Representation (Lie algebra)

A representation of a Lie algebra is a mathematical concept for studying Lie algebras . Such a representation is a homomorphism of a given Lie algebra into the Lie algebra of the endomorphisms over a vector space . In this way, abstractly given Lie algebras are related to concrete linear Lie algebras.

Motivation and Definitions

Let it be a Lie algebra, that is , a vector space together with a bilinear multiplication , called a Lie product, such that for all and for all (Jacobi identity). ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle [\cdot ,\cdot ]:L\times L\rightarrow L}$ ${\displaystyle [x,x]=0}$ ${\displaystyle x\in L}$ ${\displaystyle [x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0}$ ${\displaystyle x,y,z\in L}$

The standard example of such a Lie algebra is the vector space of linear mappings on a vector space , where the Lie product is defined by the commutator . It is easy to check that there is actually a Lie algebra, the so-called general linear Lie algebra . Sub-Lie algebras of are called linear Lie algebras . It is now obvious to want to relate general Lie algebras to linear Lie algebras. This motivates the following definition. ${\displaystyle {\mathfrak {gl}}(V)}$ ${\displaystyle V\rightarrow V}$ ${\displaystyle V}$ ${\displaystyle [x,y]:=xy-yx}$ ${\displaystyle {\mathfrak {gl}}(V)}$

A representation of a Lie algebra on a vector space is a Lie algebra homomorphism , that is to say is a linear mapping that additionally ${\displaystyle L}$ ${\displaystyle V}$ ${\displaystyle \varphi :L\rightarrow {\mathfrak {gl}}(V)}$ ${\displaystyle \varphi }$

${\displaystyle \varphi ([x,y])=[\varphi (x),\varphi (y)]}$

fulfilled for all . One calls the representation space , its dimension is called the dimension of representation . ${\displaystyle x,y\in L}$ ${\displaystyle V}$

Two representations and are called equivalent if there is a vector space isomorphism such that ${\displaystyle \varphi :L\rightarrow {\mathfrak {gl}}(V_{1})}$ ${\displaystyle \psi :L\rightarrow {\mathfrak {gl}}(V_{2})}$ ${\displaystyle T:V_{1}\rightarrow V_{2}}$

${\displaystyle \varphi (x)=T^{-1}\circ \psi (x)\circ T}$for everyone . ${\displaystyle x\in L}$

Two equivalent representations therefore behave essentially the same, only the vectors on which the image endomorphisms of the representation operate are exchanged by means of a vector space isomorphism.

Modules

As in the representation theory of groups or algebras , a Lie algebra representation can be translated into a module structure . If a Lie algebra, then a module is a vector space together with a bilinear map such that ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle V}$ ${\displaystyle \cdot :L\times V\rightarrow V}$

${\displaystyle [x,y]\cdot v=x\cdot (y\cdot v)-y\cdot (x\cdot v)}$for everyone and . ${\displaystyle x,y\in L}$ ${\displaystyle v\in V}$

If a Lie algebra representation is now on , then on is defined by a module structure on . Conversely, if a module is used, a representation is obtained by defining through . Using this relationship, statements about representations can be translated into statements about modules and vice versa, that is, representations of and modules are equivalent terms. ${\displaystyle \varphi :L\rightarrow {\mathfrak {gl}}(V)}$ ${\displaystyle V}$ ${\displaystyle x\cdot v:=\varphi (x)v}$ ${\displaystyle L}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle L}$ ${\displaystyle \varphi :L\rightarrow {\mathfrak {gl}}(V)}$ ${\displaystyle \varphi (x)\in {\mathfrak {gl}}(V)}$ ${\displaystyle \varphi (x)v:=x\cdot v}$ ${\displaystyle L}$ ${\displaystyle L}$

Examples

Zero representation

A first very simple example of a representation of a Lie algebra is the homomorphism, which maps every element to the endomorphism 0. Such a representation is called a null representation and there is such a null representation on every vector space. There is only this representation on the zero vector space .

Linear Lie algebras

Let it be a linear Lie algebra. Then is the inclusion map ${\displaystyle L\subset {\mathfrak {gl}}(V)}$

${\displaystyle i:L\rightarrow {\mathfrak {gl}}(V),\quad i(x):=x}$

apparently a representation of on . ${\displaystyle L}$ ${\displaystyle V}$

Representations induced by Lie group representations

If a representation of a Lie group , then the differential of on the neutral element is known to induce a Lie algebra homomorphism between the associated Lie algebras, that is, we get a Lie algebra representation of on . This interplay of Lie group representations and Lie algebra representations is an important instrument in the investigation of Lie groups. ${\displaystyle f:G\rightarrow GL(V)}$ ${\displaystyle f}$ ${\displaystyle e}$ ${\displaystyle D_{e}f:\,{\mathfrak {g}}\rightarrow {\mathfrak {gl}}(V)}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle V}$

The adjoint representation

If a Lie algebra, then a linear mapping is called a derivation on , if ${\displaystyle L}$ ${\displaystyle \delta :L\rightarrow L}$ ${\displaystyle L}$

${\displaystyle \delta ([y,z])=[\delta (y),z]+[y,\delta (z)]}$for everyone . ${\displaystyle y,z\in L}$

The set of all derivatives on is a Unter-Lie algebra of . Using and the Jacobi identity one can easily calculate that ${\displaystyle \mathrm {Der} \,L}$ ${\displaystyle L}$ ${\displaystyle {\mathfrak {gl}}(L)}$ ${\displaystyle [x,y]=-[y,x]}$

${\displaystyle \mathrm {ad} \,x:L\rightarrow L,\quad y\mapsto [x,y]}$

is a derivation, and by the same means that

${\displaystyle \mathrm {ad} :L\rightarrow \mathrm {Der} \,L\subset {\mathfrak {gl}}(L),\quad x\mapsto \mathrm {ad} \,x}$

is a Lie algebra homomorphism. This means that a representation of on , which is called the adjoint representation , is called the adjoint of . The adjoint representation plays an important role in the investigation of Lie algebras, among other things because of their occurrence in the killing form . ${\displaystyle \mathrm {ad} :L\rightarrow {\mathfrak {gl}}(L)}$ ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle \mathrm {ad} \,x}$ ${\displaystyle x}$

Constructions of representations

Here methods are described, how one can construct new representations from given representations of Lie algebras. The constructions can easily be translated into corresponding constructions for modules.

Partial representation

If a representation of Lie's algebra is a sub-vector space is called invariant , more precisely -invariant, if each one maps the sub-vector space in itself, i.e. if ${\displaystyle \varphi :L\rightarrow {\mathfrak {gl}}(V)}$ ${\displaystyle L}$ ${\displaystyle W\subset V}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi (x)}$

${\displaystyle \varphi (x)w\in W}$for everyone . ${\displaystyle x\in L,w\in W}$

Then the picture is

${\displaystyle L\rightarrow {\mathfrak {gl}}(W),\quad x\mapsto \varphi (x)|_{W}}$

apparently a representation , with the restriction to denote. This representation is referred to in an obvious way , even if that is not entirely correct, because the image itself is not restricted to. ${\displaystyle W}$ ${\displaystyle |_{W}}$ ${\displaystyle W}$ ${\displaystyle \varphi |W}$ ${\displaystyle \varphi }$ ${\displaystyle W}$

The invariant subspaces apparently correspond exactly to the submodules of the associated module . It has always , and even as invariant subspaces or submodules, these hot trivial , since they only lead to a zero representation or given representation. New representations different from 0 are thus only obtained for non-trivial invariant subspaces. ${\displaystyle L}$ ${\displaystyle V}$ ${\displaystyle \{0\}}$ ${\displaystyle V}$

The invariant subspaces of the adjoint representation are exactly the ideals of Lie algebra.

Direct sum

Are and representations of Lie algebra on or , so defined ${\displaystyle \varphi :L\rightarrow {\mathfrak {gl}}(V_{1})}$ ${\displaystyle \psi :L\rightarrow {\mathfrak {gl}}(V_{2})}$ ${\displaystyle L}$ ${\displaystyle V_{1}}$ ${\displaystyle V_{2}}$

${\displaystyle L\rightarrow {\mathfrak {gl}}(V_{1}\oplus V_{2}),x\mapsto \varphi (x)\oplus \psi (x)}$

a Lie algebra representation on the direct sum . This representation is referred to in an obvious way and is called the direct sum of the representations, even if that is not entirely correct, because it is not defined on. ${\displaystyle V_{1}\oplus V_{2}}$ ${\displaystyle \varphi \oplus \psi }$ ${\displaystyle L\oplus L}$

Tensor products

If and representations of the Lie algebra are on or , one can explain a representation on the tensor product as follows. ${\displaystyle \varphi :L\rightarrow {\mathfrak {gl}}(V_{1})}$ ${\displaystyle \psi :L\rightarrow {\mathfrak {gl}}(V_{2})}$ ${\displaystyle L}$ ${\displaystyle V_{1}}$ ${\displaystyle V_{2}}$ ${\displaystyle V_{1}\otimes V_{2}}$

${\displaystyle \rho :L\rightarrow {\mathfrak {gl}}(V_{1}\otimes V_{2}),\quad \rho (x)(v_{1}\otimes v_{2}):=(\varphi (x)v_{1})\otimes v_{2}\,+\,v_{1}\otimes (\psi (x)v_{2})}$

This explains the effect of initially only on elementary tensors , but this can be linearly extended to by means of the universal property of the tensor product . The representation defined in this way is also not quite correctly called the tensor product of the representations and is denoted by. ${\displaystyle \rho (x)}$ ${\displaystyle v_{1}\otimes v_{2}}$ ${\displaystyle V_{1}\otimes V_{2}}$ ${\displaystyle \varphi \otimes \psi }$

Dual representation

If the Lie algebra is represented, the following definition gives a representation on the dual space marked with : ${\displaystyle \varphi :L\rightarrow {\mathfrak {gl}}(V)}$ ${\displaystyle L}$ ${\displaystyle \varphi ^{*}}$ ${\displaystyle V^{*}}$

${\displaystyle \varphi ^{*}:L\rightarrow {\mathfrak {gl}}(V^{*}),(\varphi ^{*}(x)(f))(v):=-f(\varphi (x)(v))}$for . ${\displaystyle x\in L,f\in V^{*},v\in V}$

To define it, you have to explain which linear functional should be, i.e. how it affects vectors . This is exactly what happens with the given formula. The minus sign is required for. One calls the dual or contra-related representation. This designation is also not entirely correct, because it is not about the too dual mapping . ${\displaystyle \varphi ^{*}(x)(f)}$ ${\displaystyle \varphi ^{*}(x)(f)}$ ${\displaystyle V}$ ${\displaystyle \varphi ^{*}([x,y])=[\varphi ^{*}(x),\varphi ^{*}(y)]}$ ${\displaystyle \varphi ^{*}}$ ${\displaystyle \varphi }$

Special representations

Faithful representations

A module is true when made for all to be closed. This is equivalent to the associated representation being injective . Therefore, injective representations are also called loyal . The existence of a true representation of on means that it is isomorphic to a sub-Lie algebra of and thus to a linear Lie algebra. ${\displaystyle L}$ ${\displaystyle V}$ ${\displaystyle x\cdot v=0}$ ${\displaystyle v\in V}$ ${\displaystyle x=0}$ ${\displaystyle L}$ ${\displaystyle V}$ ${\displaystyle L}$ ${\displaystyle {\mathfrak {gl}}(V)}$

Irreducible representations

When examining representations of a Lie algebra one tries to break them down into simpler representations. Therefore, one will be interested in such representations that have no invariant subspaces, because these can be viewed as the smallest building blocks of such a decomposition. An at least one-dimensional representation is called irreducible if it has no non-trivial, invariant subspaces. The zero vector space, which only allows the zero representation, is therefore explicitly excluded as a representation space of an irreducible representation. The classification of all irreducible representations of a Lie algebra except for equivalence is an important goal in representation theory. ${\displaystyle L\rightarrow {\mathfrak {gl}}(V)}$

Completely reducible representations

A representation is said to be completely reducible if it is equivalent to a direct sum of irreducible representations. According to Weyl's theorem, all finite-dimensional representations of a semi-simple Lie algebra are completely reducible. With knowledge of all irreducible representations of a semi-simple Lie algebra, one then knows all finite-dimensional representations apart from equivalence.

See also

• Representation theory of the Lorentz group
• Representation theory of sl (2, C)

literature

• James E. Humphreys: Introduction to Lie Algebras and Representation Theory , 2nd revised edition, Graduate Texts in Mathematics, 9th Springer-Verlag, New York (1978), ISBN 0-387-90053-5