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❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ♦♥ ❧♦❝❛❧❧② ❝♦♠♣❛❝t q✉❛♥t✉♠ ❣r♦✉♣s ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ▼❛tt❤✐❛s ◆❡✉❢❛♥❣✱ ❆❞❛♠ ❙❦❛❧s❦✐ ❛♥❞ ◆✐❝♦ ❙♣r♦♥❦✮ P❡❦❦❛ ❙❛❧♠✐ ❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉ ●öt❡❜♦r❣ ✷✵✶✸ P❡❦❦❛ ❙❛❧♠✐ ✭❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ●öt❡❜♦r❣ ✷✵✶✸ ✶ ✴ ✶✸

❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥t ♠❡❛s✉r❡s ▼ ( ● ) ✐s t❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ♦❢ ❛ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❣r♦✉♣ ● ✇✐t❤ ❝♦♥✈♦❧✉t✐♦♥ ♣r♦❞✉❝t✿ �� � µ ⋆ ν, ❢ � = ❢ ( st ) ❞ µ ( s ) ❞ ν ( t ) µ, ν ∈ ▼ ( ● ); ❢ ∈ ❈ ✵ ( ● ) . ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts✿ µ ∈ ▼ ( ● ) s✉❝❤ t❤❛t µ ⋆ µ = µ ❛♥❞ � µ � = ✶✳ ❚❤❡♦r❡♠ ✭●r❡❡♥❧❡❛❢ ✬✻✺✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ✐♥ ▼ ( ● ) ❛r❡ t❤❡ ♠❡❛s✉r❡s ♦❢ t❤❡ ❢♦r♠ χ ❞♠ ❍ ✇❤❡r❡ ♠ ❍ ✐s t❤❡ ❍❛❛r ♠❡❛s✉r❡ ♦❢ ❛ ❝♦♠♣❛❝t s✉❜❣r♦✉♣ ❍ ♦❢ ● ❛♥❞ χ ✐s ❛ ❝♦♥t✐♥✉♦✉s ❝❤❛r❛❝t❡r ♦❢ ❍✳ ❚❤❡♦r❡♠ ✭❈♦❤❡♥ ✬✻✵✮ ❆❧❧ ✐❞❡♠♣♦t❡♥t ♠❡❛s✉r❡s ♦♥ ❛ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❛❜❡❧✐❛♥ ❣r♦✉♣ ❛r❡ ❣❡♥❡r❛t❡❞ ❢r♦♠ ❡❧❡♠❡♥t❛r② ✐❞❡♠♣♦t❡♥ts χ ❞♠ ❍ ✳ P❡❦❦❛ ❙❛❧♠✐ ✭❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ●öt❡❜♦r❣ ✷✵✶✸ ✷ ✴ ✶✸

❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥t ♠❡❛s✉r❡s ▼ ( ● ) ✐s t❤❡ ♠❡❛s✉r❡ ❛❧❣❡❜r❛ ♦❢ ❛ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❣r♦✉♣ ● ✇✐t❤ ❝♦♥✈♦❧✉t✐♦♥ ♣r♦❞✉❝t✿ �� � µ ⋆ ν, ❢ � = ❢ ( st ) ❞ µ ( s ) ❞ ν ( t ) µ, ν ∈ ▼ ( ● ); ❢ ∈ ❈ ✵ ( ● ) . ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts✿ µ ∈ ▼ ( ● ) s✉❝❤ t❤❛t µ ⋆ µ = µ ❛♥❞ � µ � = ✶✳ ❚❤❡♦r❡♠ ✭●r❡❡♥❧❡❛❢ ✬✻✺✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ✐♥ ▼ ( ● ) ❛r❡ t❤❡ ♠❡❛s✉r❡s ♦❢ t❤❡ ❢♦r♠ χ ❞♠ ❍ ✇❤❡r❡ ♠ ❍ ✐s t❤❡ ❍❛❛r ♠❡❛s✉r❡ ♦❢ ❛ ❝♦♠♣❛❝t s✉❜❣r♦✉♣ ❍ ♦❢ ● ❛♥❞ χ ✐s ❛ ❝♦♥t✐♥✉♦✉s ❝❤❛r❛❝t❡r ♦❢ ❍✳ ❚❤❡♦r❡♠ ✭❈♦❤❡♥ ✬✻✵✮ ❆❧❧ ✐❞❡♠♣♦t❡♥t ♠❡❛s✉r❡s ♦♥ ❛ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❛❜❡❧✐❛♥ ❣r♦✉♣ ❛r❡ ❣❡♥❡r❛t❡❞ ❢r♦♠ ❡❧❡♠❡♥t❛r② ✐❞❡♠♣♦t❡♥ts χ ❞♠ ❍ ✳ P❡❦❦❛ ❙❛❧♠✐ ✭❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ●öt❡❜♦r❣ ✷✵✶✸ ✷ ✴ ✶✸

❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ✐♥ t❤❡ ❋♦✉r✐❡r✕❙t✐❡❧t❥❡s ❛❧❣❡❜r❛ ❚❤❡ ❋♦✉r✐❡r✕❙t✐❡❧t❥❡s ❛❧❣❡❜r❛ ❇ ( ● ) ❝♦♥s✐sts ♦❢ ❝♦❡✣❝✐❡♥t ❢✉♥❝t✐♦♥s ♦❢ ✉♥✐t❛r② r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ● ✳ ❇ ( ● ) ∼ = ❈ ∗ ( ● ) ∗ ✐❢ ● ❛❜❡❧✐❛♥✱ ❇ ( ● ) ∼ = ▼ ( � ● ) ✈✐❛ ❋♦✉r✐❡r✕❙t✐❡❧t❥❡s tr❛♥s❢♦r♠ ❚❤❡♦r❡♠ ✭■❧✐❡✕❙♣r♦♥❦ ✬✵✺✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ✐♥ ❇ ( ● ) ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ✶ ❈ ✇❤❡r❡ ❈ ✐s ❛ ❝♦s❡t ♦❢ ❛♥ ♦♣❡♥ s✉❜❣r♦✉♣✳ P❡❦❦❛ ❙❛❧♠✐ ✭❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ●öt❡❜♦r❣ ✷✵✶✸ ✸ ✴ ✶✸

▲♦❝❛❧❧② ❝♦♠♣❛❝t q✉❛♥t✉♠ ❣r♦✉♣s ✭❑✉st❡r♠❛♥s✕❱❛❡s✮ ❆ ❧♦❝❛❧❧② ❝♦♠♣❛❝t q✉❛♥t✉♠ ❣r♦✉♣ G ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛ ❈✯✲❛❧❣❡❜r❛ ❈ ✵ ( G ) ❛ ❝♦♠✉❧t✐♣❧✐❝❛t✐♦♥ ∆: ❈ ✵ ( G ) → ▼ ( ❈ ✵ ( G ) ⊗ ❈ ✵ ( G )) (∆ ⊗ ✐❞ )∆ = ( ✐❞ ⊗ ∆)∆ ✭❝♦❛ss♦❝✐❛t✐✈✐t②✮ ❧❡❢t ❛♥❞ r✐❣❤t ❍❛❛r ✇❡✐❣❤ts φ ❛♥❞ ψ ✳ ❈♦♠♠✉t❛t✐✈❡ ❝❛s❡ G = ● ❈ ✵ ( G ) = ❈ ✵ ( ● ) ✱ ❋♦r ❢ ∈ ❈ ✵ ( ● ) ❛♥❞ s , t ∈ ● ✱ ∆( ❢ )( s , t ) = ❢ ( st ) . φ ❛♥❞ ψ ❛r❡ ✐♥t❡❣r❛t✐♦♥s ✇✳r✳t✳ t❤❡ ❧❡❢t ❛♥❞ r✐❣❤t ❍❛❛r ♠❡❛s✉r❡s P❡❦❦❛ ❙❛❧♠✐ ✭❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ●öt❡❜♦r❣ ✷✵✶✸ ✹ ✴ ✶✸

❈♦✲❝♦♠♠✉t❛t✐✈❡ ❝❛s❡ ❈♦✲❝♦♠♠✉t❛t✐✈❡ ❝❛s❡ G = � ● ❈ ✵ ( G ) = ❈ ∗ r ( ● ) t❤❡ r❡❞✉❝❡❞ ❣r♦✉♣ ❈✯✲❛❧❣❡❜r❛ ∆( λ ( s )) = λ ( s ) ⊗ λ ( s ) ✱ λ ✐s t❤❡ ❧❡❢t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥❀ s ∈ ● φ = ψ ✐s t❤❡ P❧❛♥❝❤❡r❡❧ ✇❡✐❣❤t ✭❢♦r ❞✐s❝r❡t❡ ● ✱ φ ( ❛ ) = � ❛ δ ❡ , δ ❡ � ✮ ✇❡ s❤❛❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ ❝♦❛♠❡♥❛❜❧❡ ▲❈◗●s ✇❤✐❝❤ ✐♥ t❤✐s ❝❛s❡ ♠❡❛♥s t❤❛t ● ✐s ❛♠❡♥❛❜❧❡ ❛♥❞ ❤❡♥❝❡ ❈ ∗ r ( ● ) = ❈ ∗ ( ● ) P❡❦❦❛ ❙❛❧♠✐ ✭❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ●öt❡❜♦r❣ ✷✵✶✸ ✺ ✴ ✶✸

❈♦♠♠♦♥ ❢♦r♠ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ✐♥ ❇ ( ● ) ✭ ● ❛♠❡♥❛❜❧❡✮ ✶ ❈ = ✶ s❍ = ✶ ❍ ( s − ✶ · ) = ✶ ❍ λ s − ✶ ✶ ❍ ✐s ❛♥ ✐❞❡♠♣♦t❡♥t st❛t❡ ✭✐✳❡✳ ♣♦s✐t✐✈❡ ✐❞❡♠♣♦t❡♥t✮ λ s − ✶ ✐s ❛ ❣r♦✉♣✲❧✐❦❡ ✉♥✐t❛r② ✐♥ ▼ ( ❈ ∗ ( ● )) ✿ ∆( λ s − ✶ ) = λ s − ✶ ⊗ λ s − ✶ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ✐♥ ▼ ( ● ) χ ❞♠ ❍ ❞♠ ❍ ✐s ❛♥ ✐❞❡♠♣♦t❡♥t st❛t❡ ✭❤❡r❡ ❛♥ ✐❞❡♠♣♦t❡♥t ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡✮ χ ✐s ❛ ❣r♦✉♣✲❧✐❦❡ ✉♥✐t❛r② ✐♥ ❈ ( ❍ ) ✱ ∆ ❍ ( χ ) = χ ⊗ χ ✳ ❲❡ ♠❛② ✈✐❡✇ χ ❛s ❛♥ ❡❧❡♠❡♥t ✐♥ ❈ ✵ ( ● ) t❤❛t ✐s ❣r♦✉♣✲❧✐❦❡ ✉♥✐t❛r② ♦♥ t❤❡ s✉♣♣♦rt ♦❢ ❞♠ ❍ ✳ P❡❦❦❛ ❙❛❧♠✐ ✭❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ●öt❡❜♦r❣ ✷✵✶✸ ✻ ✴ ✶✸

❈❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ ❝♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ▲❡t G ❜❡ ❛ ❧♦❝❛❧❧② ❝♦♠♣❛❝t q✉❛♥t✉♠ ❣r♦✉♣✳ ▼ ( G ) := ❈ ✵ ( G ) ∗ ✐s ❛ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ✉♥❞❡r µ ⋆ ν = ( µ ⊗ ν )∆ ❲❡ s✉♣♣♦s❡ t❤r♦✉❣❤♦✉t t❤❛t G ✐s ❝♦❛♠❡♥❛❜❧❡✿ t❤❡r❡ ✐s ❛ ✉♥✐t ✐♥ ▼ ( G ) ✳ ❆ ❝♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥t ♦♥ G ✐s ω ∈ ▼ ( G ) s✉❝❤ t❤❛t ω ⋆ ω = ω ❛♥❞ � ω � = ✶✳ ❚❤❡♦r❡♠ ✭◆❙❙❙✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ✐♥ ▼ ( G ) ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ❢✉♥❝t✐♦♥❛❧s ✐♥ ▼ ( G ) ♦❢ t❤❡ ❢♦r♠ ω = | ω | ( ✈ · ) ✇❤❡r❡ | ω | ✐s ❛♥ ✐❞❡♠♣♦t❡♥t st❛t❡ ❛♥❞ ✈ ∈ ❈ ✵ ( G ) s✉❝❤ t❤❛t ∆( ✈ ) − ✈ ⊗ ✈ ∈ ◆ | ω |⊗| ω | . ❍❡r❡ ◆ σ = { ❛ ∈ ❆ ; σ ( ❛ ∗ ❛ ) = ✵ } ❢♦r σ ❛ ♣♦s✐t✐✈❡ ❢✉♥❝t✐♦♥❛❧ ♦♥ ❛ ❈✯✲❛❧❣❡❜r❛ ❆ ✳ P❡❦❦❛ ❙❛❧♠✐ ✭❯♥✐✈❡rs✐t② ♦❢ ❖✉❧✉✮ ❈♦♥tr❛❝t✐✈❡ ✐❞❡♠♣♦t❡♥ts ●öt❡❜♦r❣ ✷✵✶✸ ✼ ✴ ✶✸