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Haar Measure
"... Definition 1 A group is a set Γ together with an operation · : Γ×Γ → Γ obeying i) (αβ)γ = α(βγ) for all α,β,γ ∈ Γ (Associativity) ii) ∃e ∈ Γ such that αe = eα = e for all α ∈ Γ (Existence of identity) iii) ∀α ∈ Γ ∃α −1 ∈ Γ such that αα −1 = α −1 α = e (Existence of inverses) Definition 2 A topologic ..."
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Definition 1 A group is a set Γ together with an operation · : Γ×Γ → Γ obeying i) (αβ)γ = α(βγ) for all α,β,γ ∈ Γ (Associativity) ii) ∃e ∈ Γ such that αe = eα = e for all α ∈ Γ (Existence of identity) iii) ∀α ∈ Γ ∃α −1 ∈ Γ such that αα −1 = α −1 α = e (Existence of inverses) Definition 2 A topological group is a group together with a Hausdorff topology such that the maps Γ×Γ → Γ (α,β) ↦ → αβ Γ → Γ α ↦ → α −1 are continuous. A compact group is a topological group that is a compact topological space. Example 3 (Examples of compact topological groups) i) U(1) = { z ∈ C ∣ ∣ z  = 1} = { e iθ ∣ ∣ 0 ≤ θ < 2π} with the usual multiplication in C and the usual topology in C. That is, e iθ e iϕ = e i(θ+ϕ) , the identity is e i0, the inverse is ( e iθ) −1 = e −iθ and d ( e iθ,e iϕ) = d ( cosθ +isinθ,cosϕ+isinϕ)
Haar Measure
"... Definition 1 A group is a set Γ together with an operation · : Γ×Γ → Γ obeying i) (αβ)γ = α(βγ) for all α,β,γ ∈ Γ (Associativity) ii) ∃e ∈ Γ such that αe = eα = e for all α ∈ Γ (Existence of identity) iii) ∀α ∈ Γ ∃α −1 ∈ Γ such that αα −1 = α −1 α = e (Existence of inverses) Definition 2 A topologic ..."
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Definition 1 A group is a set Γ together with an operation · : Γ×Γ → Γ obeying i) (αβ)γ = α(βγ) for all α,β,γ ∈ Γ (Associativity) ii) ∃e ∈ Γ such that αe = eα = e for all α ∈ Γ (Existence of identity) iii) ∀α ∈ Γ ∃α −1 ∈ Γ such that αα −1 = α −1 α = e (Existence of inverses) Definition 2 A topological group is a group together with a Hausdorff topology such that the maps Γ×Γ → Γ (α,β) ↦ → αβ Γ → Γ α ↦ → α −1 are continuous. A compact group is a topological group that is a compact topological space. Example 3 (Examples of compact topological groups) i) U(1) = { z ∈ C ∣ ∣ z  = 1} = { e iθ ∣ ∣ 0 ≤ θ < 2π} with the usual multiplication in C and the usual topology in C. That is, e iθ e iϕ = e i(θ+ϕ) , the identity is e i0, the inverse is ( e iθ) −1 = e −iθ and d ( e iθ,e iϕ) = d ( cosθ +isinθ,cosϕ+isinϕ)
MOTIVIC HAAR MEASURE ON REDUCTIVE GROUPS
, 2002
"... Abstract. We define a motivic analogue of the Haar measure for groups of the form G(k((t))), where k is an algebraically closed field of characteristic zero, and G is a reductive algebraic group defined over k. A classical Haar measure on such groups does not exist since they are not locally compact ..."
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Cited by 1 (0 self)
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Abstract. We define a motivic analogue of the Haar measure for groups of the form G(k((t))), where k is an algebraically closed field of characteristic zero, and G is a reductive algebraic group defined over k. A classical Haar measure on such groups does not exist since they are not locally
HAAR MEASURE ON Eq(2)
, 1996
"... The quantum E(2) group is one of the simplest known examples so far of a locally compact noncompact quantum group. The existence and uniqueness of an 'invariant measure ' on this group has been proved in this article. Using the invariant measure, we compute certain orthogonality relation ..."
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The quantum E(2) group is one of the simplest known examples so far of a locally compact noncompact quantum group. The existence and uniqueness of an 'invariant measure ' on this group has been proved in this article. Using the invariant measure, we compute certain orthogonality re
On some Haar measures on reductive groups
 Amer. J. Math
"... Abstract. We give a characteristicfree proof of a theorem of B. H. Gross comparing two Haar measures on a reductive group over a local field. As a consequence of this theorem we obtain an integration formula for adele groups over global function fields. ..."
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Cited by 3 (0 self)
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Abstract. We give a characteristicfree proof of a theorem of B. H. Gross comparing two Haar measures on a reductive group over a local field. As a consequence of this theorem we obtain an integration formula for adele groups over global function fields.
THE UNIQUENESS OF HAAR MEASURE AND SET THEORY BY
"... Let G be a group of homeomorphisms of a nondiscrete, locally compact, σcompact topological space X and suppose that a Haar measure on X exists: a regular Borel measure µ, positive on nonempty open sets, finite on compact sets and invariant under the homeomorphisms from G. Under some mild assumption ..."
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Let G be a group of homeomorphisms of a nondiscrete, locally compact, σcompact topological space X and suppose that a Haar measure on X exists: a regular Borel measure µ, positive on nonempty open sets, finite on compact sets and invariant under the homeomorphisms from G. Under some mild
Conditional Haar measures on classical compact groups
 Ann. Probab
"... We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension n. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension p. The developed method leads for the first nonzero derivative o ..."
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Cited by 1 (1 self)
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We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension n. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension p. The developed method leads for the first nonzero derivative
Field Configurations and the SU(2) Haar Measure in QCD
, 1996
"... We characterise a class of SU(2) gluonic field configurations in the modified axial gauge where a zero mode component vanishes at some space point but the global Haar measure remains nonzero. The consequence of this is that gluonic wavefunctionals need not vanish at the boundary of the fundamental ..."
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We characterise a class of SU(2) gluonic field configurations in the modified axial gauge where a zero mode component vanishes at some space point but the global Haar measure remains nonzero. The consequence of this is that gluonic wavefunctionals need not vanish at the boundary of the fundamental
Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
 Comm. Math. Phys
"... ABSTRACT. We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated po ..."
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Cited by 115 (18 self)
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ABSTRACT. We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated
Results 1  10
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583