幾何レヴィ過程に対する局所リスク最小化戦略とその数値解析的研究

早大学位記番号:新7274早稲田大

On a quaternification of complex Lie algebras

We give a definition of quaternion Lie algebra and of the quaternification of a complex Lie algebra. By our definition gl(n,H), sl(n,H), so*(2n) ans sp(n) are quaternifications of gl(n,C), sl(n,C), so(n,C) and u(n) respectively. Then we shall prove that a simple Lie algebra admits the quaternification. For the proof we follow the well known argument due to Harich-Chandra, Chevalley and Serre to construct the simple Lie algebra from its corresponding root system. The root space decomposition of this quaternion Lie algebra will be given. Each root sapce of a fundamental root is complex 2-dimensional

Quaternifications and Extensions of Current Algebras on S3

Let (mathbf{H}) be the quaternion algebra. Let (mathfrak{g}) be a complex Lie algebra and let (U(mathfrak{g})) be the enveloping algebra of (mathfrak{g}). The quaternification (mathfrak{g}^{mathbf{H}}=)(,(,mathbf{H}otimes U(mathfrak{g}),,[quad,quad]_{mathfrak{g}^{mathbf{H}}},)) of (mathfrak{g}) is defined by the bracket ( big[,mathbf{z}otimes X,,,mathbf{w}otimes Y,big]_{mathfrak{g}^{mathbf{H}}},=)(,(mathbf{z}cdot mathbf{w})otimes,(XY),- )(, (mathbf{w}cdotmathbf{z})otimes (YX),,nonumber ) for (mathbf{z},,mathbf{w}in mathbf{H}) and {the basis vectors (X) and (Y) of (U(mathfrak{g})).} Let (S^3mathbf{H}) be the ( non-commutative) algebra of (mathbf{H})-valued smooth mappings over (S^3) and let (S^3mathfrak{g}^{mathbf{H}}=S^3mathbf{H}otimes U(mathfrak{g})). The Lie algebra structure on (S^3mathfrak{g}^{mathbf{H}}) is induced naturally from that of (mathfrak{g}^{mathbf{H}}). We introduce a 2-cocycle on (S^3mathfrak{g}^{mathbf{H}}) by the aid of a tangential vector field on (S^3subset mathbf{C}^2) and have the corresponding central extension (S^3mathfrak{g}^{mathbf{H}} oplus(mathbf{C}a)). As a subalgebra of (S^3mathbf{H}) we have the algebra of Laurent polynomial spinors (mathbf{C}[phi^{pm}]) spanned by a complete orthogonal system of eigen spinors ({phi^{pm(m,l,k)}}_{m,l,k}) of the tangential Dirac operator on (S^3). Then (mathbf{C}[phi^{pm}]otimes U(mathfrak{g})) is a Lie subalgebra of (S^3mathfrak{g}^{mathbf{H}}). We have the central extension (widehat{mathfrak{g}}(a)= (,mathbf{C}[phi^{pm}] otimes U(mathfrak{g}) ,) oplus(mathbf{C}a)) as a Lie-subalgebra of (S^3mathfrak{g}^{mathbf{H}} oplus(mathbf{C}a)). Finally we have a Lie algebra (widehat{mathfrak{g}}) which is obtained by adding to (widehat{mathfrak{g}}(a)) a derivation (d) which acts on (widehat{mathfrak{g}}(a)) by the Euler vector field (d_0). That is the (mathbf{C})-vector space (widehat{mathfrak{g}}=left(mathbf{C}[phi^{pm}]otimes U(mathfrak{g})right)oplus(mathbf{C}a)oplus (mathbf{C}d)) endowed with the bracket ( bigl[,phi_1otimes X_1+ lambda_1 a + mu_1d,,phi_2otimes X_2 + lambda_2 a + mu_2d,,bigr]_{widehat{mathfrak{g}}} , =)( (phi_1phi_2)otimes (X_1,X_2) , -,(phi_2phi_1)otimes (X_2X_1)+mu_1d_0phi_2otimes X_2- ) (mu_2d_0phi_1otimes X_1 + ) ( (X_1vert X_2)c(phi_1,phi_2)a,. ) When (mathfrak{g}) is a simple Lie algebra with its Cartan subalgebra (mathfrak{h}) we shall investigate the weight space decomposition of (widehat{mathfrak{g}}) with respect to the subalgebra (widehat{mathfrak{h}}= (phi^{+(0,0,1)}otimes mathfrak{h} )oplus(mathbf{C}a) oplus(mathbf{C}d))

Quaternifications and Extensions of Current Algebras on S3

Let $\mathbf{H}$ be the quaternion algebra. Let $\mathfrak{g}$ be a complex Lie algebra and let $U(\mathfrak{g})$ be the enveloping algebra of $\mathfrak{g}$. The quaternification $\mathfrak{g}^{\mathbf{H}}=$$\,(\,\mathbf{H}\otimes U(\mathfrak{g}),\,[\quad,\quad]_{\mathfrak{g}^{\mathbf{H}}}\,)$ of $\mathfrak{g}$ is defined by the bracket $\big[\,\mathbf{z}\otimes X\,,\,\mathbf{w}\otimes Y\,\big]_{\mathfrak{g}^{\mathbf{H}}}\,=$$\,(\mathbf{z}\cdot \mathbf{w})\otimes\,(XY)\,-$$\, (\mathbf{w}\cdot\mathbf{z})\otimes (YX)\,,\nonumber$ for $\mathbf{z},\,\mathbf{w}\in \mathbf{H}$ and {the basis vectors $X$ and $Y$ of $U(\mathfrak{g})$.} Let $S^3\mathbf{H}$ be the ( non-commutative) algebra of $\mathbf{H}$-valued smooth mappings over $S^3$ and let $S^3\mathfrak{g}^{\mathbf{H}}=S^3\mathbf{H}\otimes U(\mathfrak{g})$. The Lie algebra structure on $S^3\mathfrak{g}^{\mathbf{H}}$ is induced naturally from that of $\mathfrak{g}^{\mathbf{H}}$. We introduce a 2-cocycle on $S^3\mathfrak{g}^{\mathbf{H}}$ by the aid of a tangential vector field on $S^3\subset \mathbf{C}^2$ and have the corresponding central extension $S^3\mathfrak{g}^{\mathbf{H}} \oplus(\mathbf{C}a)$. As a subalgebra of $S^3\mathbf{H}$ we have the algebra of Laurent polynomial spinors $\mathbf{C}[\phi^{\pm}]$ spanned by a complete orthogonal system of eigen spinors $\{\phi^{\pm(m,l,k)}\}_{m,l,k}$ of the tangential Dirac operator on $S^3$. Then $\mathbf{C}[\phi^{\pm}]\otimes U(\mathfrak{g})$ is a Lie subalgebra of $S^3\mathfrak{g}^{\mathbf{H}}$. We have the central extension $\widehat{\mathfrak{g}}(a)= (\,\mathbf{C}[\phi^{\pm}] \otimes U(\mathfrak{g}) \,) \oplus(\mathbf{C}a)$ as a Lie-subalgebra of $S^3\mathfrak{g}^{\mathbf{H}} \oplus(\mathbf{C}a)$. Finally we have a Lie algebra $\widehat{\mathfrak{g}}$ which is obtained by adding to $\widehat{\mathfrak{g}}(a)$ a derivation $d$ which acts on $\widehat{\mathfrak{g}}(a)$ by the Euler vector field $d_0$. That is the $\mathbf{C}$-vector space $\widehat{\mathfrak{g}}=\left(\mathbf{C}[\phi^{\pm}]\otimes U(\mathfrak{g})\right)\oplus(\mathbf{C}a)\oplus (\mathbf{C}d)$ endowed with the bracket $\bigl[\,\phi_1\otimes X_1+ \lambda_1 a + \mu_1d\,,\phi_2\otimes X_2 + \lambda_2 a + \mu_2d\,\,\bigr]_{\widehat{\mathfrak{g}}} \, =$$(\phi_1\phi_2)\otimes (X_1\,X_2) \, -\,(\phi_2\phi_1)\otimes (X_2X_1)+\mu_1d_0\phi_2\otimes X_2-$ $\mu_2d_0\phi_1\otimes X_1 +$ $(X_1\vert X_2)c(\phi_1,\phi_2)a\,.$ When $\mathfrak{g}$ is a simple Lie algebra with its Cartan subalgebra $\mathfrak{h}$ we shall investigate the weight space decomposition of $\widehat{\mathfrak{g}}$ with respect to the subalgebra $\widehat{\mathfrak{h}}= (\phi^{+(0,0,1)}\otimes \mathfrak{h} )\oplus(\mathbf{C}a) \oplus(\mathbf{C}d)$