Double-clad fiber: Difference between revisions

Latest revision as of 16:25, 13 August 2013

In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold. The algebra of a Poisson–Lie group is a Lie bialgebra.

Definition

A Poisson–Lie group is a Lie group G equipped with a Poisson bracket for which the group multiplication $μ : G \times G \to G$ with $μ (g_{1}, g_{2}) = g_{1} g_{2}$ is a Poisson map, where the manifold G×G has been given the structure of a product Poisson manifold.

Explicitly, the following identity must hold for a Poisson–Lie group:

{f_{1}, f_{2}} (g g^{'}) = {f_{1} \circ L_{g}, f_{2} \circ L_{g}} (g^{'}) + {f_{1} \circ R_{g^{'}}, f_{2} \circ R_{g^{'}}} (g)

where f₁ and f₂ are real-valued, smooth functions on the Lie group, while g and g' are elements of the Lie group. Here, L_g denotes left-multiplication and R_g denotes right-multiplication.

If $𝒫$ denotes the corresponding Poisson bivector on G, the condition above can be equivalently stated as

𝒫 (g g^{'}) = L_{g *} (𝒫 (g^{'})) + R_{g^{'} *} (𝒫 (g))

Note that for Poisson-Lie group always ${f, g} (e) = 0$ , or equivalently $𝒫 (e) = 0$ . This means that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank.

Homomorphisms

A Poisson–Lie group homomorphism $ϕ : G \to H$ is defined to be both a Lie group homomorphism and a Poisson map. Although this is the "obvious" definition, neither left translations nor right translations are Poisson maps. Also, the inversion map $ι : G \to G$ taking $ι (g) = g^{- 1}$ is not a Poisson map either, although it is an anti-Poisson map:

{f_{1} \circ ι, f_{2} \circ ι} = - {f_{1}, f_{2}} \circ ι

for any two smooth functions $f_{1}, f_{2}$ on G.

References

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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+In [[mathematics]], a '''Poisson–Lie group''' is a [[Poisson manifold]] that is also a [[Lie group]], with the group multiplication being compatible with the [[Poisson algebra]] structure on the manifold. The algebra of a Poisson–Lie group is a [[Lie bialgebra]].
+==Definition==
+A Poisson–Lie group is a Lie group ''G'' equipped with a Poisson bracket for which the group multiplication <math>\mu:G\times G\to G</math> with <math>\mu(g_1, g_2)=g_1g_2</math> is a [[Poisson map]], where the manifold ''G''×''G'' has been given the structure of a product Poisson manifold.
+Explicitly, the following identity must hold for a Poisson–Lie group:
+:<math>\{f_1,f_2\} (gg') =
+\{f_1 \circ L_g, f_2 \circ L_g\} (g') +
+\{f_1 \circ R_{g^\prime}, f_2 \circ R_{g'}\} (g)</math>
+where ''f''<sub>1</sub> and ''f''<sub>2</sub> are real-valued, smooth functions on the Lie group, while ''g'' and ''g''' are elements of the Lie group. Here, ''L<sub>g</sub>'' denotes left-multiplication and ''R<sub>g</sub>'' denotes right-multiplication.
+If <math>\mathcal{P}</math> denotes the corresponding Poisson bivector on ''G'', the condition above can be equivalently stated as
+:<math>\mathcal{P}(gg') = L_{g \ast}(\mathcal{P}(g')) + R_{g' \ast}(\mathcal{P}(g))</math>
+Note that for Poisson-Lie group always <math>\{f,g\}(e) = 0</math>, or equivalently <math>\mathcal{P}(e) = 0 </math>. This means that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank.
+==Homomorphisms==
+A Poisson–Lie group homomorphism <math>\phi:G\to H</math> is defined to be both a Lie group homomorphism and a Poisson map. Although this is the "obvious" definition, neither left translations nor right translations are Poisson maps. Also, the inversion map <math>\iota:G\to G</math> taking <math>\iota(g)=g^{-1}</math> is not a Poisson map either, although it is an anti-Poisson map:
+:<math>\{f_1 \circ \iota, f_2 \circ \iota \} =
+-\{f_1, f_2\} \circ \iota</math>
+for any two smooth functions <math>f_1, f_2</math> on ''G''.
+==References==
+*{{cite book |editor1-first=H.-D. |editor1-last=Doebner |editor2-first=J.-D. |editor2-last=Hennig |title=Quantum groups |series=Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG |year=1989 |publisher=Springer-Verlag |location=Berlin |isbn=3-540-53503-9 }}
+*{{cite book |first=Vyjayanthi |last=Chari |first2=Andrew |last2=Pressley |title=A Guide to Quantum Groups |year=1994 |publisher=Cambridge University Press |location=Cambridge |isbn=0-521-55884-0 }}
+{{DEFAULTSORT:Poisson-Lie group}}
+[[Category:Lie groups]]
+[[Category:Symplectic geometry]]

Double-clad fiber: Difference between revisions

Latest revision as of 16:25, 13 August 2013

Definition

Homomorphisms

References

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