### 统计代写|随机分析作业代写stochastic analysis代写|Current Representations of Lie Algebras

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## 统计代写|随机分析作业代写stochastic analysis代写|Current Representations of Lie Algebras

Intuitively, if ${\mathcal{L},[*,], *$,$} is a -Lie algebra, a current algebra of \mathcal{L}$ over $\mathbb{R}^{d}$ is a vector space $\mathcal{T}$ of $\mathcal{L}$-valued functions defined on $\mathbb{R}^{d}$ and closed under the pointwise operations: $$\varphi, \psi:=[\varphi(t), \psi(t)] ; \quad \varphi^{}(t):=\varphi(t)^{} ; \quad t \in \mathbb{R}, \varphi \in \mathcal{T}$$ For example, if $X_{1}, \ldots, X_{k}$ are generators of $\mathcal{L}$ one can fix a space $\mathcal{S}$, of complex valued test functions on $R$ and to each $\varphi \in \mathcal{S}$ and $j \in{1, \ldots, k}$ one can associate the $\mathcal{L}$-valued function on $\mathbb{R} X_{j}(\varphi)$ defined by: $$X_{j}(\varphi)(t):=\varphi(t) X_{j} ; \quad t \in \mathbb{R}$$ Definition 6. Let $\mathcal{G}$ be a complex-Lie algebra. A (canonical) set of generators of $\mathcal{G}$ is a linear basis of $\mathcal{G}$
$$l_{\alpha}^{+}, l_{\alpha}^{-}, l_{\beta}^{0}, \alpha \in I, \quad \beta \in I_{0}$$
where $I_{0}, I$ are sets, satzsfyng the following conditsons:
$$\begin{array}{ll} \left(l_{\beta}^{0}\right)^{+}=l_{\beta}^{0} ; & \forall \beta \in I_{0} \ \left(l_{\alpha}^{+}\right)^{*}=l_{\alpha}^{-} ; & \forall \alpha \in I \end{array}$$
and all the central elements among the generators are of $l^{0}$-type (i.e. selfadjoint).

We will denote $c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime}, \delta\right)$ the structure constants of $\mathcal{G}$ with respect to the generators $\left(l_{\alpha}^{\varepsilon}\right)$, i.e., with $\alpha, \beta \in I \cup I_{0}, \varepsilon, e^{\prime}, \delta \in{+,-, 0}$, and, assuming summation over repeated indices:
$$\begin{gathered} {\left[l_{\alpha,}^{z}, l_{\beta}^{\varepsilon^{\prime}}\right]=c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime}, \delta\right) l_{\gamma}^{\delta}=} \ :=\sum_{\gamma \in I_{0}} c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime}, 0\right) l_{\gamma}^{0}+\sum_{\gamma \in I} c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime},+\right) l_{\gamma}^{+}+\sum_{\gamma \in I} c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime},-\right) l_{\gamma}^{-} \end{gathered}$$
In the following we will consider only locally finite Lie algebras, i.e. those such that, for any pair $\alpha, \beta \in I \cup I_{0}$ only a finite number of the structure constants $c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime}, \delta\right)$ is different from zero.

## 统计代写|随机分析作业代写stochastic analysis代写|Connections with Classical Independent Increment Processes

In this section we look for some necessary conditions for the solution of the problem stated in the previous section. This will naturally lead to an interesting connection with the theory of classical independent increment processes which was first noticed in Araki’s thesis [Arak60]. We refer to the monographs of K.R. Parthasarathy and K. Schmidt [PaSch72] and of Guichardet [Gui72] for a systematic exposition. In the notations of Section (18) we consider:

• a pair $\left{\mathcal{G},\left(l_{\alpha}^{2}\right)\right}$ of a *-Lie algebra and a set of generators which admits a Fock representation.
• a measure space $(S, \mu)$
• a *-sub-algebra $\mathcal{C} \subseteq L_{\mathrm{C}}^{\infty}(S, \mathcal{B}, \mu)$
such that the current algebra
$$\left{l_{\alpha}^{c}(f): \varepsilon \in{+,-, 0}, \alpha \in I \text { or } \alpha \in I_{0}, f \in \mathcal{C}\right}$$
admits a Fock representation on some Hilbert space $\mathcal{H}$ with cyclic vector $\Phi$. We identify the elements of this current algebra with their images in this representation and we omit from the notation the symbol $\pi$ of the representation. Moreover we add the following assumptions:
(i) among the generators $\left(l_{\alpha}^{c}\right)$ there is exactly one (self-adjoint) central element, denoted $l_{0}^{0}$.
(ii) for any $f \in \mathcal{C}$ one has:
$$l_{0}^{0}(f)=\int_{S} f d \mu$$
where the scalar on the right hand side is identified to the corresponding multiple of the identity operator on $\mathcal{H}$. In particular the representation is weakly irreducible.

Under these conditions it is not difficult to see that the general principle that algebra implies statistics can be applied and that the vacuum mixed moments of the operators $l_{\alpha}^{\varepsilon}(f)$ are uniquely determined by the structure constants of the Lie algebra. Another important property is that, by fixing a measurable subset $I \subseteq S$ such that
$$\mu(I)=1$$
and denoting $\chi_{I}$ the corresponding characteristic function, the $*$-Lie algebra generated by the operators $l_{\alpha}^{\varepsilon}\left(\chi_{I}\right)$ is isomorphic to $\mathcal{G}$ and therefore it has the same vacuum statistics.

Finally the commutation relations (18.1) imply that the maps $f \mapsto l_{\alpha}^{\varepsilon}(f)$ define an independent increment process of boson type, i.e. the restriction of the vacuum state on the polynomial algebra generated by two families

$\left(l_{\alpha}^{\varepsilon}(f)\right){\varepsilon, \alpha}$ and $\left(l{\alpha}^{\varepsilon}(g)\right)_{\varepsilon, \alpha}$ with $f$ and $g$ having disjoint supports, coincides with the tensor product of the restrictions on the single algebras.

In particular, if $X(I)$ is any self-adjoint linear combination of operators of the form $l_{\alpha}^{\varepsilon}\left(\chi_{I}\right)$, then the map $I \subseteq S \mapsto X(I)$ defines an additive independent increment process on $(S, \mathcal{B}, \mu)$. Thus the law of every random variable of the form $X(I)$ will be an infinitely divisible law on $\mathbb{R}$ whenever the set $I$ can be written as a countable union of subsets of nonzero $\mu$-measure.

If $S=\mathbb{R}^{d}$ and $\mu$ is the Lebesgue measure, then any such process $X(I)$ $\left(I \subseteq \mathbb{R}^{d}\right)$ will also be translation invariant.

Combining together all the above remarks one obtains a necessary condition for the existence of the Fock representation of the current algebra of a *-Lie algebra $\mathcal{G}$ and a set of generators namely: the pair $\left{\mathcal{G},\left(l_{\alpha}^{\varepsilon}\right)\right}$ must admit a Fock representation and the vacuum distribution of any self-adjoint linear combination $X$ of generators must be infinitely divisible

Since there is no reason to expect that any pair $\left{\mathcal{G},\left(l_{\alpha}^{z}\right)\right}$ will have this property, this gives a probabilistic intuition of the reason why it might happen that a *-Lie algebra and a set of generators $\left{\mathcal{G},\left(l_{\alpha}^{z}\right)\right}$ might admit a Fock representation without this being true for the associated current algebra.
In the following section we review some progresses made in the past few years in one important special case: the full oscillator algebra.

## 统计代写|随机分析作业代写stochastic analysis代写|

We have seen how the developments reviewed in the previous sections naturally lead to the following problem: can we extend to the renormalized higher powers of quantum white noise what has been achieved for the second powers? To answer this question we start with the Heisenberg algebra
$$\left[a, a^{+}\right]=1$$
Its universally enveloping algebra is generated by the products of monomials of the form
$$a^{n}, a^{+m}$$
and their commutation relations are deduced from (20.1) and the derivation property of the commutator. The problem we want to study is the following: does there exist a current representation of this algebra over $\mathbb{R}^{d}$ for some $d>0$ ?

In order to define the current algebra of the full oscillator algebra, we have first to overcome the renormalization problem, illustrated in Section (14) in the case of the second powers of white noise. In fact, dealing with higher powers of white noise we meet higher powers of the $\delta$-function. A natural way out is to write
$$\delta^{n}=\delta^{2}\left(\delta^{n-2}\right) ; \quad n \geq 2 ; \quad \delta^{0}:=1$$

and to apply iteratively the renormalization prescription used in Section (14). This leads to the following:

Definition. The boson Fock white noise, renormalized with the prescription:
$$\delta(t)^{l}=c^{l-1} \delta(t), c>0, l=2,3, \ldots$$
simply called $R B F W N$ in the following, over a Hilbert space $\mathcal{H}$ with vacuum (unit) vector $\Phi$ is the locally finite *-Lie algebra canonically associated to the associative unital *-algebra of operator-valued distributions on $\mathcal{H}$ with generators
$$b_{t}^{+n} b_{t}^{k}, \quad k, n \in \mathbb{N}, \quad t \in \mathbb{R}^{d}$$
and relations deduced from:
$$\begin{gathered} {\left[b_{t}, b_{s}^{+}\right]=\delta(t-s)} \ {\left[b_{t}^{+}, b_{s}^{+}\right]=\left[b_{t}, b_{s}\right]=0} \ \left(b_{s}\right)^{*}=b_{s}^{+} \ b_{t} \Phi=0 \end{gathered}$$
Here locally finite méans thàt the commutator of any pair of generators is a finite linear combination of generators.

## 统计代写|随机分析作业代写stochastic analysis代写|Current Representations of Lie Algebras

[l一种,和,lbe′]=C一种bC(e,e′,d)lCd= :=∑C∈一世0C一种bC(e,e′,0)lC0+∑C∈一世C一种bC(e,e′,+)lC++∑C∈一世C一种bC(e,e′,−)lC−

## 统计代写|随机分析作业代写stochastic analysis代写|Connections with Classical Independent Increment Processes

• 一双\left{\mathcal{G},\left(l_{\alpha}^{2}\right)\right}\left{\mathcal{G},\left(l_{\alpha}^{2}\right)\right}*-Lie 代数和一组接受 Fock 表示的生成器。
• 测度空间(小号,μ)
• *-子代数C⊆大号C∞(小号,乙,μ)
使得当前代数
\left{l_{\alpha}^{c}(f): \varepsillon \in{+,-, 0}, \alpha \in I \text { 或 } \alpha \in I_{0}, f \in \mathcal{C}\右}\left{l_{\alpha}^{c}(f): \varepsillon \in{+,-, 0}, \alpha \in I \text { 或 } \alpha \in I_{0}, f \in \mathcal{C}\右}
承认某个希尔伯特空间上的 Fock 表示H带循环向量披. 我们在这个表示中用它们的图像来识别这个当前代数的元素，我们从符号中省略了符号圆周率的表示。此外，我们添加了以下假设：
（i）在生成器中(l一种C)有一个（自伴的）中心元素，记为l00.
(ii) 对于任何F∈C一个有：
l00(F)=∫小号Fdμ
其中右侧的标量被标识为对应的身份运算符的倍数H. 特别是表示是弱不可约的。

μ(一世)=1

$\left(l_{\alpha}^{\varepsilon}(f)\right) {\varepsilon, \alpha}一种nd\left(l {\alpha}^{\varepsilon}(g)\right)_{\varepsilon, \alpha}在一世吨HF一种ndg$ 具有不相交的支持，与单个代数限制的张量积一致。

## 统计代写|随机分析作业代写stochastic analysis代写|

[一种,一种+]=1

dn=d2(dn−2);n≥2;d0:=1

d(吨)l=Cl−1d(吨),C>0,l=2,3,…

b吨+nb吨ķ,ķ,n∈ñ,吨∈Rd

[b吨,bs+]=d(吨−s) [b吨+,bs+]=[b吨,bs]=0 (bs)∗=bs+ b吨披=0

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