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Also visit my web site ... hostgator1centcoupon.info In mathematics, weak bialgebras are a generalization of bialgebras that are both algebras and coalgebras but for which the compatibility conditions between the two structures have been "weakened". In the same spirit, weak Hopf algebras are weak bialgebras together with a linear map S satisfying specific conditions; they are generalizations of Hopf algebras.

These objects were introduced by Böhm, Nill and Szlachányi. The first motivations for studying them came from quantum field theory and operator algebras. Weak Hopf algebras have quite interesting representation theory; in particular modules over a semisimple finite weak Hopf algebra is a fusion category (which is a monoidal category with extra properties). It was also shown by Etingof, Nikshych and Ostrik that any fusion category is equivalent to a category of modules over a weak Hopf algebra.

Definition

A weak bialgebra $(H,\mu ,\eta ,\mathrm{\Delta },\epsilon )$ over a field $k$ is a vector space $H$ such that

• $(H,\mu ,\eta )$ forms an associative algebra with multiplication $\mu :H\otimes H\to H$ and unit $\eta :k\to H$,
• $(H,\mathrm{\Delta },\epsilon )$ forms a coassociative coalgebra with comultiplication $\mathrm{\Delta }:H\to H\otimes H$ and counit $\epsilon :H\to k$,

for which the following compatibility conditions hold :

1. Multiplicativity of the Comultiplication :

   $\mathrm{\Delta }\circ \mu =(\mu \otimes \mu )\circ ({\mathrm{i}\mathrm{d}}_H\otimes {\sigma }_{H,H}\otimes {\mathrm{i}\mathrm{d}}_H)\circ (\mathrm{\Delta }\otimes \mathrm{\Delta })$,

2. Weak Multiplicativity of the Counit :

   $\epsilon \circ \mu \circ (\mu \otimes {\mathrm{i}\mathrm{d}}_H)=(\epsilon \otimes \epsilon )\circ (\mu \otimes \mu )\circ ({\mathrm{i}\mathrm{d}}_H\otimes \mathrm{\Delta }\otimes {\mathrm{i}\mathrm{d}}_H)=(\epsilon \otimes \epsilon )\circ (\mu \otimes \mu )\circ ({\mathrm{i}\mathrm{d}}_H\otimes {\mathrm{\Delta }}^{op}\otimes {\mathrm{i}\mathrm{d}}_H)$,

3. Weak Comultiplicativity of the Unit :

   $(\mathrm{\Delta }\otimes {\mathrm{i}\mathrm{d}}_H)\circ \mathrm{\Delta }\circ \eta =({\mathrm{i}\mathrm{d}}_H\otimes \mu \otimes {\mathrm{i}\mathrm{d}}_H)\circ (\mathrm{\Delta }\otimes \mathrm{\Delta })\circ (\eta \otimes \eta )=({\mathrm{i}\mathrm{d}}_H\otimes {\mu }^{op}\otimes {\mathrm{i}\mathrm{d}}_H)\circ (\mathrm{\Delta }\otimes \mathrm{\Delta })\circ (\eta \otimes \eta )$,

where ${\sigma }_{V,W}:V\otimes W\to W\otimes V:v\otimes w\mapsto w\otimes v$ flips the two tensor factors. Moreover ${\mu }^{op}=\mu \circ {\sigma }_{H,H}$ is the opposite multiplication and ${\mathrm{\Delta }}^{op}={\sigma }_{H,H}\circ \mathrm{\Delta }$ is the opposite comultiplication. Note that we also implicitly use Mac Lane's coherence theorem for the monoidal category of vector spaces, identifying $(U\otimes V)\otimes W\cong U\otimes (V\otimes W)$ as well as $V\otimes k\cong V\cong k\otimes V$.

The definition is fairly self-explanatory, one sees that it is the compatibility between the algebra and coalgebra structures that is weaken.

A weak Hopf algebra $(H,\mu ,\eta ,\mathrm{\Delta },\epsilon ,S)$ is a weak bialgebra $(H,\mu ,\eta ,\mathrm{\Delta },\epsilon )$ with a linear map $S:H\to H$, called the antipode, that satisfies:

• $\mu \circ ({\mathrm{i}\mathrm{d}}_H\otimes S)\circ \mathrm{\Delta }=(\epsilon \otimes {\mathrm{i}\mathrm{d}}_H)\circ (\mu \otimes {\mathrm{i}\mathrm{d}}_H)\circ ({\mathrm{i}\mathrm{d}}_H\otimes {\sigma }_{H,H})\circ (\mathrm{\Delta }\otimes {\mathrm{i}\mathrm{d}}_H)\circ (\eta \otimes {\mathrm{i}\mathrm{d}}_H)$,
• $\mu \circ (S\otimes {\mathrm{i}\mathrm{d}}_H)\circ \mathrm{\Delta }=({\mathrm{i}\mathrm{d}}_H\otimes \epsilon )\circ ({\mathrm{i}\mathrm{d}}_H\otimes \mu )\circ ({\sigma }_{H,H}\otimes {\mathrm{i}\mathrm{d}}_H)\circ ({\mathrm{i}\mathrm{d}}_H\otimes \mathrm{\Delta })\circ ({\mathrm{i}\mathrm{d}}_H\otimes \eta )$,
• $S=\mu \circ (\mu \otimes {\mathrm{i}\mathrm{d}}_H)\circ (S\otimes {\mathrm{i}\mathrm{d}}_H\otimes S)\circ (\mathrm{\Delta }\otimes {\mathrm{i}\mathrm{d}}_H)\circ \mathrm{\Delta }$.

Examples

1. Hopf algebra. Of course any Hopf algebra is a weak Hopf algebra.
2. Groupoid algebra. Suppose $G=(G_0,G_1)$ is a groupoid and let $K[G]$ be the groupoid algebra, in other words, the algebra generated by the morphisms $g\in G_1$. This becomes a weak Hopf algebra if we define
   • $\mu :K[G]\otimes K[G]\to K[G]\text{by}\mu (g\otimes h)=\{\begin{array}{cl}g\circ h& \text{if target(h) = source(g)}\\ 0& \text{otherwise}\end{array}$
   • $\eta :k\to K[G]\text{by}\eta (1)=\sum _{X\in G_0}{\mathrm{i}\mathrm{d}}_X$
   • $\mathrm{\Delta }:K[G]\to K[G]\otimes K[G]\text{by}\mathrm{\Delta }(g)=g\otimes g\text{for all}g\in G_1$
   • $\epsilon :K[G]\to k\text{by}\epsilon (g)=1\text{for all}g\in G_1$
   • $S:K[G]\to K[G]\text{by}S(g)=g^{-1}\text{for all}g\in G_1$.

Note that this second example is a weak Hopf algebra but not a Hopf algebra.

Representation theory

Let H be a semisimple finite weak Hopf algebra, then modules over H form a semisimple rigid monoidal category with finitely many simple objects. Moreover the homomorphisms spaces are finite-dimensional vector spaces and the endomorphisms space of simple objects are one-dimensional. Finally, the monoidal unit is a simple object. Such a category is called a fusion category.

It can be shown that some monoidal category are not modules over a Hopf algebra. In the case of fusion categories (which are just monoidal categories with extra conditions), it was proved by Etingof, Nikshych and Ostrik that any fusion category is equivalent to a category of modules over a weak Hopf algebra.

Notes

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References

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  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang