Density wave theory

Template:No footnotes In algebra, a crystal base or canonical base is a base of a representation, such that generators of a quantum group or semisimple Lie algebra have a particularly simple action on it. Crystal bases were introduced by Template:Harvs and Template:Harvs (under the name of canonical bases).

Definition

As a consequence of the defining relations for the quantum group $U_q(G)$, $U_q(G)$ can be regarded as a Hopf algebra over $\mathbb{\mathbb{Q}}(q)$, the field of all rational functions of an indeterminate q over $\mathbb{\mathbb{Q}}$.

For simple root ${\alpha }_i$ and non-negative integer $n$, define $e_i^{(n)}=e_i^n/[n{]}_{q_i}!$ and $f_i^{(n)}=f_i^n/[n{]}_{q_i}!$ (specifically, $e_i^{(0)}=f_i^{(0)}=1$). In an integrable module $M$, and for weight $\lambda$, a vector $u\in M_{\lambda }$ (i.e. a vector $u$ in $M$ with weight $\lambda$) can be uniquely decomposed into the sums

• $u=\sum _{n=0}^{\mathrm{\infty }}{f}_i^{(n)}{u}_n=\sum _{n=0}^{\mathrm{\infty }}{e}_i^{(n)}{v}_n,$

where $u_n\in \mathrm{k}\mathrm{e}\mathrm{r}(e_i)\cap M_{\lambda +n{\alpha }_i}$, $v_n\in \mathrm{k}\mathrm{e}\mathrm{r}(f_i)\cap M_{\lambda -n{\alpha }_i}$, $u_n\ne 0$ only if $n+\frac{2(\lambda ,{\alpha }_i)}{({\alpha }_i,{\alpha }_i)}\ge 0$, and $v_n\ne 0$ only if $n-\frac{2(\lambda ,{\alpha }_i)}{({\alpha }_i,{\alpha }_i)}\ge 0$. Linear mappings ${\tilde{e}}_i:M\to M$ and ${\tilde{f}}_i:M\to M$ can be defined on $M_{\lambda }$ by

• ${\tilde{e}}_{i}u=\sum _{n=1}^{\mathrm{\infty }}{f}_i^{(n-1)}{u}_n=\sum _{n=0}^{\mathrm{\infty }}{e}_i^{(n+1)}{v}_n,$

• ${\tilde{f}}_{i}u=\sum _{n=0}^{\mathrm{\infty }}{f}_i^{(n+1)}{u}_n=\sum _{n=1}^{\mathrm{\infty }}{e}_i^{(n-1)}.v_n$

Let $A$ be the integral domain of all rational functions in $\mathbb{\mathbb{Q}}(q)$ which are regular at $q=0$ (i.e. a rational function $f(q)$ is an element of $A$ if and only if there exist polynomials $g(q)$ and $h(q)$ in the polynomial ring $\mathbb{\mathbb{Q}}[q]$ such that $h(0)\ne 0$, and $f(q)=g(q)/h(q)$). A crystal base for $M$ is an ordered pair $(L,B)$, such that

• $L$ is a free $A$-submodule of $M$ such that $M=\mathbb{\mathbb{Q}}(q){\otimes }_{A}L;$

• $B$ is a $\mathbb{\mathbb{Q}}$-basis of the vector space $L/qL$ over $\mathbb{\mathbb{Q}},$

• $L={\oplus }_{\lambda }{L}_{\lambda }$ and $B={\bigsqcup }_{\lambda }{B}_{\lambda }$, where $L_{\lambda }=L\cap M_{\lambda }$ and $B_{\lambda }=B\cap (L_{\lambda }/q{L}_{\lambda }),$

• ${\tilde{e}}_{i}L\subset L$ and ${\tilde{f}}_{i}L\subset L\text{ for all }i,$

• ${\tilde{e}}_{i}B\subset B\cup \{0\}$ and ${\tilde{f}}_{i}B\subset B\cup \{0\}\text{ for all }i,$

• $\text{for all }b\in B\text{ and }{b}^{\prime }\in B,\text{ and for all }i,{\tilde{e}}_{i}b=b^{\prime }\text{ if and only if }{\tilde{f}}_i{b}^{\prime }=b.$

To put this into a more informal setting, the actions of $e_i{f}_i$ and $f_i{e}_i$ are generally singular at $q=0$ on an integrable module $M$. The linear mappings ${\tilde{e}}_i$ and ${\tilde{f}}_i$ on the module are introduced so that the actions of ${\tilde{e}}_i{\tilde{f}}_i$ and ${\tilde{f}}_i{\tilde{e}}_i$ are regular at $q=0$ on the module. There exists a $\mathbb{\mathbb{Q}}(q)$-basis of weight vectors $\tilde{B}$ for $M$, with respect to which the actions of ${\tilde{e}}_i$ and ${\tilde{f}}_i$ are regular at $q=0$ for all i. The module is then restricted to the free $A$-module generated by the basis, and the basis vectors, the $A$-submodule and the actions of ${\tilde{e}}_i$ and ${\tilde{f}}_i$ are evaluated at $q=0$. Furthermore, the basis can be chosen such that at $q=0$, for all $i$, ${\tilde{e}}_i$ and ${\tilde{f}}_i$ are represented by mutual transposes, and map basis vectors to basis vectors or 0.

A crystal base can be represented by a directed graph with labelled edges. Each vertex of the graph represents an element of the $\mathbb{\mathbb{Q}}$-basis $B$ of $L/qL$, and a directed edge, labelled by i, and directed from vertex $v_1$ to vertex $v_2$, represents that $b_2={\tilde{f}}_i{b}_1$ (and, equivalently, that $b_1={\tilde{e}}_i{b}_2$), where $b_1$ is the basis element represented by $v_1$, and $b_2$ is the basis element represented by $v_2$. The graph completely determines the actions of ${\tilde{e}}_i$ and ${\tilde{f}}_i$ at $q=0$. If an integrable module has a crystal base, then the module is irreducible if and only if the graph representing the crystal base is connected (a graph is called "connected" if the set of vertices cannot be partitioned into the union of nontrivial disjoint subsets $V_1$ and $V_2$ such that there are no edges joining any vertex in $V_1$ to any vertex in $V_2$).

For any integrable module with a crystal base, the weight spectrum for the crystal base is the same as the weight spectrum for the module, and therefore the weight spectrum for the crystal base is the same as the weight spectrum for the corresponding module of the appropriate Kac–Moody algebra. The multiplicities of the weights in the crystal base are also the same as their multiplicities in the corresponding module of the appropriate Kac–Moody algebra.

It is a theorem of Kashiwara that every integrable highest weight module has a crystal base. Similarly, every integrable lowest weight module has a crystal base.

Tensor products of crystal bases

Let $M$ be an integrable module with crystal base $(L,B)$ and $M^{\prime }$ be an integrable module with crystal base $(L^{\prime },B^{\prime })$. For crystal bases, the coproduct $\mathrm{\Delta }$, given by $\mathrm{\Delta }(k_{\lambda })=k_{\lambda }\otimes k_{\lambda },\mathrm{\Delta }(e_i)=e_i\otimes k_i^{-1}+1\otimes e_i,\mathrm{\Delta }(f_i)=f_i\otimes 1+k_i\otimes f_i$, is adopted. The integrable module $M{\otimes }_{\mathbb{\mathbb{Q}}(q)}{M}^{\prime }$ has crystal base $(L{\otimes }_A{L}^{\prime },B\otimes B^{\prime })$, where $B\otimes B^{\prime }=\{b{\otimes }_{\mathbb{\mathbb{Q}}}{b}^{\prime }:b\in B,b^{\prime }\in B^{\prime }\}$. For a basis vector $b\in B$, define ${ϵ}_i(b)=\max \{n\ge 0:{\tilde{e}}_i^{n}b\ne 0\}$ and ${\varphi }_i(b)=\max \{n\ge 0:{\tilde{f}}_i^{n}b\ne 0\}$. The actions of ${\tilde{e}}_i$ and ${\tilde{f}}_i$ on $b\otimes b^{\prime }$ are given by

• ${\tilde{e}}_i(b\otimes b^{\prime })=\{\begin{array}{ll}{\tilde{e}}_{i}b\otimes b^{\prime },& \text{if }{\varphi }_i(b)\ge {ϵ}_i(b^{\prime }),\\ b\otimes {\tilde{e}}_i{b}^{\prime },& \text{if }{\varphi }_i(b)<{ϵ}_i(b^{\prime }),\end{array}$

• ${\tilde{f}}_i(b\otimes b^{\prime })=\{\begin{array}{ll}{\tilde{f}}_{i}b\otimes b^{\prime },& \text{if }{\varphi }_i(b)>{ϵ}_i(b^{\prime }),\\ b\otimes {\tilde{f}}_i{b}^{\prime },& \text{if }{\varphi }_i(b)\le {ϵ}_i(b^{\prime }).\end{array}$

The decomposition of the product two integrable highest weight modules into irreducible submodules is determined by the decomposition of the graph of the crystal base into its connected components (i.e. the highest weights of the submodules are determined, and the multiplicity of each highest weight is determined).

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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