity of the commutator in L(A). In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. In particular, the associative algebra of n × n matrices over a field F gives rise to the general linear Lie algebra $\mathfrak{gl}_n(F).$ The associative algebra A is called an enveloping algebra of the Lie algebra L(A). In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation 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, 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]]. In Mathematics, associativity is a property that a Binary operation can have Nonetheless, much of the terminology that was developed in the theory of associative rings or associative algebras is commonly applied to Lie algebras. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive A subspace $\mathfrak{h}$ of a Lie algebra $\mathfrak{g}$ that is closed under the Lie bracket is called a Lie subalgebra. If a subspace $I\subseteq\mathfrak{g}$ satisfies a stronger condition that

$[\mathfrak{g},I]\subseteq I,$

then I is called an ideal in the Lie algebra $\mathfrak{g}.$ ^[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:

$f: \mathfrak{g}\to\mathfrak{g'}, \quad f([x,y])=[f(x),f(y)],$

for all elements x and y in $\mathfrak{g}.$ As in the theory of associative rings, ideals are precisely the kernels of homomorphisms, given a Lie algebra $\mathfrak{g}$ and an ideal I in it, one constructs the factor algebra $\mathfrak{g}/I,$ and the first isomorphism theorem holds for Lie algebras. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural Given two Lie algebras $\mathfrak{g}$ and $\mathfrak{g'},$ their direct sum is the vector space $\mathfrak{g}\oplus\mathfrak{g'}$ consisting of the pairs $(x,x'),\, x\in\mathfrak{g}, x'\in\mathfrak{g'},$ with the operation

$[(x,x'),(y,y')]=([x,y],[x',y']), \quad x,y\in\mathfrak{g},\, x',y'\in\mathfrak{g'}.$

Categorical approach

A composition of two homomorphisms $f: \mathfrak{g}\to\mathfrak{g'}$ and $g: \mathfrak{g'}\to\mathfrak{g''}$ is a homomorphism of the Lie algebras $g\circ f: \mathfrak{g}\to\mathfrak{g''}.$ If a homomorphism $f: \mathfrak{g}\to\mathfrak{g'}$ is bijective, then it is invertible and is called an isomorphism, and these Lie algebras are called isomorphic. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective For many purposes, isomorphic Lie algebras are indistinguishable. The identity map on any Lie algebra is an isomorphism of the Lie algebra with itself. This article is about the Identity Map software design pattern

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. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

The Heisenberg algebra is a three-dimensional Lie algebra with generators x,y,z, whose commutation relations are

$[x,y]=z,\quad [x,z]=0, \quad [y,z]=0.\,$

It is explicitly exhibited as the space of 3x3 strictly upper-triangular matrices. In Mathematics, the term Heisenberg group, named after Werner Heisenberg, refers to the group of 3×3 upper triangular matrices of the

Any Lie group G defines an associated real Lie algebra $\mathfrak{g}=Lie(G).$ 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. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed In differential geometry the exponential map is a generalization of the ordinary Exponential function of mathematical analysis to all differentiable manifolds with an Affine The Lie algebra $\mathfrak{g}$ consists of those matrices X for which

$\exp(tX)\in G\,$

for all real numbers t. The Lie bracket of $\mathfrak{g}$ 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. In Mathematics, the special linear group of degree n over a field F is the set of n × n matrices with 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). In Linear algebra, a Square matrix (or more generally a linear transformation from a complex Vector space with a Sesquilinear norm to itself This is the Lie algebra of the unitary group U(n). In Mathematics, the unitary group of degree n, denoted U( n) is the group of n × n unitary matrices

An important class of infinite-dimensional real Lie algebras arises in differential topology. In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential 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. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. 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. In Mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one Vector field along the The Lie bracket [X,Y] of two vector fields is the vector field defined through its action on functions by the formula:

$L_{[X,Y]}f=L_X(L_Y f)-L_Y(L_X f).\,$

This Lie algebra is related to the pseudogroup of diffeomorphisms of M. 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 In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable

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:

$[L_x, L_y] = i \hbar L_z$

$[L_y, L_z] = i \hbar L_x$

$[L_z, L_x] = i \hbar L_y$

Structure theory and classification

Every finite-dimensional real or complex Lie algebra has a faithful representation by matrices (Ado's theorem). In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Mathematics, the complexification of a Real vector space V is a vector space V C over the Complex number This article is about rotations in three-dimensional Euclidean space In Mathematics, Ado's theorem states that every finite-dimensional Lie algebra L over a field K of Characteristic zero 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). In Mathematics, Lie's third theorem often means the result that states that any finite-dimensional Lie algebra g, over the real numbers is the Lie 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. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism 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). In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n Real and complex Lie algebras can be classified to some extent, and this is often an important step toward the classification of Lie groups.

A Lie algebra $\mathfrak{g}$ is abelian if the Lie bracket vanishes, i. e. [x,y] = 0, for all x and y in $\mathfrak{g}$ . Abelian Lie algebras correspond to commutative (or abelian) connected Lie groups. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra $\mathfrak{g}$ is nilpotent if the lower central series

$\mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],\mathfrak{g}] > [[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}] > ...$

becomes zero eventually. In Mathematics, especially in the fields of Group theory and Lie theory, a central series is a kind of Normal series of Subgroups or By Engel's theorem, a Lie algebra is nilpotent if and only if for every u in $\mathfrak{g}$ the adjoint endomorphism

$ad(u):\mathfrak{g} \to \mathfrak{g}, \quad \operatorname{ad}(u)v=[u,v]$

is nilpotent. In Representation theory, Engel's theorem is one of the basic theorems in the theory of Lie algebras it asserts that for a Lie algebra two concepts of nilpotency In Mathematics, the adjoint endomorphism or adjoint action is an Endomorphism of Lie algebras that plays a fundamental role in the development More generally still, a Lie algebra $\mathfrak{g}$ is said to be solvable if the derived series:

$\mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]] > [[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]],[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]] > ...$

becomes zero eventually. In Mathematics, a Lie algebra g is solvable if its derived series terminates in the zero subalgebra In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup Every Lie algebra has a unique maximal solvable ideal, called its radical. The radical of a Lie algebra \mathfrak{g} is a particular Ideal of \mathfrak{g} Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras.

A Lie algebra is "simple" if it has no non-trivial ideals and is not abelian. In Mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected Normal subgroups A Lie algebra $\mathfrak{g}$ is called semisimple if its radical is zero. In Mathematics, a Lie algebra is semisimple if it is a Direct sum of Simple Lie algebras i Equivalently, $\mathfrak{g}$ 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 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. In Lie theory and Representation theory, the Levi decomposition, discovered by Eugenio Elia Levi (1906 states that any finite dimensional real Lie In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can Semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems. This article discusses root systems in mathematics For root systems of Plants see Root. The classification of solvable Lie algebras is a 'wild' problem, and cannot be accomplished in general.

Cartan's criterion gives conditions for a Lie agebra to be nilpotent, solvable, or semisimple. Cartan's criterion is an important mathematical theorem in the foundations of Lie algebra theory that gives conditions for a Lie agebra to be nilpotent, solvable It is based on the notion of the Killing form, a symmetric bilinear form on $\mathfrak{g}$ defined by the formula

$K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),$

where tr denotes the trace of a linear operator. In Mathematics, the Killing form, named after Wilhelm Killing, is a Symmetric bilinear form that plays a basic role in the theories of Lie groups A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric 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 A Lie algebra $\mathfrak{g}$ is nilpotent if and only if the Killing form is identically zero, and semisimple if and only if the Killing form is nondegenerate. In Mathematics, specifically Linear algebra, a degenerate Bilinear form f(xy on a Vector space V is one such that A Lie algebra $\mathfrak{g}$ is solvable if and only if $K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0.$

The concept of semisimplicity for Lie algebras is closely related with the complete reducibility of their representations. In mathematics the term semisimple is used in a number of related ways within different subjects When the ground field F has characteristic zero, semisimplicity of a Lie algebra $\mathfrak{g}$ over F is equivalent to the complete reducibility of all finite-dimensional representations of $\mathfrak{g}.$ 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 characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Mathematics, the unitarian trick (occasionally unitary trick) is a device in the Representation theory of Lie groups introduced by Hermann

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. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since 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 Suppose that &phi: M → N is a smooth map between smooth manifolds then the differential of &phi at a point x is in some In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its 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. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

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. In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n 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. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism This group can be constructed as the image of the Lie algebra under the exponential map. In differential geometry the exponential map is a generalization of the ordinary Exponential function of mathematical analysis to all differentiable manifolds with an Affine More generally, we have that the Lie algebra is homeomorphic to a neighborhood of the identity. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. 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 Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every

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). Topological equivalence redirects here see also Topological equivalence (dynamical systems. Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.

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. In Mathematics, the Simple Lie groups were classified by Élie Cartan. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of 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. Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental in the semisimple case). 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 Vec, the category of vector spaces together with a morphism [. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, especially Category theory, the category K-Vect has all Vector spaces over a fixed field K as objects In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and ,. ]: A ⊗ A → A, where ⊗ refers to the monoidal product of Vec, such that

$[\cdot, \cdot] \circ (\mathrm{id} + \tau_{A,A}) = 0$
$[\cdot, \cdot] \circ ([\cdot, \cdot] \otimes \mathrm{id}) \circ (\mathrm{id} + \sigma + \sigma^2) = 0$

where τ (a ⊗ b) := b ⊗ a and σ is the cyclic permutation braiding (id ⊗ τ_A,A) ° (τ_A,A ⊗ id). In Mathematics, a monoidal category (or tensor category) is a category C equipped with a Bifunctor &otimes: C A cyclic Permutation is built from one or more sets of elements in Cyclic order. In diagrammatic form:

Notes

^ Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide. In Mathematics and Physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten visual depiction of Multilinear functions In Mathematics, the adjoint endomorphism or adjoint action is an Endomorphism of Lie algebras that plays a fundamental role in the development In Mathematics, an anyonic Lie algebra is a U (1 graded Vector space L over C equipped with a bilinear operator and Lie algebra cohomology is a Cohomology theory for Lie algebras. In Mathematics, a Lie bialgebra is the Lie-theoretic case of a Bialgebra: its a set with a Lie algebra and a Lie coalgebra structure which are In Mathematics a Lie coalgebra is the dual structure to a Lie algebra. In Mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z 2- grading. In Mathematics, the Killing form, named after Wilhelm Killing, is a Symmetric bilinear form that plays a basic role in the theories of Lie groups There is a natural connection first discovered by Eugene Wigner, between the properties of particles the Representation theory of Lie groups and In Mathematics, a Poisson algebra is an Associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is the bracket In Mathematics, a quasi-Lie algebra in Abstract algebra is just like a Lie algebra, but with the usual Axiom =0

References

Hall, Brian C. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, 2003. ISBN 0-387-40122-9
Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0
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/
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
Varadarajan, V. S. Lie Groups, Lie Algebras, and Their Representations, 1st edition, Springer, 2004. ISBN 0-387-90969-9

Dictionary

-noun

(mathematics) A vector space with a specific kind of binary operation on it.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents