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2014-08-13T07:46:36Z

JulioHindman:

In [[algebraic geometry]], a '''very ample [[line bundle]]''' is one with enough [[global section]]s to set up an [[embedding]] of its base [[algebraic variety|variety]] or manifold <math>M</math> into [[projective space]]. An '''ample line bundle''' is one such that some positive power is very ample. '''Globally generated sheaves''' are those with enough sections to define a morphism to projective space.

==Introduction==
===Inverse image of line bundle and hyperplane divisors===
Given a morphism <math>f\ :\ X \to Y</math>, any vector bundle <math>\mathcal F</math> on ''Y'', or more generally any sheaf in <math>\mathcal O_Y</math> modules, ''eg.'' a coherent sheaf, can be pulled back to ''X'', (see [[Inverse image functor]]). This construction preserves the condition of being a line bundle, and more generally the rank.

The notions described in this article are related to this construction in the case of morphisms to projective spaces
:<math>f : X \to \mathbb P^N, </math> and <math>\mathcal F = \mathcal O(1) \in \mathrm{Pic}(\mathbb P^N)</math>,
the line bundle corresponding to the hyperplane divisor, whose sections are the 1-homogeneous regular functions. See [[Algebraic geometry of projective spaces#Divisors and twisting sheaves]].

=== Sheaves generated by their global sections ===
{{Main|Sheaf spanned by global sections}}
Let ''X'' be a [[scheme (mathematics)|scheme]] or a complex manifold and ''F'' a sheaf on ''X''. One says that ''F'' is '''generated by (finitely many) global sections''' <math> a_i \in F(X)</math>, if every [[stalks of a sheaf|stalk]] of ''F'' is generated as a [[module]] over the stalk of the [[structure sheaf]] by the germs of the ''ai''. For example, if ''F'' happens to be a line bundle, i.e. locally free of rank 1, this amounts to having finitely many global sections, such that for any point ''x'' in ''X'', there is at least one section not vanishing at this point. In this case a choice of such global generators ''a''0, ..., ''a''''n'' gives a morphism
:''f: X'' → '''P'''''n'', ''x'' ↦ [''a''0(''x''): ... : ''a''''n''(''x'')],
such that the pullback ''f''*(''O''(1)) is ''F'' (Note that this evaluation makes sense when ''F'' is a subsheaf of the constant sheaf of rational functions on ''X''). The converse statement is also true: given such a morphism ''f'', the pullback of ''O''(1) is generated by its global sections (on ''X'').

=== Very ample line bundles ===
Given a [[scheme (mathematics)|scheme]] ''X'' over a base scheme ''S'' or a complex manifold, a line bundle (or in other words an [[invertible sheaf]], that is, a locally free sheaf of rank one) ''L'' on ''X'' is said to be '''very ample''', if there is an [[Glossary of scheme theory#Open and closed immersions|immersion]] ''i : X → '''''P'''''n''''S'', the ''n''-dimensional projective space over ''S'' for some ''n'', such that the [[inverse image functor|pullback]] of the [[Serre twist sheaf|standard twisting sheaf]] ''O''(1) on '''P'''''n''''S'' is isomorphic to ''L'':
:''i''*(O(1)) ≅ ''L''.

Hence this notion is a special case of the previous one, namely a line bundle is very ample if it is globally generated and the morphism given by some global generators is an immersion.

Given a very ample sheaf ''L'' on ''X'' and a [[coherent sheaf]] ''F'', a theorem of Serre shows that (the coherent sheaf) ''F ⊗ L⊗n'' is generated by finitely many global sections for sufficiently large ''n''. This in turn implies that global sections and higher (Zariski) [[Sheaf cohomology|cohomology]] groups
:''H''''i''(''X'', ''F'')
are finitely generated. This is a distinctive feature of the projective situation. For example, for the affine ''n''-space ''Ank'' over a field ''k'', global sections of the structure sheaf ''O'' are polynomials in ''n'' variables, thus not a finitely generated ''k''-vector space, whereas for '''P'''''n''''k'', global sections are just constant functions, a one-dimensional ''k''-vector space.

== Definitions ==
The notion of '''ample line bundles''' ''L'' is slightly weaker than very ample line bundles: ''L'' is called ample if some tensor power ''L⊗n'' is very ample. This is equivalent to the following definition: ''L'' is ample if for any coherent sheaf ''F'' on ''X'', there exists an integer ''n(F)'', such that ''F'' ⊗ ''L''⊗''n'' is generated by its global sections.

An equivalent, maybe more intuitive, definition of the ampleness of the line bundle <math>\mathcal L</math> is its having a positive tensorial power that is very ample. In other words, for <math>n \gg 0 </math> there exists a projective embedding <math>j: X \to \mathbb P^N</math> such that <math>\mathcal L^{\otimes n} = j^* (\mathcal O(1))</math>, that is the zero divisors of global sections of <math>\mathcal L^{\otimes n}</math>
are hyperplane sections.

This definition makes sense for the underlying ''divisors'' ([[Cartier divisor]]s) <math>D</math>; an ample <math>D</math> is one where <math>nD</math> ''moves in a large enough [[linear system of divisors|linear system]]''. Such divisors form a [[cone (topology)|cone]] in all divisors of those that are, in some sense, ''positive enough''. The relationship with projective space is that the <math>D</math> for a very ample <math>L</math> corresponds to the [[hyperplane section]]s (intersection with some [[hyperplane]]) of the embedded <math>M</math>.

The equivalence between the two definitions is credited to [[Jean-Pierre Serre]] in [[Faisceaux algébriques cohérents]].

==Criteria for ampleness of line bundles==
===Intersection theory===
To decide in practice when a Cartier divisor ''D'' corresponds to an ample line bundle, there are some geometric criteria.

For curves, a divisor ''D'' is very ample if and only if
''l''(''D'') = 2 + ''l''(''D'' − ''A'' − ''B'') whenever ''A'' and ''B'' are points. By the [[Riemann–Roch theorem]] every divisor of degree
at least 2''g'' + 1 satisfies this condition so is very ample. This implies that a divisor is ample if and only if it has positive degree. The [[canonical divisor]] of degree 2''g'' − 2 is very ample if and only if the curve is not
a [[hyperelliptic curve]].

The '''Nakai–Moishezon criterion''' ({{harvnb|Nakai|1963}}, {{harvnb|Moishezon|1964}}) states that a Cartier divisor ''D'' on a proper scheme ''X'' over an algebraically closed field is ample if and only if ''D''dim(''Y'').''Y'' > 0 for every closed integral subscheme ''Y'' of ''X''. In the special case of curves this says that a divisor is ample if and only if it has positive degree, and for a smooth projective [[algebraic surface]] ''S'', the Nakai–Moishezon criterion states that ''D'' is ample if and only if its [[self-intersection number]] ''D''.''D'' is strictly positive, and for any irreducible curve ''C'' on ''S'' we have ''D''.''C'' > 0.

The '''Kleiman condition''' states that for any [[projective variety|projective]] scheme ''X'', a divisor ''D'' on ''X'' is ample if and only if ''D''.''C'' > 0 for any nonzero element ''C'' in the [[closure (topology)|closure]] of NE(''X''), the [[cone of curves]] of ''X''. In other words a divisor is ample if and only if it is in the interior of the real cone generated by [[nef divisor]]s.

{{harvtxt|Nagata|1959}} constructed divisors on surfaces that have positive intersection with every curve, but are not ample.
This shows that the condition ''D''.''D'' > 0 cannot be omitted in the Nakai–Moishezon criterion, and it is necessary to use the closure of NE(''X'') rather than NE(''X'') in the Kleiman condition.

{{harvtxt|Seshadri|1972|loc=Remark 7.1, p. 549}} showed that a line bundle ''L'' on a complete algebraic scheme is ample if and only if there is some positive ε such that
deg(''L''|''C'') ≥ ε''m''(''C'') for all integral curves ''C'' in ''X'', where ''m''(''C'') is the
maximum of the multiplicities at the points of ''C''.

===Sheaves cohomology===

The theorem of [[Henri Cartan|Cartan]]-[[Jean-Pierre Serre|Serre]]-[[Grothendieck]] states that for a line bundle <math>\mathcal L</math> on a variety <math>X</math>, the following conditions are equivalent:
* <math>\mathcal L</math> is ample
* for ''m'' big enough, <math>\mathcal L^{\otimes m}</math> is very ample
* for any coherent sheaf <math>\mathcal F</math> on ''X'', the sheaf <math>\mathcal F \otimes \mathcal L^{\otimes m}</math> is generated by global sections, for ''m'' big enough
* for any coherent sheaf <math>\mathcal F</math> on ''X'', the [[sheaf cohomology|higher cohomology groups]] <math>H^i(X, \mathcal F \otimes \mathcal L^{\otimes m}), \ i \geq 1</math> vanish for ''m'' big enough.

==Generalizations==
=== Vector bundles of higher rank ===
A [[locally free sheaf]] ([[vector bundle]]) <math>F</math> on a variety is called '''ample''' if the invertible sheaf <math>\mathcal{O}(1)</math> on <math>\mathbb{P}(F)</math> is ample {{harvtxt|Hartshorne|1966}}.

Ample vector bundles inherit many of the properties of ample line bundles.

===Big line bundles===
{{main| Iitaka dimension}}
An important generalization, notably in [[birational geometry]], is that of a '''big line bundle'''. A line bundle <math>\mathcal L</math> on ''X'' is said to be big if the equivalent following conditions are satisfied:
*<math>\mathcal L</math> is the tensor product of an ample line bundle and an effective line bundle
*the [[Hilbert polynomial]] of the finitely generated [[graded ring]] <math>\bigoplus_{k=0}^\infty \Gamma (X, \mathcal L ^{\otimes k})</math> has degree the dimension of ''X''
*the rational mapping of the [[linear system of divisors|total system of divisors]] <math>X \to \mathbb P \Gamma (X, \mathcal L^{\otimes k})</math> is [[birational]] on its image for <math>k \gg 0</math>.
The interest of this notion is its stability with respect to rational transformations.

==See also==
===General algebraic geometry===
*[[Cartier divisor]]
*[[Algebraic geometry of projective spaces]]
*[[Fano variety]]: a variety whose [[Canonical line bundle]] is anti-ample

===Ampleness in complex geometry===
*[[Holomorphic vector bundle]]
*The [[Chern class]] is a characteristic form that detects ampleness of line bundles, this is the
*[[Kodaira embedding theorem]]: for compact complex manifolds, ampleness and positivity coincide.
*[[Lefschetz hyperplane theorem]]: the study of very ample line bundles on complex projective manifolds gives strong topological information

==References==
===Study references===
* {{Citation | last1=Hartshorne | first1=Robin | author1-link= Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | mr=0463157 | year=1977}}
* {{Citation | last1=Lazarsfeld | first1=Robert | author1-link= Robert Lazarsfeld | title=[[Positivity in Algebraic Geometry (book)|Positivity in Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin | year=2004}}
* The slides on ampleness in Vladimir Lazić's [http://www2.imperial.ac.uk/~vlazic/AGlect11.pdf Lectures on algebraic geometry]

===Research texts===
*{{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Ample vector bundles | url=http://www.numdam.org/item?id=PMIHES_1966__29__63_0 | mr=0193092 | year=1966 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=29 | pages=63–94}}
*{{Citation | doi=10.2307/1970447 | last1=Kleiman | first1=Steven L. | author1-link=Steven Kleiman | title=Toward a numerical theory of ampleness | jstor=1970447 | mr=0206009 | year=1966 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=84 | pages=293–344 | issue=3 | publisher=Annals of Mathematics}}
*{{Citation | last1=Moishezon | first1=B. G. | authorlink1 = Boris Moishezon | title=A projectivity criterion of complete algebraic abstract varieties | mr=0160782 | year=1964 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=28 | pages=179–224}}
* {{Citation | last1=Nagata | first1=Masayoshi | author1-link= Masayoshi Nagata | title=On the 14th problem of Hilbert | mr=0154867 | year=1959 | journal=[[American Journal of Mathematics]] | volume=81 | pages=766–772 | doi=10.2307/2372927 | jstor=2372927 | issue=3 | publisher=The Johns Hopkins University Press}}
*{{Citation | doi=10.2307/2373180 | last1=Nakai | first1=Yoshikazu | title=A criterion of an ample sheaf on a projective scheme | jstor=2373180 | mr=0151461 | year=1963 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=85 | pages=14–26 | issue=1 | publisher=The Johns Hopkins University Press}}
*{{Citation | doi=10.2307/1970870 | last1=Seshadri | first1=C. S. | title=Quotient spaces modulo reductive algebraic groups | jstor=1970870 | mr=0309940 | year=1972 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=95 | pages=511–556 | issue=3 | publisher=Annals of Mathematics}}

[[Category:Vector bundles]]
[[Category:Algebraic geometry]]
[[Category:Geometry of divisors]]

[[ko:넉넉한 선다발]]

Main Page

2014-08-13T01:20:49Z

JulioHindman:

The '''Wigner–Eckart theorem''' is a [[theorem]] of [[representation theory]] and [[quantum mechanics]]. It states that [[Matrix (mathematics)|matrix]] elements of [[spherical tensor]] [[Operator (physics)|operator]]s on the basis of [[angular momentum]] [[eigenstate]]s can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a [[Clebsch-Gordan coefficient]]. The name derives from physicists [[Eugene Wigner]] and [[Carl Eckart]] who developed the formalism as a link between the symmetry transformation groups of space (applied to the Schrödinger equations) and the laws of conservation of energy, momentum, and angular momentum.<ref name="Eckart Biography">[http://orsted.nap.edu/openbook.php?record_id=571&page=194 Eckart Biography]– The National Academies Press</ref>

The Wigner–Eckart theorem reads:

:<math>\langle jm|T^k_q|j'm'\rangle =\langle j||T^k||j'\rangle C^{jm}_{kqj'm'}</math>

where ''Tqk'' is a rank ''k'' spherical tensor, <math>|jm\rangle</math> and <math>|j'm'\rangle</math> are eigenkets of total angular momentum ''J''2 and its z-component ''Jz'', <math>\langle j||T^k||j'\rangle</math> has a value which is independent of ''m'' and ''q'', and <math>C^{jm}_{kqj'm'}=\langle j'm';kq|jm \rangle</math> is the Clebsch-Gordan coefficient for adding ''j''′ and ''k'' to get ''j''.

In effect, the Wigner–Eckart theorem says that operating with a spherical tensor operator of rank ''k'' on an angular momentum eigenstate is like adding a state with angular momentum ''k'' to the state. The matrix element one finds for the spherical tensor operator is proportional to a Clebsch-Gordan coefficient, which arises when considering adding two angular momenta. When stated another way, one can say that the Wigner-Eckart theorem is a theorem that tells you how vector operators behave in a subspace. Within a given subspace, a component of a vector operator will behave in a way proportional to the same component of the angular momentum operator. This definition is given in the book "Quantum Mechanics" by Cohen-Tannoudji, Diu and Laloe.

==Proof==

Starting with the definition of a [[spherical tensor]], we have that

<math>[J_{\pm}, T_q^{(k)}]=\hbar \sqrt{(k\mp q)(k\pm q+1)}T_{q\pm 1}^{(k)}</math>

which we use to then calculate

<math>\langle \alpha',j'm'|[J_{\pm}, T_q^{(k)}]|\alpha,jm\rangle=\hbar \sqrt{(k\mp q)(k\pm q+1)}\langle \alpha',j'm'|T_{q\pm 1}^{(k)}|\alpha,jm\rangle </math>.

If we expand the commutator on the LHS by calculating the action of the ''J''± on the bra and ket, then we get

<math> \begin{align}
\langle \alpha',j'm'|[J_{\pm}, T_q^{(k)}]|\alpha,jm\rangle
& = \sqrt{(j'\pm m')(j'\mp m'+1)}\langle \alpha',j'm'\mp1 |T_{q}^{(k)}|\alpha,jm\rangle\\
& \qquad -\sqrt{(j\mp m)(j\pm m+1)}\langle \alpha',j'm' |T_{q}^{(k)}|\alpha,jm\pm 1\rangle
\end{align} </math>.

We may combine these two results to get

<math> \begin{align}
\sqrt{(j'\pm m')(j'\mp m'+1)}\langle \alpha',j'm'\mp1 |T_{q}^{(k)}|\alpha,jm\rangle
& = \sqrt{(j\mp m)(j\pm m+1)}\langle \alpha',j'm' |T_{q}^{(k)}|\alpha,jm\pm 1\rangle\\
& \qquad +\sqrt{(k\mp q)(k\pm q+1)}\langle \alpha',j'm'|T_{q\pm 1}^{(k)}|\alpha,jm\rangle
\end{align} </math>.

This recursion relation for the matrix elements closely resembles that of the [[Clebsch-Gordan coefficient]]. In fact, both are of the form <math>\sum_j a_{ij}x_j=0</math>. We therefore have two sets of linear homogeneous equations

:<math>\sum_j a_{ij}x_j=0,\qquad \sum_j a_{ij}y_j=0</math>

one for the Clebsch-Gordan coefficients (''xj'') and one for the matrix elements (''yj''). It is not possible to exactly solve for the ''xj''. We can only say that the ratios are equal, that is

:<math>\frac{x_j}{x_k}=\frac{y_j}{y_k}</math>

or that ''xj'' = ''cyj'', where ''c'' is a coefficient of proportionality independent of the indices. Hence, by comparing recursion relations, we can identify the Clebsch-Gordan coefficient <math>\langle j_1 j_2; m_1,m_2\pm 1|j_1 j_2; jm \rangle</math> with the matrix element <math>\langle \alpha', j'm'|T_{q\pm 1}^{(k)}|\alpha, jm\rangle</math>, then we may write

:<math>\langle \alpha', j'm'|T_{q\pm 1}^{(k)}|\alpha, jm\rangle=\text{(proportionality constant)}\langle jk; mq\pm 1|jk;j'm'\rangle</math>.

By convention the proportionality constant is written as <math>\langle \alpha'j'||T^{(k)}||\alpha j\rangle \frac{1}{\sqrt{2j+1}}</math>, where the denominator is a normalizing factor.

==Example==

Consider the position expectation value <math>\langle njm|x|njm\rangle</math>. This matrix element is the expectation value of a Cartesian operator in a spherically-symmetric hydrogen-atom-eigenstate [[Basis (linear algebra)|basis]], which is a nontrivial problem. However, using the Wigner–Eckart theorem simplifies the problem. (In fact, we could obtain the solution quickly using [[Parity (physics)|parity]], although a slightly longer route will be taken.)

We know that ''x'' is one component of {{vec|''r''}}, which is a vector. Vectors are rank-1 tensors, so ''x'' is some linear combination of ''T''1q for ''q'' = -1, 0, 1. In fact, it can be shown that

:<math>x=\frac{T_{-1}^{1}-T^1_1}{\sqrt{2}}\,,</math>

where we defined the
[[spherical tensor]]s<ref name="J. Sakurai 1994">J. J. Sakurai: "Modern quantum mechanics" (Massachusetts, 1994, Addison-Wesley)</ref>
''T''10 = ''z''
and
:<math>T^1_{\pm1}=\mp (x \pm i y)/{\sqrt{2}}</math>
(the pre-factors have to be chosen according to the definition<ref name="J. Sakurai 1994"/> of a [[spherical tensor]] of rank ''k''. Hence, the ''T''1''q'' are only proportional to the [[ladder operators]]).
Therefore
:<math>\langle njm|x|n'j'm'\rangle = \langle njm|\frac{T_{-1}^{1}-T^1_1}{\sqrt{2}}|n'j'm'\rangle = \frac{1}{\sqrt{2}}\langle nj||T^1||n'j'\rangle (C^{jm}_{1(-1)j'm'}-C^{jm}_{11j'm'})</math>
The above expression gives us the matrix element for ''x'' in the <math>|njm\rangle</math> basis. To find the expectation value, we set ''n''′ = ''n'', ''j''′ = ''j'', and ''m''′ = ''m''. The selection rule for ''m''′ and ''m'' is <math>m\pm1=m'</math> for the <math>T_{\mp1}^{(1)}</math> spherical tensors. As we have ''m''′ = ''m'', this makes the Clebsch-Gordan Coefficients zero, leading to the expectation value to be equal to zero.

==References==

<references/>

==See Also==

*[[Tensor operator]]

==External links==

*J. J. Sakurai, (1994). "Modern Quantum Mechanics", Addison Wesley, ISBN 0-201-53929-2.
*{{mathworld|urlname=Wigner-EckartTheorem|title= Wigner–Eckart theorem}}
*[http://electron6.phys.utk.edu/qm2/modules/m4/wigner.htm Wigner–Eckart theorem]
*[http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/TensorOperators.htm Tensor Operators]

{{DEFAULTSORT:Wigner-Eckart theorem}}
[[Category:Quantum mechanics]]
[[Category:Representation theory of Lie groups]]
[[Category:Theorems in quantum physics]]
[[Category:Theorems in representation theory]]

Main Page

2014-08-12T22:05:12Z

JulioHindman:

The '''Virtual Router Redundancy Protocol''' ('''VRRP''') is a computer networking protocol that provides for automatic assignment of available [[Internet Protocol]] (IP) routers to participating hosts. This increases the availability and reliability of routing paths via automatic [[default gateway]] selections on an IP [[subnetwork]].

The protocol achieves this by creation of virtual routers, which are an abstract representation of multiple routers, i.e. master and backup [[router (computing)|router]]s, acting as a group. The default gateway of a participating host is assigned to the virtual router instead of a physical router. If the physical router that is [[routing]] packets on behalf of the virtual router fails, another physical router is selected to automatically replace it. The physical router that is forwarding packets at any given time is called the master router.

VRRP provides information on the state of a router, not the routes processed and exchanged by that router. Each VRRP instance is limited, in scope, to a single subnet. It does not advertise [[Internet Protocol|IP]] routes beyond that subnet or affect the [[routing]] table in any way.

VRRP can be used in [[Ethernet]], [[Multiprotocol Label Switching|MPLS]] and [[token ring]] networks with [[IPv4|Internet Protocol Version 4]] (IPv4), as well as [[IPv6]].

The protocol is described in IETF publication RFC 5798, which is an open standard, but a similar protocol with essentially the same facility is allegedly patented and licensed.<ref>[http://www.ietf.org/ietf-ftp/IPR/VRRP-CISCO IETF source]</ref>

==Implementation==
A virtual router must use 00-00-5E-00-01-XX as its [[Media Access Control]] (MAC) address. The last byte of the address (XX) is the Virtual Router IDentifier (VRID), which is different for each virtual router in the network. This address is used by only one physical router at a time, and it will reply with this MAC address when an ARP request is sent for the virtual router's IP address. Physical routers within the virtual router must communicate within themselves using packets with [[Multicast address|multicast]] [[Internet Protocol|IP]] address 224.0.0.18 and IP protocol number 112.<ref>[http://tools.ietf.org/html/rfc3768#section-5.2 Section 5.2.4. Protocol]</ref>

Routers have a priority of between 1-255 and the router with the highest priority will become the master. When a planned withdrawal of a master router is to take place, its priority can be lowered which means a backup router will pre-empt the master router status rather than having to wait for the hold time to expire. This reduces the black hole period.

==Elections of master routers==
A failure to receive a multicast packet from the master router for a period longer than three times the advertisement timer causes the backup routers to assume that the master router is dead. The virtual router then transitions into an unsteady state and an election process is initiated to select the next master router from the backup routers. This is fulfilled through the use of multicast packets.

Backup router(s) are only supposed to send multicast packets during an election process. One exception to this rule is when a physical router is configured with a higher priority than the current master, which means that on connection to the network it will preempt the master status. This allows a system administrator to force a physical router to the master state immediately after [[booting]], for example when that particular router is more powerful than others within the virtual router. The backup router with the highest priority becomes the master router by raising its priority above that of the current master. It will then take responsibility for routing packets sent to the virtual gateway's MAC address. In cases where backup routers all have the same priority, the backup router with the highest IP address becomes the master router.

All physical routers acting as a virtual router must be in the same LAN segment. Communication within the virtual router takes place periodically. This period can be adjusted by changing advertisement interval timers. The shorter the advertisement interval, the shorter the black hole period, though at the expense of more traffic in the network. Security is achieved by responding only to first hop packets, though other mechanisms are provided to reinforce this, particularly against local attacks. Election process is made orderly through the use of [[skew time]], derived from a router's priority and used to reduce the chance of the [[thundering herd problem]] occurring during election. The [[skew time]] is given by the formula <math>1 - \frac{Priority}{256}</math> (expressed in milliseconds).

Backup router utilization can be improved by load sharing. For more on this, see RFC 3768.

==History==
VRRP is based on Cisco's proprietary [[Hot Standby Router Protocol]] (HSRP) concepts. The protocols, while similar in concept, are not compatible. Therefore, on newer installations VRRP is usually implemented, because it is the standard and is supported by many router and switch products.

* (Cisco Example) '''VLAN Tagging'''

track 1 interface Serial0/0/0.1 ip routing ! Points at the interface that needs to be Prioritized
interface fastethernet0/0.1 ! VLANs have to be on a Sub-Interface, It is best practice to match the Sub-Interface # and the VLAN #
encapsulation dot1q 1 ! Enables IEEE 802.1Q VLAN frame tagging, followed by the VLAN # that this sub-interface will route
ip address x.x.x.x 255.255.255.0 ! Make sure the IP is on the same subnet as the virtual Gateway1
vrrp 1 priority 110 ! The Priority of the Gateway1
vrrp 1 ip <Gateway1> ! The Virtual Gateway for the VLAN 1
vrrp 1 preempt delay minimum 20 ! If the other router fails it will wait 20 sec before becoming the master
vrrp 1 track 1 decrement 15 ! If the S0/0/0.1 Link fails, This command drops the priority by 15
!
interface fastethernet0/0.5 ! VLANs have to be on a Sub-Interface, It is best practice to match the Sub-Interface # and the VLAN #
encapsulation dot1q 5 ! Enables IEEE 802.1Q VLAN frame tagging, followed by the VLAN # that this sub-interface will route
ip address x.x.x.x 255.255.255.0 ! Make sure the IP is on the same subnet as the virtual Gateway2
vrrp 5 priority 110 ! The Priority of the Gateway2
vrrp 5 ip <Gateway2> ! The Virtual Gateway for the VLAN 5
vrrp 5 preempt delay minimum 20 ! If the other router fails it will wait 20 sec before becoming the master
vrrp 5 track 1 decrement 15 ! If the Fa0/0.5 Link fails, This command drops the priority by 15
!
router bgp <ASN>
network <Gateway1> mask 255.255.255.0 ! Broadcasts Gateway1 out the WAN through BGP
network <Gateway2> mask 255.255.255.0 ! Broadcasts Gateway2 out the WAN through BGP

==See also==
* [[Common Address Redundancy Protocol]] (CARP) - A non-proprietary, patent-free, and unrestricted alternative to HSRP and VRRP.
* [[Gateway Load Balancing Protocol]] - A [[Cisco Systems]] proprietary router redundancy protocol providing load balancing
* [[Hot Standby Routing Protocol]] - A [[Cisco Systems]] proprietary router redundancy protocol
* [[R-SMLT]] (Routed Split Multilink Trunking) - An [[Avaya]] proprietary router redundancy and router load balancing protocol - replacement for VRRP in Avaya core networks
* [[SMLT]] An [[Avaya]] redundancy protocol
* [[First Hop Redundancy Protocols]] - Lists of default gateway redundancy protocols

==References==
{{reflist}}

==External links==
* [http://www.keepalived.org/ Keepalived 1.2.x adds ipv6 support]
* [http://tools.ietf.org/html/rfc5798 The current VRRP RFC (RFC 5798 - VRRPv3 for IPv4 and IPv6) which obsoletes RFC3768]
* [http://www.ietf.org/mail-archive/web/vrrp/current/maillist.html The IETF VRRP mailing list archive]
* [http://www.redbooks.ibm.com/redpapers/pdfs/redp3657.pdf A detailed VRRP article]
* [http://kerneltrap.org/comment/reply/477/1567 Controversy involving VRRP and Cisco patents]
* [http://web.archive.org/web/20080625055935/http://www.hanetworks.com/networks/nokia/vrrp/analysis_of_vrrpv2.htm Analysis of VRRPv 2 Issues and Solutions]
* Implementations
** [http://sourceforge.net/projects/vrrpd/ A GPL licensed implementation of VRRP designed for Linux operating systems]
** [http://sourceforge.net/projects/svrrpd/ A BSD licensed implementation of VRRP for Unix-like operating systems] (described as "not functional yet")
** [http://www.keepalived.org A GPL licensed implementation of VRRPv2 for Linux operating systems]
** [http://www.cisco.com/en/US/docs/ios/ipapp/configuration/guide/ipapp_vrrp.html Configuring VRRP on Cisco IOS]
** [http://support.3com.com/infodeli/tools/bridrout/u_guides/html/nb111/family/features/vrrp.htm Configuring VRRP on 3com NETBuilder]
** [[Vyatta]], a commercial open-source router / firewall with VRRP functionality.
** [http://www.jbm-web.com/cart/index.php?main_page=product_info&cPath=67&products_id=184 JBM C120 - A cellular enabled enterprise class router]

[[Category:Internet protocols]]
[[Category:Routing protocols]]

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