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| In [[mathematics]], the '''tensor algebra''' of a [[vector space]] ''V'', denoted ''T''(''V'') or ''T''<sup> •</sup>(''V''), is the [[algebra over a field|algebra]] of [[tensor]]s on ''V'' (of any rank) with multiplication being the [[tensor product]]. It is the [[free algebra]] on ''V'', in the sense of being [[left adjoint]] to the [[forgetful functor]] from algebras to vector spaces: it is the "most general" algebra containing ''V'', in the sense of the corresponding [[universal property]] (see [[#Adjunction and universal property|below]]).
| | You may find easy techniques to speed up computer by making the most out of the built in tools inside your Windows also as getting the Service Pack updates-speed up your PC and fix error. Simply follow a few protocols to immediately make the computer rapidly than ever.<br><br>Document files enable the consumer to input information, images, tables plus other ingredients to improve the presentation. The just issue with this format compared to different file kinds including .pdf for example is its ability to be readily editable. This signifies that anyone watching the file can change it by accident. Also, this file structure could be opened by alternative programs but it does not guarantee that what you see inside the Microsoft Word application might nevertheless be the same when you see it utilizing another program. However, it is actually nevertheless preferred by many computer users for its ease of employ and attributes.<br><br>Windows is pretty dumb. It only knows how to follow commands plus instructions, which means which when you install a program, which system has to tell Windows precisely what to do. This really is done by storing an "instruction file" inside the registry of the system. All the computer programs put these "manuals" into the registry, allowing the computer to run a wide range of programs. Whenever we load up one of those programs, Windows merely looks up the program file in the registry, and carries out its instructions.<br><br>Analysis a files and clean it up regularly. Destroy all the unwanted plus unused files considering they only jam the computer program. It might surely enhance the speed of your computer plus be careful that a computer never infected by a virus. Remember always to update the antivirus software each time. If you never utilize a computer fairly often, you are able to take a free antivirus.<br><br>In a word, to accelerate windows XP, Vista startup, it's very significant to disable certain startup items and clean and optimize the registry. You can follow the procedures above to disable unwanted programs. To optimize the registry, I recommend we employ a [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities 2014] software. Because it's especially risky for you to edit the registry by oneself.<br><br>The leading reason why I can not make my PC run quicker was the program registry plus it being fragmented. So software to defragment or clean the registry are needed. Such software are called registry cleaners. Like all additional software, there are paid ones and free ones with their advantages and disadvantages. To choose between the 2 is the user's choice.<br><br>Perfect Optimizer is a good Registry Product, updates consistently and has lots of qualities. Despite its price, there are that the update are truly practical. They provide plenty of support through telephone, mail and forums. You could need to pay a visit to the free trial to check it out for oneself.<br><br>Registry products will enable your computer run in a more efficient mode. Registry cleaners ought to be piece of the normal scheduled maintenance program for the computer. You don't have to wait forever for a computer or the programs to load and run. A little repair may bring back the speed you lost. |
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| The tensor algebra also has two [[#Coalgebra structures|coalgebra structures]]; one simple one, which does not make it a bialgebra, and a more complicated one, which yields a [[bialgebra]], and can be extended with an [[Antipode_(algebra)#Properties_of_the_antipode|antipode]] to a [[Hopf algebra]] structure. | |
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| ''Note'': In this article, all algebras are assumed to be [[unital algebra|unital]] and [[associative algebra|associative]].
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| ==Construction==
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| Let ''V'' be a [[vector space]] over a [[field (mathematics)|field]] ''K''. For any nonnegative [[integer]] ''k'', we define the '''''k''<sup>th</sup> tensor power''' of ''V'' to be the [[tensor product]] of ''V'' with itself ''k'' times:
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| :<math>T^kV = V^{\otimes k} = V\otimes V \otimes \cdots \otimes V.</math>
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| That is, ''T''<sup>''k''</sup>''V'' consists of all tensors on ''V'' of [[Tensor#Tensor rank|rank]] ''k''. By convention ''T''<sup>0</sup>''V'' is the [[ground field]] ''K'' (as a one-dimensional vector space over itself).
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| We then construct ''T''(''V'') as the [[direct sum of vector spaces|direct sum]] of ''T''<sup>''k''</sup>''V'' for ''k'' = 0,1,2,…
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| :<math>T(V)= \bigoplus_{k=0}^\infty T^kV = K\oplus V \oplus (V\otimes V) \oplus (V\otimes V\otimes V) \oplus \cdots.</math>
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| The multiplication in ''T''(''V'') is determined by the canonical isomorphism
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| :<math>T^kV \otimes T^\ell V \to T^{k + \ell}V</math>
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| given by the tensor product, which is then extended by linearity to all of ''T''(''V''). This multiplication rule implies that the tensor algebra ''T''(''V'') is naturally a [[graded algebra]] with ''T''<sup>''k''</sup>''V'' serving as the grade-''k'' subspace. This grading can be extended to a '''Z''' grading by appending subspaces <math>T^{k}V=\{0\}</math> for negative integers ''k''.
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| The construction generalizes in straightforward manner to the tensor algebra of any [[module (mathematics)|module]] ''M'' over a [[commutative ring|''commutative'' ring]]. If ''R'' is a [[non-commutative ring]], one can still perform the construction for any ''R''-''R'' [[bimodule]] ''M''. (It does not work for ordinary ''R''-modules because the iterated tensor products cannot be formed.)
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| ==Adjunction and universal property==
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| The tensor algebra ''T''(''V'') is also called the '''[[free algebra]]''' on the vector space ''V'', and is functorial. As with other [[free object|free constructions]], the functor ''T'' is [[adjoint functor|left adjoint]] to some [[forgetful functor]]. In this case, it's the functor which sends each ''K''-algebra to its underlying vector space.
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| Explicitly, the tensor algebra satisfies the following [[universal property]], which formally expresses the statement that it is the most general algebra containing ''V'':
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| : Any [[linear transformation]] ''f'' : ''V'' → ''A'' from ''V'' to an algebra ''A'' over ''K'' can be uniquely extended to an [[algebra homomorphism]] from ''T''(''V'') to ''A'' as indicated by the following [[commutative diagram]]:
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| [[Image:TensorAlgebra-01.png|center|Universal property of the tensor algebra]]
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| Here ''i'' is the [[Inclusion map|canonical inclusion]] of ''V'' into ''T''(''V'') (the unit of the adjunction). One can, in fact, define the tensor algebra ''T''(''V'') as the unique algebra satisfying this property (specifically, it is unique [[up to]] a unique isomorphism), but one must still prove that an object satisfying this property exists.
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| The above universal property shows that the construction of the tensor algebra is ''functorial'' in nature. That is, ''T'' is a [[functor]] from the '''''K''-Vect''', [[category of vector spaces]] over ''K'', to '''''K''-Alg''', the category of ''K''-algebras. The functoriality of ''T'' means that any linear map from ''V'' to ''W'' extends uniquely to an algebra homomorphism from ''T''(''V'') to ''T''(''W'').
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| ==Non-commutative polynomials==
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| If ''V'' has finite dimension ''n'', another way of looking at the tensor algebra is as the "algebra of polynomials over ''K'' in ''n'' non-commuting variables". If we take [[basis vector]]s for ''V'', those become non-commuting variables (or [[Indeterminate (variable)|''indeterminants'']]) in ''T''(''V''), subject to no constraints beyond [[associativity]], the [[distributive law]] and ''K''-linearity.
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| Note that the algebra of polynomials on ''V'' is not <math>T(V)</math>, but rather <math>T(V^*)</math>: a (homogeneous) linear function on ''V'' is an element of <math>V^*,</math> for example coordinates <math>x^1,\dots,x^n</math> on a vector space are covectors, as they take in a vector and give out a scalar (the given coordinate of the vector).
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| ==Quotients==
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| Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e. by constructing certain [[quotient algebra]]s of ''T''(''V''). Examples of this are the [[exterior algebra]], the [[symmetric algebra]], [[Clifford algebra]]s and [[universal enveloping algebra]]s.
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| ==Coalgebra structures==
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| The tensor algebra has two coalgebra structures; one simple one, which does not make it a bialgebra, and a more complicated one, which yields a bialgebra, and can be extended with an antipode to a Hopf algebra structure.
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| ===Simple coalgebra structure===
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| The simple [[coalgebra]] structure on the tensor algebra is given as follows. The [[coproduct]] Δ is defined by
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| :<math>\Delta(v_1 \otimes \dots \otimes v_m ) := \sum_{i=0}^{m}
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| (v_1 \otimes \dots \otimes v_i) \otimes (v_{i+1} \otimes \dots \otimes v_m)</math>
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| extended by linearity to all of ''TV''. The counit is given by
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| :<math>\varepsilon\left(v\right)=v</math> for every <math>v\in T^0\left(V\right)</math> and
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| :<math>\varepsilon\left(v\right)=0</math> for every <math>v\in T^k\left(V\right)</math> for every <math>k > 0</math>.
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| Note that Δ : ''TV'' → ''TV'' ⊗ ''TV'' respects the grading
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| :<math>T^mV \to \bigoplus_{i+j=m} T^iV \otimes T^jV</math>
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| and ε is also compatible with the grading.
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| The tensor algebra is ''not'' a [[bialgebra]] with this coproduct.
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| ===Bialgebra and Hopf algebra structure===
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| However, the following more complicated coproduct does yield a bialgebra:
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| :<math>\Delta(x_1\otimes\dots\otimes x_m) = \sum_{p=0}^m \sum_{\sigma\in\mathrm{Sh}_{p,m-p}} \left(x_{\sigma(1)}\otimes\dots\otimes x_{\sigma(p)}\right)\otimes\left(x_{\sigma(p+1)}\otimes\dots\otimes x_{\sigma(m)}\right)</math>
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| where the summation is taken over all [[(p,q) shuffle|(p,m-p)-shuffles]].
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| Finally, the tensor algebra becomes a [[Hopf algebra]] with antipode given by
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| :<math>S(x_1\otimes\dots\otimes x_m) = (-1)^mx_m\otimes\dots\otimes x_1</math>
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| extended linearly to all of ''TV''.
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| This is just the standard Hopf algebra structure on a free algebra, where one defines the comultiplication on <math>T^1(V)=V</math> by
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| :<math>\Delta(x)=x\otimes1+1\otimes x</math>
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| and then extends to <math>T^m(V)</math> via
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| :<math>\Delta(x_1\otimes\dots\otimes x_m) = \Delta(x_1)\Delta(x_2)\cdots\Delta(x_m).</math>
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| Similarly one defines the antipode on <math>T^1(V)=V</math> by
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| :<math>S(x)=-x</math>
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| and then extends the antipode as the unique [[antiautomorphism]] of <math>T(V)</math> with this property, i.e. we define the antipode on <math>T^m(V)</math> via | |
| :<math>S(x_1\otimes\dots\otimes x_m) = S(x_m)S(x_{m-1})\cdots S(x_2)S(x_1).</math>
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| ==See also==
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| *[[Symmetric algebra]]
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| *[[Exterior algebra]]
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| *[[Monoidal category]]
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| *[[Multilinear subspace learning]]
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| *[[q:Stanisław Lem#Love_and_Tensor_Algebra|Stanisław Lem's ''Love and Tensor Algebra'']]
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| ==References==
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| *{{cite book
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| | last = Bourbaki
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| | first = Nicolas
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| | title = [http://books.google.ca/books/about/Algebra.html?id=STS9aZ6F204C&redir_esc=y Algebra I. Chapters 1-3]
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| | chapter= Algebra, Chapter 3 §5
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| | publisher = [[Springer-Verlag]]
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| | series = Elements of Mathematics
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| | year = 1989
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| | isbn = 3-540-64243-9
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| }}
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| * {{citation | author=Serge Lang | authorlink=Serge Lang | title=Algebra | series=[[Graduate Texts in Mathematics]] | volume=211 | edition=3rd | publisher=[[Springer Verlag]] | year=2002 | isbn=978-0-387-95385-4 }}
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| {{tensor}}
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| [[Category:Algebras]]
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| [[Category:Multilinear algebra]]
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| [[Category:Tensors]]
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| [[Category:Hopf algebras]]
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You may find easy techniques to speed up computer by making the most out of the built in tools inside your Windows also as getting the Service Pack updates-speed up your PC and fix error. Simply follow a few protocols to immediately make the computer rapidly than ever.
Document files enable the consumer to input information, images, tables plus other ingredients to improve the presentation. The just issue with this format compared to different file kinds including .pdf for example is its ability to be readily editable. This signifies that anyone watching the file can change it by accident. Also, this file structure could be opened by alternative programs but it does not guarantee that what you see inside the Microsoft Word application might nevertheless be the same when you see it utilizing another program. However, it is actually nevertheless preferred by many computer users for its ease of employ and attributes.
Windows is pretty dumb. It only knows how to follow commands plus instructions, which means which when you install a program, which system has to tell Windows precisely what to do. This really is done by storing an "instruction file" inside the registry of the system. All the computer programs put these "manuals" into the registry, allowing the computer to run a wide range of programs. Whenever we load up one of those programs, Windows merely looks up the program file in the registry, and carries out its instructions.
Analysis a files and clean it up regularly. Destroy all the unwanted plus unused files considering they only jam the computer program. It might surely enhance the speed of your computer plus be careful that a computer never infected by a virus. Remember always to update the antivirus software each time. If you never utilize a computer fairly often, you are able to take a free antivirus.
In a word, to accelerate windows XP, Vista startup, it's very significant to disable certain startup items and clean and optimize the registry. You can follow the procedures above to disable unwanted programs. To optimize the registry, I recommend we employ a tuneup utilities 2014 software. Because it's especially risky for you to edit the registry by oneself.
The leading reason why I can not make my PC run quicker was the program registry plus it being fragmented. So software to defragment or clean the registry are needed. Such software are called registry cleaners. Like all additional software, there are paid ones and free ones with their advantages and disadvantages. To choose between the 2 is the user's choice.
Perfect Optimizer is a good Registry Product, updates consistently and has lots of qualities. Despite its price, there are that the update are truly practical. They provide plenty of support through telephone, mail and forums. You could need to pay a visit to the free trial to check it out for oneself.
Registry products will enable your computer run in a more efficient mode. Registry cleaners ought to be piece of the normal scheduled maintenance program for the computer. You don't have to wait forever for a computer or the programs to load and run. A little repair may bring back the speed you lost.