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mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, a Lie algebra representation or representation of a Lie algebra is a way of writing a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
as a set of
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
(or
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
s of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
) in such a way that the Lie bracket is given by the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
. In the language of physics, one looks for a vector space V together with a collection of operators on V satisfying some fixed set of commutation relations, such as the relations satisfied by the
angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum p ...
s. The notion is closely related to that of a
representation of a Lie group In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the ve ...
. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of a Lie group are the integrated form of the representations of its Lie algebra. In the study of representations of a Lie algebra, a particular ring, called the
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the represent ...
, associated with the Lie algebra plays an important role. The universality of this ring says that the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
of representations of a Lie algebra is the same as the category of
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
s over its enveloping algebra.


Formal definition

Let \mathfrak g be a Lie algebra and let V be a vector space. We let \mathfrak(V) denote the space of endomorphisms of V, that is, the space of all linear maps of V to itself. We make \mathfrak(V) into a Lie algebra with bracket given by the commutator: rho,\sigma\rho \circ \sigma-\sigma \circ \rho for all ''ρ,σ'' in \mathfrak(V). Then a representation of \mathfrak g on V is a
Lie algebra homomorphism In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
:\rho\colon \mathfrak g \to \mathfrak(V). Explicitly, this means that \rho should be a linear map and it should satisfy :\rho( ,Y=\rho(X)\rho(Y)-\rho(Y)\rho(X) for all ''X, Y'' in \mathfrak g. The vector space ''V'', together with the representation ''ρ'', is called a \mathfrak g-module. (Many authors abuse terminology and refer to ''V'' itself as the representation). The representation \rho is said to be faithful if it is injective. One can equivalently define a \mathfrak g-module as a vector space ''V'' together with a bilinear map \mathfrak g \times V\to V such that : ,Ycdot v = X\cdot(Y\cdot v) - Y\cdot(X\cdot v) for all ''X,Y'' in \mathfrak g and ''v'' in ''V''. This is related to the previous definition by setting ''X'' ⋅ ''v'' = ''ρ''(''X'')(''v'').


Examples


Adjoint representations

The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra \mathfrak on itself: :\textrm:\mathfrak \to \mathfrak(\mathfrak), \quad X \mapsto \operatorname_X, \quad \operatorname_X(Y) = , Y Indeed, by virtue of the Jacobi identity, \operatorname is a Lie algebra homomorphism.


Infinitesimal Lie group representations

A Lie algebra representation also arises in nature. If \phi: ''G'' → ''H'' is a
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
of (real or complex)
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s, and \mathfrak g and \mathfrak h are the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s of ''G'' and ''H'' respectively, then the differential d_e \phi: \mathfrak g \to \mathfrak h on
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s at the identities is a Lie algebra homomorphism. In particular, for a finite-dimensional vector space ''V'', a
representation of Lie groups In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the ve ...
:\phi: G\to \operatorname(V)\, determines a Lie algebra homomorphism :d \phi: \mathfrak g \to \mathfrak(V) from \mathfrak g to the Lie algebra of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
GL(''V''), i.e. the endomorphism algebra of ''V''. For example, let c_g(x) = gxg^. Then the differential of c_g: G \to G at the identity is an element of \operatorname(\mathfrak). Denoting it by \operatorname(g) one obtains a representation \operatorname of ''G'' on the vector space \mathfrak. This is the
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...
of ''G''. Applying the preceding, one gets the Lie algebra representation d\operatorname. It can be shown that d_e\operatorname = \operatorname, the adjoint representation of \mathfrak g. A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.


In quantum physics

In quantum theory, one considers "observables" that are self-adjoint operators on a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. The commutation relations among these operators are then an important tool. The
angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum p ...
s, for example, satisfy the commutation relations : _x,L_yi\hbar L_z, \;\; _y,L_zi\hbar L_x, \;\; _z,L_xi\hbar L_y,. Thus, the span of these three operators forms a Lie algebra, which is isomorphic to the Lie algebra so(3) of the
rotation group SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is ...
. Then if V is any subspace of the quantum Hilbert space that is invariant under the angular momentum operators, V will constitute a representation of the Lie algebra so(3). An understanding of the representation theory of so(3) is of great help in, for example, analyzing Hamiltonians with rotational symmetry, such as the
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen cons ...
. Many other interesting Lie algebras (and their representations) arise in other parts of quantum physics. Indeed, the history of representation theory is characterized by rich interactions between mathematics and physics.


Basic concepts


Invariant subspaces and irreducibility

Given a representation \rho:\mathfrak\rightarrow\operatorname(V) of a Lie algebra \mathfrak, we say that a subspace W of V is invariant if \rho(X)w\in W for all w\in W and X\in\mathfrak. A nonzero representation is said to be irreducible if the only invariant subspaces are V itself and the zero space \. The term ''simple module'' is also used for an irreducible representation.


Homomorphisms

Let \mathfrak be a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
. Let ''V'', ''W'' be \mathfrak-modules. Then a linear map f: V \to W is a homomorphism of \mathfrak-modules if it is \mathfrak-equivariant; i.e., f(X\cdot v) = X\cdot f(v) for any X \in \mathfrak,\, v \in V. If ''f'' is bijective, V, W are said to be equivalent. Such maps are also referred to as intertwining maps or morphisms. Similarly, many other constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc.


Schur's lemma

A simple but useful tool in studying irreducible representations is Schur's lemma. It has two parts: *If ''V'', ''W'' are irreducible \mathfrak-modules and f: V \to W is a homomorphism, then f is either zero or an isomorphism. *If ''V'' is an irreducible \mathfrak-module over an algebraically closed field and f: V \to V is a homomorphism, then f is a scalar multiple of the identity.


Complete reducibility

Let ''V'' be a representation of a Lie algebra \mathfrak. Then ''V'' is said to be completely reducible (or semisimple) if it is isomorphic to a direct sum of irreducible representations (cf.
semisimple module In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself ...
). If ''V'' is finite-dimensional, then ''V'' is completely reducible if and only if every invariant subspace of ''V'' has an invariant complement. (That is, if ''W'' is an invariant subspace, then there is another invariant subspace ''P'' such that ''V'' is the direct sum of ''W'' and ''P''.) If \mathfrak is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and ''V'' is finite-dimensional, then ''V'' is semisimple; this is Weyl's complete reducibility theorem. Thus, for semisimple Lie algebras, a classification of irreducible (i.e. simple) representations leads immediately to classification of all representations. For other Lie algebra, which do not have this special property, classifying the irreducible representations may not help much in classifying general representations. A Lie algebra is said to be reductive if the adjoint representation is semisimple. Certainly, every (finite-dimensional) semisimple Lie algebra \mathfrak g is reductive, since ''every'' representation of \mathfrak g is completely reducible, as we have just noted. In the other direction, the definition of a reductive Lie algebra means that it decomposes as a direct sum of ideals (i.e., invariant subspaces for the adjoint representation) that have no nontrivial sub-ideals. Some of these ideals will be one-dimensional and the rest are simple Lie algebras. Thus, a reductive Lie algebra is a direct sum of a commutative algebra and a semisimple algebra.


Invariants

An element ''v'' of ''V'' is said to be \mathfrak-invariant if x\cdot v = 0 for all x \in \mathfrak. The set of all invariant elements is denoted by V^\mathfrak.


Basic constructions


Tensor products of representations

If we have two representations of a Lie algebra \mathfrak, with ''V''1 and ''V''2 as their underlying vector spaces, then the tensor product of the representations would have ''V''1 ⊗ ''V''2 as the underlying vector space, with the action of \mathfrak uniquely determined by the assumption that :X\cdot(v_1\otimes v_2)=(X\cdot v_1)\otimes v_2+v_1\otimes (X\cdot v_2) . for all v_1\in V_1 and v_2\in V_2. In the language of homomorphisms, this means that we define \rho_1\otimes\rho_2:\mathfrak\rightarrow\mathfrak(V_1\otimes V_2) by the formula :(\rho_1\otimes\rho_2)(X)=\rho_1(X)\otimes \mathrm+\mathrm\otimes\rho_2(X). In the physics literature, the tensor product with the identity operator is often suppressed in the notation, with the formula written as :(\rho_1\otimes\rho_2)(X)=\rho_1(X)+\rho_2(X), where it is understood that \rho_1(x) acts on the first factor in the tensor product and \rho_2(x) acts on the second factor in the tensor product. In the context of representations of the Lie algebra su(2), the tensor product of representations goes under the name "addition of angular momentum." In this context, \rho_1(X) might, for example, be the orbital angular momentum while \rho_2(X) is the spin angular momentum.


Dual representations

Let \mathfrak be a Lie algebra and \rho:\mathfrak\rightarrow\mathfrak(V) be a representation of \mathfrak. Let V^* be the dual space, that is, the space of linear functionals on V. Then we can define a representation \rho^*:\mathfrak\rightarrow\mathfrak(V^*) by the formula :\rho^*(X)=-(\rho(X))^\operatorname, where for any operator A:V\rightarrow V, the transpose operator A^\operatorname:V^*\rightarrow V^* is defined as the "composition with A" operator: :(A^\operatorname\phi)(v)=\phi(Av) The minus sign in the definition of \rho^* is needed to ensure that \rho^* is actually a representation of \mathfrak, in light of the identity (AB)^\operatorname=B^\operatornameA^\operatorname. If we work in a basis, then the transpose in the above definition can be interpreted as the ordinary matrix transpose.


Representation on linear maps

Let V, W be \mathfrak-modules, \mathfrak a Lie algebra. Then \operatorname(V, W) becomes a \mathfrak-module by setting (X \cdot f)(v) = X f(v) - f (X v). In particular, \operatorname_\mathfrak(V, W) = \operatorname(V, W)^\mathfrak; that is to say, the \mathfrak-module homomorphisms from V to W are simply the elements of \operatorname(V, W) that are invariant under the just-defined action of \mathfrak on \operatorname(V, W). If we take W to be the base field, we recover the action of \mathfrak on V^* given in the previous subsection.


Representation theory of semisimple Lie algebras

See Representation theory of semisimple Lie algebras.


Enveloping algebras

To each Lie algebra \mathfrak over a field ''k'', one can associate a certain ring called the universal enveloping algebra of \mathfrak and denoted U(\mathfrak). The universal property of the universal enveloping algebra guarantees that every representation of \mathfrak gives rise to a representation of U(\mathfrak). Conversely, the PBW theorem tells us that \mathfrak sits inside U(\mathfrak), so that every representation of U(\mathfrak) can be restricted to \mathfrak. Thus, there is a one-to-one correspondence between representations of \mathfrak and those of U(\mathfrak). The universal enveloping algebra plays an important role in the representation theory of semisimple Lie algebras, described above. Specifically, the finite-dimensional irreducible representations are constructed as quotients of Verma modules, and Verma modules are constructed as quotients of the universal enveloping algebra. The construction of U(\mathfrak) is as follows. Let ''T'' be the tensor algebra of the vector space \mathfrak. Thus, by definition, T = \oplus_^\infty \otimes_1^n \mathfrak and the multiplication on it is given by \otimes. Let U(\mathfrak) be the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
of ''T'' by the ideal generated by elements of the form : , Y- (X \otimes Y - Y \otimes X). There is a natural linear map from \mathfrak into U(\mathfrak) obtained by restricting the quotient map of T \to U(\mathfrak) to degree one piece. The PBW theorem implies that the canonical map is actually injective. Thus, every Lie algebra \mathfrak can be embedded into an associative algebra A=U(\mathfrak)in such a way that the bracket on \mathfrak is given by ,YXY-YX in A. If \mathfrak is
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
, then U(\mathfrak) is the symmetric algebra of the vector space \mathfrak. Since \mathfrak is a module over itself via adjoint representation, the enveloping algebra U(\mathfrak) becomes a \mathfrak-module by extending the adjoint representation. But one can also use the left and right regular representation to make the enveloping algebra a \mathfrak-module; namely, with the notation l_X(Y) = XY, X \in \mathfrak, Y \in U(\mathfrak), the mapping X \mapsto l_X defines a representation of \mathfrak on U(\mathfrak). The right regular representation is defined similarly.


Induced representation

Let \mathfrak be a finite-dimensional Lie algebra over a field of characteristic zero and \mathfrak \subset \mathfrak a subalgebra. U(\mathfrak) acts on U(\mathfrak) from the right and thus, for any \mathfrak-module ''W'', one can form the left U(\mathfrak)-module U(\mathfrak) \otimes_ W. It is a \mathfrak-module denoted by \operatorname_\mathfrak^\mathfrak W and called the \mathfrak-module induced by ''W''. It satisfies (and is in fact characterized by) the universal property: for any \mathfrak-module ''E'' :\operatorname_\mathfrak(\operatorname_\mathfrak^\mathfrak W, E) \simeq \operatorname_\mathfrak(W, \operatorname^\mathfrak_\mathfrak E). Furthermore, \operatorname_\mathfrak^\mathfrak is an exact functor from the category of \mathfrak-modules to the category of \mathfrak-modules. These uses the fact that U(\mathfrak) is a free right module over U(\mathfrak). In particular, if \operatorname_\mathfrak^\mathfrak W is simple (resp. absolutely simple), then ''W'' is simple (resp. absolutely simple). Here, a \mathfrak-module ''V'' is absolutely simple if V \otimes_k F is simple for any field extension F/k. The induction is transitive: \operatorname_\mathfrak^\mathfrak \simeq \operatorname_\mathfrak^\mathfrak \circ \operatorname_\mathfrak^\mathfrak for any Lie subalgebra \mathfrak \subset \mathfrak and any Lie subalgebra \mathfrak \subset \mathfrak'. The induction commutes with restriction: let \mathfrak \subset \mathfrak be subalgebra and \mathfrak an ideal of \mathfrak that is contained in \mathfrak. Set \mathfrak_1 = \mathfrak/\mathfrak and \mathfrak_1 = \mathfrak/\mathfrak. Then \operatorname^\mathfrak_\mathfrak \circ \operatorname_\mathfrak \simeq \operatorname_\mathfrak \circ \operatorname^\mathfrak_\mathfrak.


Infinite-dimensional representations and "category O"

Let \mathfrak be a finite-dimensional semisimple Lie algebra over a field of characteristic zero. (in the solvable or nilpotent case, one studies primitive ideals of the enveloping algebra; cf. Dixmier for the definitive account.) The category of (possibly infinite-dimensional) modules over \mathfrak turns out to be too large especially for homological algebra methods to be useful: it was realized that a smaller subcategory category O is a better place for the representation theory in the semisimple case in zero characteristic. For instance, the category O turned out to be of a right size to formulate the celebrated BGG reciprocity.Why the BGG category O?
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(g,K)-module

One of the most important applications of Lie algebra representations is to the representation theory of real reductive Lie groups. The application is based on the idea that if \pi is a Hilbert-space representation of, say, a connected real semisimple linear Lie group ''G'', then it has two natural actions: the complexification \mathfrak and the connected maximal compact subgroup ''K''. The \mathfrak-module structure of \pi allows algebraic especially homological methods to be applied and K-module structure allows harmonic analysis to be carried out in a way similar to that on connected compact semisimple Lie groups.


Representation on an algebra

If we have a Lie superalgebra ''L'', then a representation of ''L'' on an algebra is a (not necessarily
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
) Z2 graded
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
''A'' which is a representation of ''L'' as a Z2
graded vector space In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces. Integer gradation Let \mathbb be th ...
and in addition, the elements of ''L'' acts as
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
s/
antiderivation In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring or a field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies Le ...
s on ''A''. More specifically, if ''H'' is a pure element of ''L'' and ''x'' and ''y'' are pure elements of ''A'', :''H'' 'xy''= (''H'' 'x''''y'' + (−1)''xH''''x''(''H'' 'y'' Also, if ''A'' is unital, then :''H'' = 0 Now, for the case of a representation of a Lie algebra, we simply drop all the gradings and the (−1) to the some power factors. A Lie (super)algebra is an algebra and it has an
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...
of itself. This is a representation on an algebra: the (anti)derivation property is the
super Super may refer to: Computing * SUPER (computer program), or Simplified Universal Player Encoder & Renderer, a video converter / player * Super (computer science), a keyword in object-oriented programming languages * Super key (keyboard butt ...
Jacobi identity. If a vector space is both an
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
and a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
and the adjoint representation of the Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central i ...
. The analogous observation for Lie superalgebras gives the notion of a
Poisson superalgebra In mathematics, a Poisson superalgebra is a Z2- graded generalization of a Poisson algebra. Specifically, a Poisson superalgebra is an (associative) superalgebra ''A'' with a Lie superbracket : cdot,\cdot: A\otimes A\to A such that (''A'', �,· i ...
.


See also

*
Representation of a Lie group In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the ve ...
* Weight (representation theory) * Weyl's theorem on complete reducibility *
Root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
*
Weyl character formula In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...
* Representation theory of a connected compact Lie group * Whitehead's lemma (Lie algebras) * Kazhdan–Lusztig conjectures *
Quillen's lemma In algebra, Quillen's lemma states that an endomorphism of a simple module over the enveloping algebra of a finite-dimensional Lie algebra over a field ''k'' is algebraic over ''k''. In contrast to a version of Schur's lemma In mathematics, ...
- analog of Schur's lemma


Notes


References

*Bernstein I.N., Gelfand I.M., Gelfand S.I., "Structure of Representations that are generated by vectors of highest weight," Functional. Anal. Appl. 5 (1971) *. *A. Beilinson and J. Bernstein, "Localisation de g-modules," Comptes Rendus de l'Académie des Sciences, Série I, vol. 292, iss. 1, pp. 15–18, 1981. * * * * D. Gaitsgory
Geometric Representation theory, Math 267y, Fall 2005
* * * * Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki, ''D-modules, perverse sheaves, and representation theory''; translated by Kiyoshi Takeuch * *N. Jacobson, ''Lie algebras'', Courier Dover Publications, 1979. * * * (elementary treatment for SL(2,C)) *


Further reading

* {{DEFAULTSORT:Lie Algebra Representation