Speaker
Description
For $N\in \{ 1,2,3 \}$ there is considered the semi-direct sum $ \tilde {\mathcal G} \propto \tilde {\mathcal G} _{reg}^*$ of the loop Lie algebra $ \tilde {\mathcal G}$, consisting of the even left superconformal vector fields on a supercircle $\mathbb S^{1|N}$ in the form $\tilde a := a \partial / \partial x + \frac 12 \sum \nolimits _{i=1}^N (D_{\vartheta_i} a)D_{\vartheta_i},$ where $ a:=a(x,\vartheta;\lambda) \in $ $C ^{\infty}(\mathbb S^{1|N} \times (\mathbb D^1_+ \cup \mathbb D^1_-);\Lambda_0) $ is holomorphic in the "spectral" parameter $\lambda\in $ $\mathbb D^1_+\cup \mathbb D^1_- \subset \mathbb C$, $ \mathbb D^1_+, \mathbb D^1_-$ are the interior and exterior regions of the unit centrally located disk $\mathbb D^1 \subset \mathbb C$ respectively, $a(x,\vartheta; \infty) = 0 $, $(x,\vartheta )\in \mathbb S^{1|N} \simeq \mathbb S^1 \times \Lambda _1^N$, $\Lambda: = \Lambda_0 \oplus \Lambda_1$ is a commutative Banach superalgebra over the field $\mathbb C \subset \Lambda _0$, $\partial / \partial x$ is a partial derivative by the commuting variable $x$, $\vartheta:=(\vartheta_1, \vartheta_2, \ldots , \vartheta_N)$, $\partial / \partial \vartheta_i$ is a left partial derivative by the anticommuting variable $\vartheta_i \in \Lambda _1$, $D_{\vartheta_i}:= $ $\partial / \partial \vartheta_i + \vartheta_i \partial / \partial x $, $i=\overline{1,N}$, and its regular dual space $\tilde {\mathcal G} _{reg}^*$ with respect to the parity $(\tilde a, \tilde l )_0 = \mbox{res}_{\lambda \in \mathbb C} \, \int \nolimits _{\mathbb S ^{1|N}} dx d\vartheta_1 \ldots d\vartheta_N \, ( a l) ,$ where $\mbox{res}_{\lambda \in \mathbb C}$ denotes the coefficient at $\lambda ^{-1}$ in the corresponding Laurent series, $\tilde l \in \tilde {\mathcal G} _{reg}^*$ is a right superdifferential 1-form on $\mathbb S ^{1|N}$ such as $ \tilde l: = (dx - \sum \nolimits _{i=1}^N ( d\vartheta _i ) \vartheta_i ) l(x,\vartheta ;\lambda ) \in \tilde {\mathcal G} _{reg}^* ,$ $l:=l(x,\vartheta;\lambda)\in $ $ C^{\infty }(\mathbb S^{1|N}\times (\mathbb D^1_+\cup \mathbb D^1_-);\Lambda _{s})$ is holomorphic in the "spectral" parameter $\lambda\in $ $\mathbb D^1_+ \cup \mathbb D^1_-$, $l(x,\vartheta;\infty )=0$, $s=1$ if $N$ is an odd natural number and $s=0$ if $N$ is an even one. The loop Lie algebra $\tilde {\mathcal G}$ is splitting into the direct sum $ \tilde {\mathcal G} = \tilde {\mathcal G}_+ \oplus \tilde {\mathcal G}_-$ of its Lie subalgebras for which $\tilde {\mathcal G}_{+,reg}^* \simeq \tilde {\mathcal G}_- $, $\tilde {\mathcal G}_{-,reg}^* \simeq \tilde {\mathcal G}_+$, where $ a(x,\vartheta ;\infty )=0$ for any $\tilde a \in \tilde {\mathcal G}_- $. On $ \tilde {\mathcal G} \propto \tilde {\mathcal G} _{reg}^*$ one determines the commutator $ [\tilde a \propto \tilde l , \tilde b \propto \tilde m ]: = $ $ [\tilde a ,\tilde b] \propto (ad^*_{\tilde a} \tilde m - ad^*_{\tilde b}\tilde l)$ for any $ \tilde a , \tilde b \in \tilde {\mathcal G}$ and $\tilde l , \tilde m \in \tilde {\mathcal G} _{reg}^*$, where $ [\tilde a , \tilde b] : = $ $ \tilde c $, $ \tilde c \in \tilde {\mathcal G} $, $c:= a (\partial b /\partial x ) - b (\partial a /\partial x) + \frac 12 \sum \nolimits _{i=1}^N (D_{\vartheta_i} a)(D_{\vartheta_i}b)$, $ ad^*$ is the coadjoint action of $\tilde {\mathcal G}$ with respect to the parity $(.,.)_0$, as well as the symmetric bilinear form $(\tilde a \propto \tilde l, \tilde b \propto \tilde m ) _0 = (\tilde a , \tilde m )_0 + (\tilde b, \tilde l)_0 $. One constructs the central extensions $\hat {\mathfrak G}:= \tilde {\mathfrak G} \oplus \mathbb C^2$ of the Lie algebra $\tilde {\mathfrak G}:= \prod_{z\in\mathbb{S}^{1}} ( \tilde {\mathcal G} \propto \tilde {\mathcal G} _{reg}^*) $ by the superanalogs of the Ovsienko-Roger 2-cocycle such as $\omega_2 (\tilde a \propto \tilde l, \tilde b \propto \tilde m ): = $ $ (\omega^1_2 (\tilde a \propto \tilde l, \tilde b \propto \tilde m ), $ $ \omega^2_2 (\tilde a \propto \tilde l, \tilde b \propto \tilde m )),$ where $ \omega^1_2 (\tilde a \propto \tilde l, \tilde b \propto \tilde m ) = \mbox{res}\, \int \nolimits _{\mathbb S ^1 } dz \, \int \nolimits _{\mathbb S ^{1|N}} dx d^N\vartheta \, ( a (\mathcal P b)) ,$ $\omega^2_2 (\tilde a \propto \tilde l, \tilde b \propto \tilde m ) = \int \nolimits _{\mathbb S ^1 } dz \, ( ( a , \partial m /\partial z )_0 - ( b , (\partial l /\partial z )_0 ),$ $(\tilde a \propto \tilde l), (\tilde b \propto \tilde m) \in $ $ \tilde {\mathfrak G}$, $z\in \mathbb S ^1$, and $\mathcal P=D_{\vartheta_1}\partial^2 /\partial^2 x$ when $N=1$, $\mathcal P=D_{\vartheta_1} D_{\vartheta_2} \partial /\partial x $ when $N=2$, $\mathcal P=D_{\vartheta_1} D_{\vartheta_2} D_{\vartheta_3}$ when $N=3$.
Since the Lie algebra $\tilde {\mathfrak G}$ permits the standard splitting $\tilde {\mathfrak G}:= \tilde {\mathfrak G}_+ \oplus
\tilde {\mathfrak G}_-$ into a direct sum of its Lie subalgebras $\tilde {\mathfrak G}_+: = \prod_{z\in\mathbb{S}^{1}} (\tilde
{\mathcal G}_+ \propto \tilde {\mathcal G} _{-,reg}^*)$ and $\tilde {\mathfrak G}_-: = $ $ \prod_{z\in\mathbb{S}^{1}} (\tilde {\mathcal G}_- \propto \tilde {\mathcal G} _{+,reg}^*))$, on its dual space $\tilde {\mathfrak G}^* $ with respect to the symmetric bilinear form $\langle .,. \rangle _0 : = \int \nolimits _{\mathbb S ^1 } dz \, (.,.) _0 $ one can introduce for any smooth by Frechet functionals $\mu, \nu \in \mathcal D(\tilde {\mathfrak G}^*)$ the $\mathcal R$-deformed Lie-Poisson bracket $ \{\mu , \nu \}_{\mathcal R} (\tilde a \propto \tilde l) = $ $
\langle \tilde a \propto \tilde l, [R \nabla_r \mu (\tilde a
\propto \tilde l), \nabla_l \nu (\tilde a \propto \tilde l)] +
[\nabla_r \mu (\tilde a \propto \tilde l), R\nabla_l \nu (\tilde a
\propto \tilde l)] \rangle _0 + $ $
< e, \omega_2 (R \nabla_r \mu (\tilde a \propto \tilde l),
\nabla_l \nu (\tilde a \propto \tilde l)) + \omega_2 (\nabla_r \mu
(\tilde a \propto \tilde l), R\nabla_l \nu (\tilde a \propto
\tilde l)) >$,
where $e=(e_1,e_2) \in \mathbb C^2$, the brackets $<.,.>$ denote the scalar product on $\mathbb C^2$, $\mathcal R=(P_+-P_-)/2$, $P_+$ and $P_-$ are projectors on $\tilde {\mathfrak G} _+$ and $\tilde {\mathfrak G} _-$ respectively, $\nabla_l h (\tilde a \propto \tilde l):= (\nabla_l h_{\tilde l} \propto \nabla_l h_{\tilde a}) \in \tilde {\mathfrak G}$ and $\nabla_r h (\tilde a \propto \tilde l):= (\nabla_r h_{\tilde l} \propto \nabla_r
h_{\tilde a}) \in \tilde {\mathfrak G}$ are left and right gradients of an arbitrary smooth functional $h\in \mathcal D(\tilde {\mathfrak G}^*)$ at $(\tilde a \propto \tilde l) \in \tilde {\mathfrak G}^*$, which due to the Adler-Kostant-Symes theory generates the hierarchy of Hamiltonian flows on $\tilde {\mathfrak G}^* \simeq \tilde {\mathfrak G}$ in the form $\partial (\tilde a \propto \tilde l)/\partial t_p = $ $ \{ \tilde a \propto \tilde l , h^{(p)} (\tilde a \propto \tilde l) \}_{\mathcal R}, $ $p\in \mathbb Z_+,$ where $h^{(p)} (\tilde a \propto \tilde l ) = \lambda ^{p} h (\tilde a \propto \tilde l )$, for the Casimir invariant $h\in I(\hat {\mathfrak G}^*)$. The reductions of this hierarchy on polynomial type coadjoint orbits of the Lie algebra $\hat {\mathfrak G}$ are shown to lead to hierarchies of compatibly bi-Hamiltonian $(2|N+1)$-dimensional systems on functional supermanifolds.
