Representation theory

This article covers basic theory - for other meanings, see Representation Theory (disambiguation) .

In representation theory , elements of groups or, more generally, of algebras are mapped onto linear mappings of vector spaces ( matrices ) using homomorphisms .

Representation theory has applications in almost all areas of mathematics and theoretical physics . A representation theory theorem by Robert Langlands was an essential step for Andrew Wiles ' proof of Fermat's Great Theorem , and representation theory also provided the theoretical background for the prediction that quarks exist. The representation by means of matrices is also often useful for the purely algebraic investigation of groups or algebras.

Types of representations

Classically, representation theory dealt with homomorphisms for groups and vector spaces (where the general linear group denotes above ), see ${\displaystyle \rho \colon G\rightarrow GL(V)}$ ${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle GL(V)}$ ${\displaystyle V}$

• Representation (group) .

The representation theory of rings and algebras is considered more generally, which contains the representation theory of groups as a special case (because every representation of a group induces a representation of its group ring )

• Representation (algebra) .

In physics, in addition to the discrete groups of solid state physics , representations of Lie groups are particularly important, for example in the case of the rotating group and the groups of the standard model . Here one also demands that representations should be smooth homomorphisms , see ${\displaystyle \rho \colon G\rightarrow GL(V)}$

• Representation (Lie group) .

The Lie'schen sentences convey a correspondence between representations of Lie groups and the induced representations of its Lie algebra . For the representation theory of Lie algebras see ${\displaystyle \pi =D_{e}\rho \colon {\mathfrak {g}}\rightarrow {\mathfrak {g}}l(V)}$

• Representation (Lie algebra) .

Lie algebras are not associative, which is why their representation theory is not a special case of the representation theory of associative algebras. But each Lie algebra can be assigned its universal enveloping algebra , which is an associative algebra.

Basic concepts

In the following a group, Lie group or algebra and a representation of , i.e. a group, Lie group or algebra homomorphism into the algebra of the linear mappings of a vector space (whose image in the case of groups or Lie groups Isomorphisms of course even lies in). ${\displaystyle A}$ ${\displaystyle \pi \colon A\rightarrow L(V)}$ ${\displaystyle A}$ ${\displaystyle L(V)}$ ${\displaystyle V}$ ${\displaystyle GL(V)}$

The vector space dimension of is 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 . 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. ${\displaystyle \pi _{1}\colon A\rightarrow L(V_{1})}$ ${\displaystyle \pi _{2}\colon A\rightarrow L(V_{2})}$ ${\displaystyle T\colon V_{1}\rightarrow V_{2}}$ ${\displaystyle \pi _{1}(a)=T^{-1}\circ \pi _{2}(a)\circ T}$ ${\displaystyle a\in A}$ ${\displaystyle \pi _{1}\sim \pi _{2}}$

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 the following applies , where the equivalence is mediated by the isomorphism . ${\displaystyle W^{c}}$ ${\displaystyle W}$ ${\displaystyle \pi \sim \pi |_{W}\oplus \pi |_{W^{c}}}$ ${\displaystyle W\oplus W^{c}\rightarrow V,(w_{1},w_{2})\mapsto w_{1}+w_{2}}$

Direct sums

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

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

again a representation of , operating component-wise on the direct sum , i.e. 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 be generalized for direct sums of any number of summands. Is a family of representations, so too ${\displaystyle (\pi _{i})_{i\in I}}$

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

Irreducibility, complete reducibility, reduction

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. The “product” (better: tensor product ) of two (irreducible) representations is generally reducible and can be “reduced” according to the components of the irreducible representations, with special coefficients such as B. the Clebsch-Gordan coefficients of angular momentum physics arise. This is a particularly important aspect for applications in physics. ${\displaystyle \pi \colon A\rightarrow L(V)}$ ${\displaystyle \{0\}}$ ${\displaystyle V}$ ${\displaystyle V}$

history

In the 18th and 19th centuries, representation theory and harmonic analysis (in the form of the decomposition of functions into multiplicative characters ) Abelian groups such as , or, for example, in connection with Euler products or Fourier transformations occurred. In doing so, one did not work with the representations, but with their multiplicative characters. In 1896 Frobenius first defined (without explicitly referring to representations) a concept of multiplicative characters also for non-Abelian groups, Burnside and Schur then redeveloped his definitions on the basis of matrix representations and Emmy Noether finally gave the current definition essentially using linear maps of a vector space, which later enabled the investigation of infinite-dimensional representations required in quantum mechanics. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} /2\pi \mathbb {Z} }$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$

Around 1900 the representation theory of symmetrical and alternating groups was worked out by Frobenius and Young. In 1913 Cartan proved the Theorem of Most Weight , which classifies the irreducible representations of complex semi-simple Lie algebras. Schur observed in 1924 that one can use invariant integration to extend the representation theory of finite groups to compact groups. Weyl then developed the representation theory of compact, connected Lie groups. The existence and uniqueness of the Haar measure , as proved by Haar and von Neumann, allowed this theory to be extended to compact topological groups in the early 1930s. Further developments then concerned the application of representation theory of locally compact groups such as the Heisenberg group in quantum mechanics, the theory of locally compact Abelian groups with applications in algebraic number theory (harmonic analysis on Adelen ) and later the Langlands program .

literature

• Etingof, Golberg, Hensel, Liu, Schwendner, Vaintrob, Yudovina: Introduction to Representation Theory . AMS, 2011. ISBN 978-0-8218-5351-1 .
• Roe Goodman, Nolan R. Wallach: Symmetry, representations, and invariants. (= Graduate Texts in Mathematics. 255). Springer, Dordrecht 2009, ISBN 978-0-387-79851-6 .
• Brian C. Hall: Lie groups, Lie algebras, and representations. An elementary introduction. (= Graduate Texts in Mathematics. 222). Springer-Verlag, New York 2003, ISBN 0-387-40122-9 .
• Theodor Bröcker, Tammo tom Dieck: Representations of compact Lie groups. (= Graduate Texts in Mathematics. 98). Translated from the German manuscript. Corrected reprint of the 1985 translation. Springer-Verlag, New York 1995, ISBN 0-387-13678-9 .
• JL Alperin, Rowen B. Bell: Groups and representations. (= Graduate Texts in Mathematics. 162). Springer-Verlag, New York 1995, ISBN 0-387-94525-3 .
• William Fulton, Joe Harris: Representation theory. A first course. (= Graduate Texts in Mathematics. 129). Readings in Mathematics. Springer-Verlag, New York 1991, ISBN 0-387-97527-6 ; 0-387-97495-4
• VS Varadarajan: Lie groups, Lie algebras, and their representations. (= Graduate Texts in Mathematics. 102). Reprint of the 1974 edition. Springer-Verlag, New York 1984, ISBN 0-387-90969-9 .
• James E. Humphreys: Introduction to Lie algebras and representation theory. (= Graduate Texts in Mathematics. 9). Second printing, revised. Springer-Verlag, New York / Berlin 1978, ISBN 0-387-90053-5 .
• Charles W. Curtis: Pioneers of representation theory: Frobenius, Burnside, Schur, and Brauer. (= History of Mathematics. 15). American Mathematical Society, Providence, RI / London Mathematical Society, London 1999, ISBN 0-8218-9002-6 .

Web links

On the history of representation theory:

• Anthony W. Knapp: Group representations and harmonic analysis from Euler to Langlands. In: Notices of the American Mathematical Society. 43, 4, 1996, part 1 ; 43, 5, 1996, part 2 .

Individual evidence

1. ↑ Introduction to Knapp ( op.cit. )
2. ↑ Part 2 by Knapp (op.cit.)