{"id":3230,"date":"2011-10-08T14:28:43","date_gmt":"2011-10-08T12:28:43","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=3230"},"modified":"2014-12-19T19:40:28","modified_gmt":"2014-12-19T18:40:28","slug":"aspects-of-the-heisenberg-group","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2011\/10\/08\/aspects-of-the-heisenberg-group\/","title":{"rendered":"Aspects of the Heisenberg group"},"content":{"rendered":"

\"Unit<\/a><\/p>\n

The Heisenberg group<\/a> is a remarkable simple mathematical object, with interesting algebraic, geometric, and probabilistic aspects. It is available in tow flavors: discrete and continuous. The (continuous) Heisenberg group \\( {\\mathbb{H}} \\) is formed by the real \\( {3\\times 3} \\) matrices of the form<\/p>\n

\\[ \\begin{pmatrix} 1 & x & z \\\\ 0 & 1 & y \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix}, \\quad x,y,z\\in\\mathbb{R}. \\]<\/p>\n

The Heisenberg group is a non-commutative sub-group of \\( {\\mathrm{GL}_3(\\mathbb{R})} \\):<\/p>\n

\\[ \\begin{pmatrix} 1 & x & z \\\\ 0 & 1 & y \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix} \\begin{pmatrix} 1 & x' & z' \\\\ 0 & 1 & y' \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix} = \\begin{pmatrix} 1 & x+x' & z+z'+xy' \\\\ 0 & 1 & y+y' \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix} \\]<\/p>\n

The inverse is given by<\/p>\n

\\[ \\begin{pmatrix} 1 & x & z \\\\ 0 & 1 & y \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix}^{-1} = \\begin{pmatrix} 1 & -x & -z+xy \\\\ 0 & 1 & -y \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix}. \\]<\/p>\n

(the discrete Heisenberg group is the discrete sub-group of \\( {\\mathbb{H}} \\) formed by the elements of \\( {\\mathbb{H}} \\) with integer coordinates). The Heisenberg group \\( {\\mathbb{H}} \\) is a Lie group<\/a>. Its Lie algebra<\/a> \\( {\\mathfrak{H}} \\) is the sub-algebra of \\( {\\mathcal{M}_3(\\mathbb{R})} \\) given by the \\( {3\\times 3} \\) real matrices of the form<\/p>\n

\\[ \\begin{pmatrix} 0 & a & c \\\\ 0 & 0 & b \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix}, \\quad a,b,c\\in\\mathbb{R} \\]<\/p>\n

The exponential map \\( {\\exp:A\\in\\mathcal{L}\\mapsto\\exp(A)\\in\\mathbb{H}} \\) is a diffeomorphism. This allows to identify the group \\( {\\mathbb{H}} \\) with the algebra \\( {\\mathfrak{H}} \\). Let us define<\/p>\n

\\[ X= \\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix}, \\quad Y= \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix}, \\quad\\text{and}\\quad Z= \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix}. \\]<\/p>\n

We have then<\/p>\n

\\[ [X,Y]=XY-YX=Z\\quad\\text{and}\\quad [X,Z]=[Y,Z]=0. \\]<\/p>\n

The Lie algebra \\( {\\mathfrak{H}} \\) is nilpotent of order \\( {2} \\):<\/p>\n

\\[ \\mathfrak{H}=\\mathrm{span}(X,Y)\\oplus\\mathrm{span}(Z). \\]<\/p>\n

This makes the Baker-Campbell-Hausdorff formula<\/a> on \\( {\\mathfrak{H}} \\) particularly simple:<\/p>\n

\\[ \\exp(A)\\exp(B)=\\exp\\left(A+B+\\frac{1}{2}[A,B]\\right). \\]<\/p>\n

The names Heisenberg group<\/em> and Heisenberg algebra<\/em> come from the fact that in quantum physics, and following Werner Heisenberg<\/a> and Hermann Weyl<\/a>, the algebra generated by the position operator and the momentum operator is exactly \\( {\\mathfrak{H}} \\). The identification of \\( {\\mathbb{H}} \\) with \\( {\\mathfrak{H}} \\)<\/p>\n

\\[ \\begin{pmatrix} 1 & a & c \\\\ 0 & 1 & b \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix} \\equiv \\exp \\begin{pmatrix} 0 & x & z \\\\ 0 & 0 & y \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix} =\\exp(xX+yY+zZ) \\]<\/p>\n

allows to identify \\( {\\mathbb{H}} \\) with \\( {\\mathbb{R}^3} \\) equipped with the group structure<\/p>\n

\\[ (x,y,z)(x',y',z')=(x+x',y+y',z+z'+\\frac{1}{2}(xy'-yx')) \\]<\/p>\n

and<\/p>\n

\\[ (x,y,z)^{-1}=(-x,-y,-z). \\]<\/p>\n

This is the exponential coordinates of \\( {\\mathbb{H}} \\). Geometrically, the quantity \\( {\\frac{1}{2}(xy'-yx')} \\) is the algebraic area<\/b> in \\( {\\mathbb{R}^2} \\) between the piecewise linear path<\/p>\n

\\[ [(0,0),(x,y)]\\cup[(x,y),(x+x',y+y')] \\]<\/p>\n

and its chord<\/p>\n

\\[ [(0,0),(x+x',y+y')]. \\]<\/p>\n

This area is zero if \\( {(x,y)} \\) and \\( {(x',y')} \\) are colinear. The group product encodes the sum of increments<\/b> in \\( {\\mathbb{R}^2} \\) and computes automatically the generated area<\/b>.<\/p>\n

The Heisenberg group \\( {\\mathbb{H}} \\) is topologically homeomorphic to \\( {\\mathbb{R}^3} \\) and the Lebesgue measure on \\( {\\mathbb{R}^3} \\) is a Haar measure<\/a> on \\( {\\mathbb{H}} \\). However, as a manifold, its geometry is sub-Riemannian<\/b>: the tangent space (at the origin and thus everywhere) is of dimension \\( {2} \\) instead of \\( {3} \\), putting a constraint on the geodesics (due to the lack of vertical speed vector, some of them are helices instead of straight lines). The Heisenberg group \\( {\\mathbb{H}} \\) is also a metric space for the so called Carnot-Carath\u00e9odory sub-Riemannian<\/a> distance. The Heisenberg group is a Carnot group<\/a>. Its Hausdorff dimension with respect to the Carnot-Carath\u00e9odory metric is \\( {4} \\), in contrast with its dimension as a topological manifold which is \\( {3} \\).<\/p>\n

The dilation semigroup of automorphisms \\( {(\\mathrm{dil}_t)_{t\\geq0}} \\) on \\( {\\mathbb{H}} \\) is defined by<\/p>\n

\\[ \\mathrm{dil}_t \\exp \\begin{pmatrix} 0 & x & z \\\\ 0 & 0 & y \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix} = \\exp \\begin{pmatrix} 0 & tx & t^2z \\\\ 0 & 0 & ty \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix} . \\]<\/p>\n

Let \\( {(x_n,y_n)_{n\\geq0}} \\) be the simple random walk<\/b> on \\( {\\mathbb{Z}^2} \\) starting from the origin. If one embed \\( {\\mathbb{Z}^2} \\) into \\( {\\mathbb{H}} \\) by \\( {(x,y)\\mapsto xX+yY} \\) then one can consider the position at time \\( {n} \\) in the group by taking the product of increments in the group:<\/p>\n

\\[ \\begin{array}{rcl} S_n=(x_1,y_1)\\cdots(x_n,y_n) &=&(s_{n,1},s_{n,2},s_{n,3}) \\\\ &=&(x_1+\\cdots+x_n,y_1+\\cdots+y_n,s_{n,3}). \\end{array} \\]<\/p>\n

These increments are commutative for the first two coordinates (called the horizontal coordinates) and non commutative for the third coordinate. The first two coordinates of \\( {S_n} \\) form the position in \\( {\\mathbb{Z}^2} \\) of the random walk while the third coordinate is exactly the algebraic area between the random walk path and its chord on the time interval \\( {[0,n]} \\). We are now able to state the Central Limit Theorem<\/b> on the Heisenberg group:<\/p>\n

\\[ \\left(\\mathrm{dil}_{1\/\\sqrt{n}}(S_{\\lfloor nt\\rfloor})\\right)_{t\\geq0} \\quad \\underset{n\\rightarrow\\infty}{\\overset{\\text{law}}{\\longrightarrow}} \\quad \\left(B_t,A_t\\right)_{t\\geq0} \\]<\/p>\n

where \\( {(B_t)_{t\\geq0}} \\) is a simple Brownian motion<\/b> on \\( {\\mathbb{R}^2} \\) and where \\( {(A_t)_{t\\geq0}} \\) is its Lévy area<\/b> (algebraic area between the Brownian path and its chord, seen as a stochastic integral). The stochastic process \\( {(\\mathbb{B}_t)_{t\\geq0}=((B_t,A_t))_{t\\geq0}} \\) is the sub-Riemannian Brownian motion on \\( {\\mathbb{H}} \\).<\/p>\n

\\[ \\begin{array}{rcl} \\mathbb{B}_t &=& (B_t,A_t) \\\\ &=&(B_{t,1},B_{t,2},A_t) \\\\ &=& \\exp \\begin{pmatrix} 0 & B_{t,1} & \\frac{1}{2}\\left(\\int_0^t\\!B_{s,1}dB_{s,2}-\\int_0^t\\!B_{t,2}dB_{t,1}\\right) \\\\ 0 & 0 & B_{t,2} \\\\ 0 & 0 & 0 \\end{pmatrix} \\\\ &=& \\begin{pmatrix} 1 & B_{t,1} & \\int_0^t\\!B_{s,1}dB_{s,2} \\\\ 0 &1 & B_{t,2} \\\\ 0 &0 &1 \\end{pmatrix}. \\end{array} \\]<\/p>\n

The process \\( {(\\mathbb{B}_t)_{t\\geq0}} \\) has independent and stationary (non-commutative) increments and belong the class of Lévy processes, associated to (non-commutative) convolution semigroups on \\( {\\mathbb{H}} \\). The law of \\( {\\mathbb{B}_t} \\) is infinitely divisible and maybe seen as a sort of Gaussian measure on \\( {\\mathbb{H}} \\). The process \\( {(\\mathbb{B}_t)_{t\\geq0}} \\) is also a Markov diffusion process on \\( {\\mathbb{R}^3} \\) admitting the Lebesgue measure as an invariant reversible measure, and with infinitesimal generator<\/p>\n

\\[ L=\\frac{1}{2}(V_1^2+V_2^2)=\\frac{1}{2}\\left((\\partial_x-\\frac{1}{2}y\\partial_z)^2+(\\partial_y+\\frac{1}{2}x\\partial_z)^2\\right). \\]<\/p>\n

We have \\( {V_3:=[V_1,V_2]=\\partial_z} \\) and \\( {[V_1,V_3]=[V_2,V_3]=0} \\). The linear differential operator \\( {L} \\) on \\( {\\mathbb{R}^3} \\) is hypoelliptic<\/b> but is not elliptic<\/b>. It is called the sub-Laplacian<\/b> on \\( {\\mathbb{H}} \\). A formula for its heat kernel was computed by Lévy using Fourier analysis (it is an oscillatory integral).<\/p>\n

Note that \\( {L} \\) acts like the two dimensional Laplacian on functions depending only on \\( {x,y} \\). Note also that the Riemannian Laplacian on \\( {\\mathbb{H}} \\) is given by<\/p>\n

\\[ L+\\frac{1}{2}V_3^2=\\frac{1}{2}\\left(V_1^2+V_2^2+V_3^3\\right) =\\frac{1}{2}\\left((\\partial_x-\\frac{1}{2}y\\partial_z)^2+(\\partial_y+\\frac{1}{2}x\\partial_z)^2+(\\partial_z)^2\\right). \\]<\/p>\n

Open question.<\/b> Use the CLT to obtain a sub-Riemannian Poincaré inequality or even a logarithmic Sobolev inequality on \\( {\\mathbb{H}} \\) for the heat kernel. It is tempting to try to adapt to the sub-Riemannian case the strategy used by L. Gross<\/a> (for Riemannian Lie groups). This question is naturally connected to my previous work on gradient bounds for the heat kernel on the Heisenberg group<\/a>, in collaboration with D. Bakry, F. Baudoin, and M. Bonnefont.<\/p>\n

Related reading.<\/b> (among many other references)<\/p>\n