Representation (algebra)

The representation theory of algebras is a branch of mathematics that deals with the representation of algebras on vector spaces. In this way, any associative algebras are related to the algebras of operators by means of homomorphisms . The object of investigation is the structure of such homomorphisms and their classification. The representation theory of an algebra is equivalent to the theory of its modules . More specific representation theories deal with groups , Lie algebras or C * algebras .

In the following, for the sake of simplicity, we consider algebras with one element 1. If one has an algebra without one element, one adjoint one.

Definitions

Let there be a body and an algebra. A representation of is an algebra homomorphism , where a vector space and the algebra of all linear operators is on , more precisely one speaks of a representation of on . ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle \pi \colon A\rightarrow L(V)}$ ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle L(V)}$ ${\displaystyle V}$ ${\displaystyle A}$ ${\displaystyle V}$

The vector space dimension of is as a dimension of designated. Finite-dimensional representations are also called matrix representations, because every element can be written out as a matrix by choosing a vector space basis. Injective representations are called faithful . ${\displaystyle V}$ ${\displaystyle \pi }$ ${\displaystyle L(V)}$

Two representations and are called equivalent if there is a vector space isomorphism with for all . One also writes abbreviated for this . ${\displaystyle \pi _{1}\colon A\rightarrow L(V_{1})}$ ${\displaystyle \pi _{2}\colon A\rightarrow L(V_{2})}$ ${\displaystyle T\colon V_{1}\rightarrow V_{2}}$ ${\displaystyle \pi _{1}(a)=T^{-1}\pi _{2}(a)T}$ ${\displaystyle a\in A}$ ${\displaystyle \pi _{1}\sim \pi _{2}}$

The equivalence so defined is an equivalence relation on the class of all representations. The conceptualizations in representation theory are designed in such a way that they are retained when switching to an equivalent representation, dimension and fidelity are first examples.

Examples

• The zero homomorphism, which maps every algebra element to the zero operator , is called the zero representation or trivial representation. ${\displaystyle A\rightarrow L(V)}$
• The identical figure is a faithful representation of on . ${\displaystyle L(V)\rightarrow L(V)}$ ${\displaystyle L(V)}$ ${\displaystyle V}$
• Let it be the -algebra of real-valued (continuous) functions . Then ${\displaystyle C[0,1]}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle a\colon [0,1]\rightarrow \mathbb {R} }$

${\displaystyle \pi \colon C[0,1]\rightarrow L(\mathbb {R} ^{2}),\quad a\mapsto {\begin{pmatrix}a(0)&0\\0&a(1)\end{pmatrix}}}$

a two-dimensional, non-faithful representation of C [0,1].

• If an -algebra, then , where be defined by , is a representation of . This special representation is also called the left-regular representation , since it maps to the set of all left multiplications with elements from . The formula shows the fidelity of the left-regular representation, in particular every algebra has a true representation. ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle \pi _{l}\colon A\rightarrow L(A)}$ ${\displaystyle \pi _{l}(a)\in L(A)}$ ${\displaystyle \pi _{l}(a)b:=ab}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle \pi _{l}(a)1=a}$

The multiplicativity of the left-regular representation means for everyone and that means for everyone and that is nothing other than for everyone . This consideration makes the role of the associative law clear. ${\displaystyle \pi _{l}(ab)=\pi _{l}(a)\pi _{l}(b)}$ ${\displaystyle a,b\in A}$ ${\displaystyle \pi _{l}(ab)c=\pi _{l}(a)(\pi _{l}(b)c)}$ ${\displaystyle a,b,c\in A}$ ${\displaystyle (ab)c=a(bc)}$ ${\displaystyle a,b,c\in A}$

Direct sums

Are and two representations, so defined ${\displaystyle \pi _{1}\colon A\rightarrow L(V_{1})}$ ${\displaystyle \pi _{2}\colon A\rightarrow L(V_{2})}$

${\displaystyle \pi \colon A\rightarrow L(V_{1}\oplus V_{2}),\quad a\mapsto \pi _{1}(a)\oplus \pi _{2}(a)}$

apparently again a representation of , with component-wise operating on the direct sum , that is, for all . This representation is called the direct sum of and and denotes it with . ${\displaystyle A}$ ${\displaystyle \pi _{1}(a)\oplus \pi _{2}(a)}$ ${\displaystyle V_{1}\oplus V_{2}}$ ${\displaystyle (\pi _{1}(a)\oplus \pi _{2}(a))(\xi _{1}\oplus \xi _{2}):=\pi _{1}(a)\xi _{1}\oplus \pi _{2}(a)\xi _{2}}$ ${\displaystyle \xi _{i}\in V_{i}}$ ${\displaystyle \pi _{1}}$ ${\displaystyle \pi _{2}}$ ${\displaystyle \pi _{1}\oplus \pi _{2}}$

This construction can obviously be generalized for direct sums of any number of summands. Is a family of representations, so too ${\displaystyle (\pi _{i})_{i\in I}}$

${\displaystyle \bigoplus _{i\in I}\pi \colon A\rightarrow L(\bigoplus _{i\in I}V_{i}),\,a\mapsto \bigoplus _{i\in I}\pi _{i}(a)}$.

Partial representations

Be a representation. A subspace is called invariant (more precisely -invariant), if for all . ${\displaystyle \pi \colon A\rightarrow L(V)}$ ${\displaystyle W\subset V}$ ${\displaystyle \pi }$ ${\displaystyle \pi (a)W\subset W}$ ${\displaystyle a\in A}$

Apparently it is

${\displaystyle {\tilde {\pi }}\colon A\rightarrow L(W),\quad a\mapsto {\tilde {\pi }}(a):=\pi (a)|_{W}}$

again a representation of , which is called the restriction of to and is denoted by. ${\displaystyle A}$ ${\displaystyle \pi }$ ${\displaystyle W}$ ${\displaystyle \pi |_{W}}$

If a subspace is too complementary and is also invariant, then it obviously holds that the equivalence is mediated by the isomorphism . ${\displaystyle W^{c}}$ ${\displaystyle W}$ ${\displaystyle \pi \sim \pi |_{W}\oplus \pi |_{W^{c}}}$ ${\displaystyle W\oplus W^{c}\rightarrow V,(w_{1},w_{2})\mapsto w_{1}+w_{2}}$

The invariant subspaces of the left regular representation of an algebra are exactly the left ideals of the algebra.

Further representations

An important subject of investigation in representation theory is the decomposition of representations as the sum of partial representations. Of course, you are interested in representations that cannot be broken down further. That casually leads to the following term:

Irreducible representations

A representation is called irreducible if there are no other invariant subspaces of besides and . For an equivalent characterization see Schur's Lemma . A representation is said to be completely reducible if it is equivalent to a direct sum of irreducible representations. ${\displaystyle \pi \colon A\rightarrow L(V)}$ ${\displaystyle \{0\}}$ ${\displaystyle V}$ ${\displaystyle V}$

The above example of a two-dimensional representation of is obviously equivalent to the direct sum of two one-dimensional and therefore irreducible representations. The identical representation of the matrix algebra on is a -dimensional irreducible representation, of which one can show that it is the only one except for equivalence. A common goal of representation theory is to classify all equivalence classes of irreducible representations of a given algebra. ${\displaystyle C[0,1]}$ ${\displaystyle L(K^{n})\rightarrow L(K^{n})}$ ${\displaystyle K^{n}}$ ${\displaystyle n}$

Non-degenerate representations

A representation of an algebra on vector space is called non-degenerate if from always follows for all . ${\displaystyle \pi }$ ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle \pi (a)\xi =0}$ ${\displaystyle a\in A}$ ${\displaystyle \xi =0}$

If there is any representation, then are ${\displaystyle \pi \colon A\rightarrow L(V)}$

${\displaystyle V_{0}:=\{\xi \in V;\,\pi (a)\xi =0\,\forall a\in A\}=\mathrm {ker} (\pi (1))}$ and

${\displaystyle V_{1}:=\{\sum _{i=1}^{n}\pi (a_{i})\xi _{i};\,n\in \mathbb {N} ,\,a_{1},\ldots a_{n}\in A,\,\xi _{1},\ldots \xi _{n}\in V\}}$

apparently invariant subspaces, also called the null space of the representation. It is the projection onto and the associated complementary space . Since the null representation and is non-degenerate, we have the result that each representation is the sum of a non-degenerate and a null representation. Therefore, one often only considers non-degenerate representations and accepts them without restriction . ${\displaystyle V_{0}}$ ${\displaystyle \pi (1)\in L(V)}$ ${\displaystyle V_{1}}$ ${\displaystyle V_{0}=(\mathrm {id} _{V}-\pi (1))V}$ ${\displaystyle \pi |_{V_{0}}}$ ${\displaystyle \pi |_{V_{1}}}$ ${\displaystyle \pi (1)=\mathrm {id} _{V}}$

Cyclical representations

A representation is called cyclic , if there is one with , the vector is called a cyclic vector . Is an arbitrary representation and , then is obviously an invariant subspace and is a cyclic representation with as a cyclic vector. Often one demands that it is not in the null space in order to avoid trivial matters. ${\displaystyle \pi \colon A\rightarrow L(V)}$ ${\displaystyle \xi \in V}$ ${\displaystyle V=\{\pi (a)\xi ;\,a\in A\}}$ ${\displaystyle \xi }$ ${\displaystyle \pi \colon A\rightarrow L(V)}$ ${\displaystyle \xi \in V}$ ${\displaystyle V_{\xi }:=\{\pi (a)\xi ;\,a\in A\}}$ ${\displaystyle \pi |_{V_{\xi }}}$ ${\displaystyle \pi (1)\xi }$ ${\displaystyle \xi }$

Connection with modules

→ Main article : Module (mathematics)

If it is a non-degenerate representation, it becomes a module through the definition . The non-degeneracy is needed for all , the other modulus axioms are easily attributed to the homomorphism properties of . ${\displaystyle \pi \colon A\rightarrow L(V)}$ ${\displaystyle V}$ ${\displaystyle a\cdot \xi :=\pi (a)\xi }$ ${\displaystyle A}$ ${\displaystyle 1\cdot \xi =\xi }$ ${\displaystyle \xi \in V}$ ${\displaystyle \pi }$

Conversely, if a module is, then with the scalar multiplication explained by there is a vector space. If you define an endomorphism by the formula , you obviously get a representation . ${\displaystyle V}$ ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle k\cdot \xi :=(k\cdot 1)\cdot \xi ,k\in K}$ ${\displaystyle K}$ ${\displaystyle a\in A}$ ${\displaystyle \pi (a)\in L(V)}$ ${\displaystyle \pi (a)\xi :=a\cdot \xi }$ ${\displaystyle \pi \colon A\rightarrow L(V)}$

In this construction, two representations are equivalent if and only if the associated modules are isomorphic. The representation theory of -algebra is therefore equivalent to the theory of -modules. The partial representations correspond to the sub-modules, an irreducible representation corresponds to a simple module, a completely reducible representation corresponds to a semi- simple module . Cyclic representations correspond to modules generated by an element. The module belonging to the left-regular representation is nothing other than itself. ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A=A^{1}}$

If you only have one ring without the operation of a body, you can only talk about modules. The theory of modules over a ring is in this sense a generalization of the representation theory of algebras on rings. ${\displaystyle \mathbb {Z} }$

Group representations

→ Main article : Representation theory (group theory)

If a group is, then the group algebra is an -algebra that contains in the group of invertible elements with an isomorphic subgroup, which is identified with . Any non-degenerate representation of group algebra therefore delivers by restricting it to a representation of the group . Conversely, if a group representation is given, then a representation of group algebra is given. In this sense, the representation theory of the groups is subordinate to the representation theory of algebras discussed here. ${\displaystyle G}$ ${\displaystyle KG}$ ${\displaystyle K}$ ${\displaystyle \{1\cdot g;\,g\in G\}}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \lambda \colon G\rightarrow L(V)}$ ${\displaystyle \pi (\sum _{i=1}^{n}k_{i}g_{i}):=\sum _{i=1}^{n}k_{i}\lambda (g_{i})}$

Representations of Lie algebras

Lie algebras are not associative, but one is interested in homomorphisms on subalgebras of , where the Lie bracket is mapped to the commutator , i.e. where applies to all . An associated universal construction leads to the universal enveloping algebra , with which the representations of Lie algebras are related to the representations of associative algebras dealt with here. ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \pi }$ ${\displaystyle L(V)}$ ${\displaystyle \pi ([x_{1},x_{2}])\,=\,\pi (x_{1})\pi (x_{2})-\pi (x_{2})\pi (x_{1})}$ ${\displaystyle x_{1},x_{2}\in {\mathfrak {g}}}$

Hilbert space representations

→ Main article : Hilbert space representation

To investigate Banach - * algebras , especially C * algebras and group algebras of locally compact groups , one looks for representations that also reflect the topological relationships and the involution . This easily leads to the investigation of representations on Hilbert spaces , which in turn leads to classes of such algebras, for example the important concept of type IC * algebra , which can be defined by the representation theory of C * algebra. The fact that C * algebras have faithful Hilbert space representations is known as the Gelfand-Neumark Theorem . ${\displaystyle L^{1}(G)}$

literature

• Bartel Leendert van der Waerden : Algebra II . Springer Verlag, Berlin 1967, ISBN 3-540-03869-8 (using lectures by Emil Artin and Emmy Noether ).
• Jacques Dixmier : Les C * -algèbres et leurs représentations (Cahiers scientifiques; Vol. 29). Gauthier-Villars, Paris 1969.