Unimodular Group is a redirect to this article. For groups of matrices with determinant 1 see
special linear group .
The modular function is a term from harmonic analysis , i.e. from the theory of locally compact groups . The modular function measures a left-right asymmetry of the group.
definition
Let it be a locally compact group. Then there is known to be a linksinvariantes Haar measure on . Left invariance means that for all and all Borel sets . In general, it does not follow from this that is also right invariant, that is, it can absolutely apply.
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![{\ displaystyle \ mu (tA) = \ mu (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b7b953a0182ca6fd1d7da9bc4d0c2ddbe759e2)
![A \ subset G](https://wikimedia.org/api/rest_v1/media/math/render/svg/215c399a77f078e27c15cf95c5e27c53b9d93aae)
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![{\ displaystyle \ mu (At) \ not = \ mu (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e91dba98f0f32857d8abead123963b145046217e)
For solid , the figure is also a left-invariant Haar's measure, as can easily be confirmed. Since this is uniquely determined except for a constant, there is a number with , that is, for all measurable ones .
![t \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8c2125ac6c76f186eca27c0a7d215269b90838)
![{\ displaystyle \ mu _ {t}: \, A \ mapsto \ mu (At)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/925888efa89f3af06f792a22f76c943546dee91d)
![{\ displaystyle \ Delta _ {G} (t) \ in \ mathbb {R} ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc11f36f82f1dfcc31fa880333d7fcf0c4d70e0b)
![{\ displaystyle \ mu _ {t} \, = \, \ Delta _ {G} (t) \ mu}](https://wikimedia.org/api/rest_v1/media/math/render/svg/577bb07638656cbf742b14e238bbd4911e435580)
![{\ displaystyle \ mu (At) \, = \, \ Delta _ {G} (t) \ mu (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb5f7a717d687b909d9766cde0cb82b2c4463b4)
![A \ subset G](https://wikimedia.org/api/rest_v1/media/math/render/svg/215c399a77f078e27c15cf95c5e27c53b9d93aae)
In this way, a mapping is obtained that proves to be independent of the choice of the left-invariant Haar measure and is a continuous homomorphism of in the multiplicative group . is the modular function of![{\ displaystyle \ Delta _ {G}: G \ rightarrow \ mathbb {R} ^ {+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf16db7eaebd878b6151ec0208ee7cf3f23dd41)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![\ R ^ +](https://wikimedia.org/api/rest_v1/media/math/render/svg/97dc5e850d079061c24290bac160c8d3b62ee139)
![{\ displaystyle \ Delta _ {G}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2173e28a4dfa449e4b6dabad389c4d161627f01f)
Unimodular groups
Groups for which the modular function is the same as the constant function for all are called unimodular . These are exactly those groups for which a left-invariant Haar's measure is also right-invariant. Three important types of locally compact groups are automatically unimodular:
![{\ displaystyle \ Delta _ {G} (t) = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d9d306fa3f63d2977171c53ef6250fbb47b1957)
![t \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8c2125ac6c76f186eca27c0a7d215269b90838)
- Commutative locally compact groups are unimodular, because due to the commutativity, left-invariant measures are naturally also right-invariant.
- Compact groups are bound one, because the image of the modular function has a compact subgroup in his, and there is only the question.
![\ R ^ +](https://wikimedia.org/api/rest_v1/media/math/render/svg/97dc5e850d079061c24290bac160c8d3b62ee139)
![\{1\}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124)
-
Discrete groups are unimodular, because the multiples of the counting measure are exactly the left and right invariant Haarsche measures.
An example of a unimodular, locally compact group that does not fall under any of these three types is the general linear group . A left and right invariant measure is through
![GL (n, \ mathbb {R})](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f7197960fac26cadfe027d3045154b9972f8d3)
given, where the Lebesgue measure is on .
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![\ mathbb {R} ^ {{n ^ {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b62b12287038d00c20d18abf0ba48f871c0396)
example
Here we give an example of a non-trivial modular function. Let it be the locally compact group of all matrices with . A left-invariant Haar's measure is
given by, a right-invariant one by
. This results in
.
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![2 \ times 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80)
![{\ displaystyle {\ begin {pmatrix} a & b \\ 0 & 1 \ end {pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6ca1e107a0563331fe8bfaf1c84744b1a1e6769)
![{\ displaystyle a, b \ in \ mathbb {R}, a> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a9816c75690fe2264f862acc7b6f920a9aab4d)
![{\ displaystyle \ mu (A) = \ int _ {\ mathbb {R}} \ int _ {\ mathbb {R} ^ {+}} {\ frac {1} {a ^ {2}}} \, \ mathrm {d} a \ mathrm {d} b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72c165149674536765f42c816a1be3587ce3d80f)
![{\ displaystyle \ nu (A) = \ int _ {\ mathbb {R}} \ int _ {\ mathbb {R} ^ {+}} {\ frac {1} {a}} \, \ mathrm {d} a \ mathrm {d} b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf17e68ce225eb644fa2919a7d0ffeb653a6353)
![{\ displaystyle \ Delta _ {G} ({\ begin {pmatrix} a & b \\ 0 & 1 \ end {pmatrix}}) \, = \, {\ frac {1} {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b34192db1b64ce4a0a1e3e16b5ee37c4243a86f)
Calculation rules
Let it be a locally compact group with left invariant Haar's measure . For a function, let the so-called translation of um .
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![{\ displaystyle f: G \ rightarrow R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f71d3c82c1d161106f1da3c9ce238b7dcee39abe)
![{\ displaystyle f_ {s} (t) \,: = \, f (ts ^ {- 1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f19c2f6e89fe08e5a78a0d70d12ca51f6d58347)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![s](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632)
If the characteristic function is the Borel set , then and therefore according to the construction of the modular function
![\ chi _ {A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49baf1caaa804f2d77bfc7570d102ee4a3cafa26)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle (\ chi _ {A}) _ {s} = \ chi _ {As}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9599cbe3cb1468cef004f776979a1bf5fc14b8d)
.
With the usual mass theoretic inferences we get for every -integratable function :
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
.
The modular function also occurs if one integrates via inverted arguments. For -integratable functions on applies
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
.
Finally, the modular function occurs in the definition of the involution on convolutional algebra
. In the space above, define for function![L ^ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9)
![(G, \ mu)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9532aaf819d85a16101f8b37a60d29e9dee698bf)
.
It is only defined almost everywhere , namely where the integral exists, and the dash stands for the complex conjugation . With the so-called convolution product defined by and the mapping , a Banach algebra with isometric involution becomes . The study of this Banach algebra is an important tool in harmonic analysis .
![f \ star g](https://wikimedia.org/api/rest_v1/media/math/render/svg/371d3161cd7e094182a184d7601b49880228385c)
![\star](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd316a21eeb5079a850f223b1d096a06bfa788c0)
![f \ mapsto f ^ *](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1802f3097b85fd3f2d6e086eb857a82e696df04)
![L ^ 1 (G)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e10e36f941f2b8fec8f9c058ebefd8ac69609b97)
Individual evidence
-
^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA 1980, ISBN 3-7643-3003-1 , set 9.3.4.
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^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA 1980, ISBN 3-7643-3003-1 , chapter 9.3, exercise 2
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^ Donald L. Cohn: Measure Theory. Birkhäuser, Boston MA 1980, ISBN 3-7643-3003-1 , text after sentence 9.3.3.
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↑ Lynn H. Loomis : An Introduction to Abstract Harmonic Analysis. D. van Nostrand Co., Princeton NJ et al. 1953, § 30B.
-
^ Jacques Dixmier : C * -algebras (= North-Holland Mathematical Library. Vol. 15). North Holland Publishing Company, Amsterdam et al. 1977, ISBN 0-7204-2450-X , Chapter 13.2.