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| | "Why does my computer keep freezing up?" I was asked by a great deal of individuals the cause of their computer freeze problems. And I am fed up with spending much time in answering the query time and time again. This article is to tell you the real cause of your PC Freezes.<br><br>So one day my computer suddenly started being strange. I was so frustrated, considering my files were missing, plus I cannot open the files that I required, and then, suddenly, everything stopped working!<br><br>With the Internet, the danger to the registry is a bit more and windows XP error messages may appear frequently. Why? The malicious wares like viruses, Trojans, spy-wares, ad wares, plus the like gets recorded too. Cookies are best examples. You get to save passwords, and stuff, appropriate? That is a easy illustration of the register functioning.<br><br>The 1328 error is a widespread issue caused by your system being unable to correctly process different changes for the program or Microsoft Office. If you have this error, it commonly means which the computer is either unable to read the actual update file or the computer has issues with the settings it's using to run. To fix this problem, we initially should change / fix any issues that your computer has with its update files, plus then repair some of the issues that the program may have.<br><br>The [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities] could come because standard with a back up and restore center. This ought to be an simple to implement task.That signifies which in the event you encounter a problem with your PC after using a registry cleaning you are able to simply restore the settings.<br><br>Files with the DOC extension are also susceptible to viruses, yet this is solved by superior antivirus programs. Another problem is the fact that .doc files can be corrupted, unreadable or damaged due to spyware, adware, plus malware. These situations usually avoid consumers from properly opening DOC files. This is when effective registry cleaners become beneficial.<br><br>By restoring the state of your program to an earlier date, error 1721 could not appear in Windows 7, Vista plus XP. There is a tool called System Restore which we have to utilize in this process.<br><br>A system and registry cleaner could be downloaded from the web. It's easy to use and the process refuses to take long. All it does is scan plus then whenever it finds mistakes, it may fix plus clean those errors. An error free registry may protect the computer from mistakes and provide we a slow PC fix. |
| In [[abstract algebra]], a '''representation of a Hopf algebra''' is a [[algebra representation|representation]] of its underlying [[associative algebra]]. That is, a representation of a Hopf algebra ''H'' over a field ''K'' is a ''K''-[[vector space]] ''V'' with an [[group action|action]] ''H'' × ''V'' → ''V'' usually denoted by juxtaposition (that is, the image of (''h'',''v'') is written ''hv''). The vector space ''V'' is called an ''H''-module.
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| ==Properties==
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| The module structure of a representation of a Hopf algebra ''H'' is simply its structure as a module for the underlying associative algebra. The main use of considering the additional structure of a Hopf algebra is when considering all ''H''-modules as a category. The additional structure is also used to define invariant elements of an ''H''-module ''V''. An element ''v'' in ''V'' is [[Invariant (mathematics)|invariant]] under ''H'' if for all ''h'' in ''H'', ''hv'' = ε(''h'')''v'', where ε is the [[counit]] of ''H''. The subset of all invariant elements of ''V'' forms a submodule of ''V''.
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| ==Categories of representations as a motivation for Hopf algebras==
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| For an associative algebra ''H'', the [[tensor product]] ''V''<sub>1</sub> ⊗ ''V''<sub>2</sub> of two ''H''-modules ''V''<sub>1</sub> and ''V''<sub>2</sub> is a vector space, but not necessarily an ''H''-module. For the tensor product to be a [[functor]]ial product operation on ''H''-modules, there must be a linear binary operation Δ : ''H'' → ''H'' ⊗ ''H'' such that for any ''v'' in ''V''<sub>1</sub> ⊗ ''V''<sub>2</sub> and any ''h'' in ''H'',
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| :<math>hv=\Delta h(v_{(1)}\otimes v_{(2)})=h_{(1)}v_{(1)}\otimes h_{(2)}v_{(2)},</math>
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| and for any ''v'' in ''V''<sub>1</sub> ⊗ ''V''<sub>2</sub> and ''a'' and ''b'' in ''H'',
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| :<math>\Delta(ab)(v_{(1)}\otimes v_{(2)})=(ab)v=a[b[v]]=\Delta a[\Delta b(v_{(1)}\otimes v_{(2)})]=(\Delta a )(\Delta b)(v_{(1)}\otimes v_{(2)}).</math>
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| using sumless [[Sweedler's notation]], which is somewhat like an index free form of [[Einstein's summation convention]]. This is satisfied if there is a Δ such that Δ(''ab'') = Δ(''a'')Δ(''b'') for all ''a'', ''b'' in ''H''.
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| For the category of ''H''-modules to be a strict [[monoidal category]] with respect to ⊗, <math>V_1\otimes(V_2\otimes V_3)</math> and <math>(V_1\otimes V_2)\otimes V_3</math> must be equivalent and there must be unit object ε<sub>''H''</sub>, called the trivial module, such that ε<sub>''H''</sub> ⊗ ''V'', ''V'' and ''V'' ⊗ ε<sub>''H''</sub> are equivalent.
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| This means that for any ''v'' in
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| :<math>V_1\otimes(V_2\otimes V_3)=(V_1\otimes V_2)\otimes V_3</math>
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| and for ''h'' in ''H'', | |
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| :<math>((\operatorname{id}\otimes \Delta)\Delta h)(v_{(1)}\otimes v_{(2)}\otimes v_{(3)})=h_{(1)}v_{(1)}\otimes h_{(2)(1)}v_{(2)}\otimes h_{(2)(2)}v_{(3)}=hv=((\Delta\otimes \operatorname{id}) \Delta h) (v_{(1)}\otimes v_{(2)}\otimes v_{(3)}).</math>
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| This will hold for any three ''H''-modules if Δ satisfies
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| :<math>(\operatorname{id}\otimes \Delta)\Delta A=(\Delta \otimes \operatorname{id})\Delta A.</math>
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| The trivial module must be one dimensional, and so an [[algebra homomorphism]] ε : ''H'' → ''F'' may be defined such that ''hv'' = ε(''h'')''v'' for all ''v'' in ε<sub>''H''</sub>. The trivial module may be identified with ''F'', with 1 being the element such that 1 ⊗ ''v'' = ''v'' = ''v'' ⊗ 1 for all ''v''. It follows that for any ''v'' in any ''H''-module ''V'', any ''c'' in ε<sub>''H''</sub> and any ''h'' in ''H'', | |
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| :<math>(\varepsilon(h_{(1)})h_{(2)})cv=h_{(1)}c\otimes h_{(2)}v=h(c\otimes v)=h(cv)=(h_{(1)}\varepsilon(h_{(2)}))cv.</math>
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| The existence of an algebra homomorphism ε satisfying | |
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| :<math>\varepsilon(h_{(1)})h_{(2)} = h = h_{(1)}\varepsilon(h_{(2)})</math> | |
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| is a sufficient condition for the existence of the trivial module.
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| It follows that in order for the category of ''H''-modules to be a monoidal category with respect to the tensor product, it is sufficient for ''H'' to have maps Δ and ε satisfying these conditions. This is the motivation for the definition of a [[bialgebra]], where Δ is called the [[comultiplication]] and ε is called the [[counit]].
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| In order for each ''H''-module ''V'' to have a [[dual representation]] ''V'' such that the underlying vector spaces are dual and the operation * is functorial over the monoidal category of ''H''-modules, there must be a linear map ''S'' : ''H'' → ''H'' such that for any ''h'' in ''H'', ''x'' in ''V'' and ''y'' in ''V*'',
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| :<math>\langle y, S(h)x\rangle = \langle hy, x \rangle.</math>
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| where <math>\langle\cdot,\cdot\rangle</math> is the usual [[pairing]] of dual vector spaces. If the map <math>\varphi:V\otimes V^*\rightarrow \varepsilon_H</math> induced by the pairing is to be an ''H''-homomorphism, then for any ''h'' in ''H'', ''x'' in ''V'' and ''y'' in ''V*'',
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| :<math>\varphi\left(h(x\otimes y)\right)=\varphi\left(x\otimes S(h_{(1)})h_{(2)}y\right)=\varphi\left(S(h_{(2)})h_{(1)}x\otimes y\right)=h\varphi(x\otimes y)=\varepsilon(h)\varphi(x\otimes y),</math>
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| which is satisfied if
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| :<math>S(h_{(1)})h_{(2)}=\varepsilon(h)=h_{(1)}S(h_{(2)})</math>
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| for all ''h'' in ''H''.
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| If there is such a map ''S'', then it is called an ''antipode'', and ''H'' is a Hopf algebra. The desire for a monoidal category of modules with functorial tensor products and dual representations is therefore one motivation for the concept of a Hopf algebra.
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| ==Representations on an algebra==
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| A Hopf algebra also has representations which carry additional structure, namely they are algebras.
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| Let ''H'' be a Hopf algebra. If ''A'' is an [[algebra over a field|algebra]] with the product operation μ : ''A'' ⊗ ''A'' → ''A'', and ρ : ''H'' ⊗ ''A'' → ''A'' is a representation of ''H'' on ''A'', then ρ is said to be a representation of ''H'' on an algebra if μ is ''H''-[[equivariant]]. As special cases, Lie algebras, Lie superalgebras and groups can also have representations on an algebra.
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| ==See also==
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| *[[Tannaka–Krein reconstruction theorem]]
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| {{DEFAULTSORT:Representation Theory Of Hopf Algebras}}
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| [[Category:Hopf algebras]]
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| [[Category:Representation theory]]
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"Why does my computer keep freezing up?" I was asked by a great deal of individuals the cause of their computer freeze problems. And I am fed up with spending much time in answering the query time and time again. This article is to tell you the real cause of your PC Freezes.
So one day my computer suddenly started being strange. I was so frustrated, considering my files were missing, plus I cannot open the files that I required, and then, suddenly, everything stopped working!
With the Internet, the danger to the registry is a bit more and windows XP error messages may appear frequently. Why? The malicious wares like viruses, Trojans, spy-wares, ad wares, plus the like gets recorded too. Cookies are best examples. You get to save passwords, and stuff, appropriate? That is a easy illustration of the register functioning.
The 1328 error is a widespread issue caused by your system being unable to correctly process different changes for the program or Microsoft Office. If you have this error, it commonly means which the computer is either unable to read the actual update file or the computer has issues with the settings it's using to run. To fix this problem, we initially should change / fix any issues that your computer has with its update files, plus then repair some of the issues that the program may have.
The tuneup utilities could come because standard with a back up and restore center. This ought to be an simple to implement task.That signifies which in the event you encounter a problem with your PC after using a registry cleaning you are able to simply restore the settings.
Files with the DOC extension are also susceptible to viruses, yet this is solved by superior antivirus programs. Another problem is the fact that .doc files can be corrupted, unreadable or damaged due to spyware, adware, plus malware. These situations usually avoid consumers from properly opening DOC files. This is when effective registry cleaners become beneficial.
By restoring the state of your program to an earlier date, error 1721 could not appear in Windows 7, Vista plus XP. There is a tool called System Restore which we have to utilize in this process.
A system and registry cleaner could be downloaded from the web. It's easy to use and the process refuses to take long. All it does is scan plus then whenever it finds mistakes, it may fix plus clean those errors. An error free registry may protect the computer from mistakes and provide we a slow PC fix.