Monthly Archives: February, 2012

Fractional Sobolev Spaces

Recall for a function f\in L^1(\Omega) for \Omega\subseteq\mathbb{R}^n we define the Fourier transform of f by

\displaystyle\hat{f}(x)=F(f)(x)=\int_\Omega e^{-2\pi ix\cdot\xi}f(\xi)\,d\xi

and the inverse Fourier transform by

\displaystyle \check{f}(x)=F^{-1}(f)(x)=\int_\Omega e^{2\pi i x\cdot\xi}f(\xi)\,d\xi.

If f,\hat{f}\in L^1(\Omega), then \check{\hat{f}}=\hat{\check{f}}=f a.e..

Now if f\in L^1 and D^\alpha (f)\in L^1 with \alpha\in\mathbb{Z}^n, we say D^\alpha f is the weak derivative of f provided

\displaystyle\int_\Omega f D^\alpha\phi\,dx=(-1)^{|\alpha|}\int_\Omega D^\alpha(f)\phi\,dx

with \phi\in C_C^\infty(\Omega), |\alpha|=\sum_i\alpha_i, and

\displaystyle D^\alpha\phi=\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}\phi.

The Sobolev space W^{k,p}(\Omega) is then defined as the set of all functions f\in L^p(\Omega) such that D^\alpha(f)\in L^p(\Omega) for all |\alpha|\leq k.  These become Banach spaces under the norm

\displaystyle\|f\|_{k,p}=\left(\sum_{|\alpha|\leq k}\int_\Omega \left|D^\alpha(f)\right|^p\,dx\right)^{1/p}

and Hilbert when p=2, whence we denote H^k(\Omega)=W^{k,2}(\Omega).

For 1<p<\infty we can say that W^{k,p}(\mathbb{R}^n) consists of f\in L^p(\mathbb{R}^n) such that

\displaystyle F^{-1}\circ(1+|2\pi x|^2)^{k/2}F\circ f=(1-\Delta)^{k/2}f(x)\in L^p(\mathbb{R}^n)

(using Fourier multipliers).

Bessel Potential Spaces:

We can generalize with r\in\mathbb{R} and 1<p<\infty and define the Bessel potential space W^{r,p}(\mathbb{R}^n) as all f\in L^p such that \|f\|_{r,p}:=\|(1-\Delta)^{r/2}f\|_p<\infty.  Note the fractional power of the Laplacian is defined by virtue of the fact that L^p\cap L^1 is dense in L^p, so we say

\displaystyle(1-\Delta)^{r/2}(f):=(F^{-1}\circ (1+|2\pi x|^2)\circ F)(f).

 These spaces satisfy our desire of being Banach (and Hilbert when p=2).

Sobolev-Slobodeckij Spaces:

There is an alternative approach.  Recall that the Holder space C^{k,\gamma}(\Omega) is defined as all functions f\in C^k(\Omega) such that

\displaystyle\|f\|_{C^{k,\gamma}}=\sum_{|\alpha|\leq k}\sup_{x\in\Omega}|D^\alpha|+\sum_{|\alpha|=k}\sup_{x,y\in\Omega}\frac{|D^\alpha f(x)-D^\alpha f(y)|}{|x-y|^\gamma}<\infty.

That is, it is the set of functions on \Omega which are C^k and whose k-th partial derivatives are bounded and Holder continuous of degree \gamma.  These spaces are Banach under the above norm.  We can generalize the Sobolev spaces to incorporate similar properties.  Let us define the Slobodeckij norm for f\in L^p(\Omega) with 1\leq p<\infty and \theta\in (0,1) by

\displaystyle [f]_{\theta,p}=\int_\Omega\int_\Omega\frac{|f(x)-f(y)|^p}{|x-y|^{\theta p+n}}\,dxdy.

The corresponding Sobolev-Slobodeckij space W^{s,p}(\Omega) is defined as all functions f\in W^{\lfloor s\rfloor,p}(\Omega) such that

\displaystyle\sup_{|\alpha|=\lfloor s\rfloor}[D^\alpha f]_{\theta,p}<\infty

where \theta=s-\lfloor s\rfloor\in(0,1).  This becomes a Banach space under the norm

\displaystyle\|f\|_{s,p}=\|f\|_{W^{\lfloor s\rfloor,p}}+\sup_{|\alpha|=\lfloor s\rfloor}[D^\alpha f]_{\theta,p}.

[1] http://en.wikipedia.org/wiki/Sobolev_space (unclear text references)

[2]  Lieb, Elliot and Michael Loss.  Analysis.  2nd Edition.  Graduate Studies in Mathematics.  Vol. 14.  American Mathematical Society.  2001.

[3]  Evans, Lawrence.  Partial Differential Equations.  Graduate Studies in Mathematics.  Vol. 19.  American Mathematical Society.  1998.

Lie Groups/Algebras, Notes 4: Lie Groups

Lie Groups:

Definition 1.  A Lie group is a manifold and group whose binary operation is a C^\omega map between the manifolds G\times G and G.

We will assume our Lie groups are Banach manifolds unless otherwise stated (locally homeomorphic to a Banach space E).  It turns out a Lie group is metrizable, and the metric space is complete.  Lie subgroups are subgroups which are also submanifolds and Lie groups.

Proposition 2.  Let G be an E-Lie group (that is, a Lie group which is an E-manifold).  Then

  1. If E\in\{\mathbb{R},\mathbb{C}\}, then G is locally connected.
  2. If E\notin\{\mathbb{R},\mathbb{C}\}, then \dim(G)=\dim_F(E)=0.
  3. Suppose E is locally compact, then G is locally compact iff \dim(G)<\infty.
  4. If G is generated by a subspace whose topology admits a countable base, then the topology on G admits a countable base.

Now suppose X is a C^\omega-manifold with an analytic associative binary operation with identity.  Then the subset G\subseteq X of invertible elements of X is open in X and is a Lie group.

Definition 3.  A Lie group morphism between Lie groups G and H is a C^\omega group homomorphism f:G\to H.  Aut(G) will denote the group of Lie group automorphisms of G.  A Lie group representation on a Banach space E is a Lie group morphism \rho:G\to Aut(E).  (Note if E is a Banach space over the field F, then Aut(E) is a Lie group which is a manifold over F)

Definition 4.  Let G be a topological group and X be an analytic manifold.  We say that X is a homogeneous space for G if X is a G-space (that is, continuous action) and (g_1g_2)x=g_1(g_2x).

This induces an equivalence relation on X into orbits under the action of G and a corresponding quotient space X/G, called the orbit space of X by G.

Proposition 5.  Let X be Lie group and G be a Lie subgroup.  Then:

  1. The orbit space X/G has a unique analytic manifold structure such that the canonical map \pi:X\to X/G is a submersion.
  2. If G\unlhd X, then X/G is a Lie group and \pi is a Lie group morphism.

In case 1 above, we call the quotient X/G the Lie homogeneous space of X by G, and in case 2 we call the quotient the Lie quotient group.

Let G and H be Lie groups.  Then there is a canonical correspondence \tau between T(G)\times T(H) and T(G\times H).  In particular, suppose we have two tangent vectors u\in T_x(G), v\in T_y(G), and f\in C^\omega(G).  Let us then define

(uv)(f)=\tau(u,v)(f,0)+\tau(u,v)(0,f).

It follows that uv\in T_{xy}(G) for

\begin{array}{lcl} (uv)(fg)&=&\tau(u,v)(fg,0)+\tau(u,v)(0,fg)\\&=&\tau(u,v)(f,0)\cdot g(xy)+f(xy)\tau(u,v)(g,0)+\tau(u,v)(0,f)\cdot g(xy)+f(xy)\tau(u,v)(0,g)\\&=&g(xy)\left(\tau(u,v)(f,0)+\tau(u,v)(0,f)\right)+f(xy)\left(\tau(u,v)(g,0)+\tau(u,v)(0,g)\right)\\&=&(uv)(f)g(xy)+f(xy)(uv)(g).\end{array}

Thus we have a product on the tangent bundle of G (note that G being a group makes xy an arbitrary element of G).  Hence T(G) is a Lie group.

The Lie Algebra of a Lie Group:

Recall a vector field on a manifold M is section of the vector fiber bundle T(M) with M as its base.  Hence it is a map X:M\to T(M) satisfying \pi\circ X=1 where \pi is the projection of the vector fiber bundle to its base.  Now suppose X and Y are two vector fields on M, then we can define the vector fields

(X+Y)(p)(f)=(X(p)+Y(p))(f)

(XY)(p)(f)=X(p)[Y\circ f]-Y(p)[X\circ f]

where (Y\circ f)(x)=Y(x)(f).  This turns V(M), the set of vector fields of M, into a Lie algebra.

Now let G be a Lie group and \lambda_g:G\to G be left multiplication by g\in G.  These are certainly diffeomorphisms of G

Definition 6.  A vector field X is left-invariant if

(\lambda_g)_*(X)=X

for all g\in G where (\lambda_g)_*(X)(x)=X(gx).

Proposition 7.  The set L(G) of left-invariant vector fields of G is a Lie subalgebra of V(G).

Proof.  Let X and Y be left-invariant and g\in G.  Then

\begin{array}{lcl}(\lambda_g)_*(X+Y)(x)(f)&=&(\lambda_g)_*X(x)(f)+(\lambda_g)_*Y(x)(f)\\&=&X(gx)(f)+Y(gx)(f)\\&=&(X+Y)(gx)(f)\end{array}

and

\displaystyle \begin{array}{lcl}(\lambda_g)_*(XY)(x)(f)&=&(\lambda_g)_*\left(X(x)[Y\circ f]-Y(x)[X\circ f]\right)\\&=&X(gx)[Y\circ f]-Y(gx)[X\circ f]\\&=&(XY)(gx)(f)\end{array}.

The Lie subalgebra L(G) of V(G) is called the Lie algebra of G.  Note an obvious consequence is that if v\in T_1(G), X\in L(G), and X(1)=v.  Then

X(g)=(\lambda_g)_*(X)(1).

Hence it follows that left-invariant vector fields of G are completely determined by where they send the identity element.  In this sense, we obtain the equivalence of the vector spaces L(G) and T_1(G).

Lie Group of a Lie Algebra:

The standard convention is to discuss an “inverse” map from the Lie algebra of a Lie group back to the Lie group.  Let X\in L(G).  Then this left-invariant vector field corresponds to a tangent vector v\in T_1(G).  Let \gamma_X:\mathbb{R}\to G be a path defined by

\gamma_X(0)=1

X(\gamma_X(0))=X(1)=v

\gamma_X(t_1+t_2)=\gamma_X(t_1)\gamma_X(t_2).

Note these uniquely define \gamma_X, called a one-parameter subgroup of G.  We in turn define the map \exp:L(G)\to G by

\exp(X)=\gamma_X(1)

or more generally

\exp(tX)=\gamma_X(t).

Hence \exp(0)=1. It follows that L(\exp(L(G)))=L(G).

Suppose G=GL_n(\mathbb{R}).  Then L(G)=M_n(\mathbb{R}) and

\displaystyle\exp(X)=\sum_{n=0}^\infty\frac{X^n}{n!}.

[1]  Bourbaki, Nicholas.  Lie Groups and Lie Algebras, Chapters 1-3.  Springer-Verlag.  1971.

[2]  Robert Milson, Thomas Foregger, Mike Fikes. “Lie group” (version 12). PlanetMath.org. Freely available at http://planetmath.org/?op=getobj;from=objects;id=1112

Lie Groups/Algebras, Notes 3: Bialgebras and Free Lie Algebras

Recall the construction of an associative algebra A over a field K.  We have a module homomorphism \nabla:A\otimes A\to A such that

\nabla\circ (1\otimes\nabla)=\nabla\circ(\nabla\otimes 1).

If the algebra is unital, then we also a canonical map \eta:K\to A with \eta(1) defined as the identity of A and corresponding assumption that

\nabla\circ(1\otimes\eta)=\nabla\circ(\eta\otimes 1)=1

on A (with canonical identifications A\otimes K=K\otimes A=A).

We can dually define a Kcoalgebra as a K-module C together with a comultiplication map \Delta:C\to C\otimes C such that

(1\otimes\Delta)\circ\Delta=\Delta\circ(\Delta\otimes 1).

We also call it unital if there is a map \varepsilon:C\to K such that

(1\otimes\varepsilon)\circ\Delta=(\varepsilon\otimes 1)\circ\Delta=1

on C (with the canonical identification C\otimes K=K\otimes C=C).

Definition 1.  A Kbialgebra is a unital K-algebra and unital K-coalgebra such that

  1. \Delta\circ\nabla=(\nabla\otimes\nabla)\circ(1\otimes\tau\otimes 1)\circ(\Delta\otimes\Delta) (\tau(x\otimes y)=y\otimes x),
  2. \varepsilon\circ\nabla=\varepsilon\otimes\varepsilon,
  3. \Delta\circ\eta=\eta\otimes\eta, and
  4. \varepsilon\circ\eta=1.

Definition 2.  An element x of a coalgebra is uprimitive if \Delta(x)=x\otimes u+u\otimes x.  Elements are primitive if they are 1-primitive.

If \sigma:\mathfrak{g}\to U(\mathfrak{g}) is the embedding into the enveloping algebra, then the map \Delta(\sigma(x))=\sigma(x)\otimes 1+1\otimes\sigma(x) gives U(\mathfrak{g}) a K-bialgebra structure.

Definition 3.  Let X be a set and A_K(X) be the free algebra over X.  We define the free Lie Kalgebra over X as the quotient A_K(X)/I where I is the ideal generated by elements of the form xy+yx and x(yz)+z(xy)+y(zx).  We will denote this Lie algebra by L(X) and its product by [x,y].

The enveloping algebra U(L(X)) of the free Lie algebra L(X) ends up being the free associative algebra over X.

[1]  Bourbaki, Nicholas.  Lie Groups and Lie Algebras, Chapters 1-3.  Springer-Verlag.  1971.

Lie Groups/Algebras, Notes 2: Representations and Types

Recall a representation of an algebra A on an R-module M is an algebra morphism \rho:A\to End_R(M).  When we have a Lie algebra \mathfrak{g}, a representation of it on an R-module M is an algebra morphism

\rho:\mathfrak{g}\to L(End_R(M)).

Hence in particular we have \rho(xy)(m)=\left(\rho(x)\rho(y)-\rho(y)\rho(x)\right)(m).  We correspondingly call M a \mathfrak{g}module.  A representation is faithful if it is injective.  It is simple/irreducible if its \mathfrak{g}-module is simple.

Let \rho_1 and \rho_2 be representations of Lie algebras \mathfrak{g}_1 and \mathfrak{g}_2 on M_1 and M_2 respectively.  Then \rho_1\otimes\rho_2 is a representation of \mathfrak{g}_1\times\mathfrak{g}_2 on M_1\otimes M_2 defined by

\begin{array}{lcl}(\rho_1\otimes\rho_2)(x_1,x_2)(m_1\otimes m_2)&=&(\sigma_1(x_1)\otimes 1+1\otimes\sigma_2(x_2))(m_1\otimes m_2)\\&=&\sigma_1(x_1)(m_1)\otimes m_2+m_1\otimes\sigma_2(x_2)(m_2)\end{array}

where \sigma_1,\sigma_2 are the canonical inclusions of \mathfrak{g}_1 in U(\mathfrak{g}_1) and \mathfrak{g}_2 in U(\mathfrak{g}_2).  Hence M_1\otimes M_2 is both a \mathfrak{g}_1\times\mathfrak{g}_2-module and a U(\mathfrak{g}_1)\otimes U(\mathfrak{g}_2)-module.

Definition 1\mathfrak{g} is solvable if its derived series D^n\mathfrak{g}=0 for some n.  It is nilpotent if its lower central series C_n\mathfrak{g}=0 for some n.  The radical of \mathfrak{g}, \sqrt{\mathfrak{g}}, is the largest solvable ideal in \mathfrak{g}.  We say \mathfrak{g} is semisimple if it has no nonzero abelian ideals.

Hence it follows that the center Z(\mathfrak{g}) of a nilpotent Lie algebra is nontrivial.  Also note that a nilpotent Lie algebra is solvable.  \mathfrak{r}=\sqrt{\mathfrak{g}} is the smallest ideal such that \mathfrak{g}/\mathfrak{r} has radical 0.

Theorem 2 (Engel).  Let V be a vector space and \mathfrak{g} be a finite dimensional subalgebra of \mathfrak{gl}(V)..  If V\neq 0, then there exists a u\neq 0 with u\in\mathfrak{g} such that xu=0 for all x\in\mathfrak{g}.

It follows that a Lie algebra \mathfrak{g} is nilpotent iff \mbox{adj}\,x is nilpotent for all x\in\mathfrak{g}.

Note that B(x,y)=\mbox{tr}(\mbox{adj}(x)\mbox{adj}(y)) defines a symmetric bilinear form on \mathfrak{g}, called the Killing form of \mathfrak{g}.  The Killing form also satisfies B(xy,z)=B(x,yz).

Proposition 3.  \mathfrak{g} is solvable iff B(D\mathfrak{g},\mathfrak{g})=0.

Proposition 4.  \mathfrak{g} is semisimple iff \mathfrak{g}=0 iff B is nondegenerate.

Theorem 5 (Weyl).  Every finite dimensional representation of a semisimple Lie algebra is completely reducible (i.e. the \mathfrak{g}-module is semisimple).

A Lie algebra is simple if its only ideals are \mathfrak{g} and 0 and noncommutative.

Proposition 6.  \mathfrak{g} is semisimple iff it is a product of simple algebras.

A Lie algebra is reductive if its adjoint representation is semisimple.  That is, it is reductive if it can be written \mathfrak{g}=\mathfrak{s}\oplus\mathfrak{a} where \mathfrak{s} is semisimple and \mathfrak{a} is abelian (since here we want \mathfrak{g} as a module to be semisimple–so that as an algebra, it has an an abelian term).  This also turns out to be equivalent to D\mathfrak{g} being semisimple.

Definition 7.  The nilpotent radical of \mathfrak{g} is the intersection of the kernels of all finite dimensional simple representations of \mathfrak{g}.

For an element x in the nilpotent radical, it follows that \mbox{adj}(x) is nilpotent.  Now for any nilpotent endomorphism u of an algebra, we have that

\displaystyle e^u=\sum_{n=0}^\infty\frac{u^n}{n!}

has a finite number of nonzero terms, and is hence well-defined.  If we can write \mathfrak{g}=\sqrt{\mathfrak{g}}\ltimes_{e^{u\cdot}}\mathfrak{s} with \mathfrak{s} semisimple and u nilpotent, then we will call \mathfrak{s} a Levi subalgebra of \mathfrak{g}.

Theorem 8 (Levi-Malcev).  Every Lie algebra has a Levi subalgebra (that is, it can be written \mathfrak{g}=\sqrt{\mathfrak{g}}\ltimes\mathfrak{s} as above).  Moreover, every other Levi subalgebra \mathfrak{s}' is the image of \mathfrak{s} under \exp(\mbox{adj}(x)) for some x in the nilpotent radical of \mathfrak{g}.

It turns out that a subalgebra of \mathfrak{g} is a Levi subalgebra iff it is a maximal semisimple subalgebra.

Theorem 9 (Ado).  Let \mathfrak{n} be the largest nilpotent ideal in \mathfrak{g}.  Then \mathfrak{g} admits a finite dimensional faithful representation \rho such that every element of \rho(\mathfrak{n}) is nilpotent.

[1]  Bourbaki, Nicholas.  Lie Groups and Lie Algebras, Chapters 1-3.  Springer-Verlag.  1971.