|
|
| Line 1: |
Line 1: |
| In [[mathematics]], a '''Kac–Moody algebra''' (named for [[Victor Kac]] and [[Robert Moody]], who independently discovered them) is a [[Lie algebra]], usually infinite-dimensional, that can be defined by generators and relations through a [[generalized Cartan matrix]]. These algebras form a generalization of finite-dimensional [[semisimple Lie algebra]]s, and many properties related to the structure of a Lie algebra such as its [[root system]], [[representation of a Lie algebra|irreducible representations]], and connection to [[flag manifold]]s have natural analogues in the Kac–Moody setting.
| | Rogozin's reaction is a dig at the most recent sanctions leveled by the U.S. in response to the Ukraine disaster. The White Home issued a press release that they may " deny export license functions for any high-technology gadgets that would contribute to Russia's army capabilities. These Departments also will revoke any present export licenses that meet these conditions." This immediately impacts the Russian tech sector, particularly satellites and the protection industry.<br><br>Prepared-fitted. If you go to a new launch show flat, you'll be awed at how posh the interior designs are. However you may be a little too naive to think that the fittings, furniture, electronics, and so on all come along with your purchase. The good factor is that a lot of the principal objects are included. Just ask the gross sales staff about it and they might make clear what gadgets are included and what aren't. You may even have the luxurious to pick out the type of flooring and tiles to use amongst others. You'll unlikely need to spend a huge sum on renovations. If you happen to do, perhaps you will have bought the wrong property.<br><br>The Panorama, positioned along Ang Mo Kio Ave 2, would be the next upcoming new launch with strolling distance to approaching new MRT Station, Mayflower, and likewise surrounded with Training Establishment with only a stone throw away Latest improvement by Hong Leong Holdings Pte Ltd location at Flora Drive. eight-storey condominium improvement comprising 1BR, 2BR, 2+Study, 3BR, 4BR, 4BR Twin Key & 3BR Roof Terrace Flats. Items are effectively-designed & useful layout fully-geared up with free oven, fridge, washer/ dryer (Microwave oven for selected items) Respond to the rental listings of new projects in the same district to check the recognition of properties for lease there. Feb Gross sales Plunge – Has the Storm Arrived? (at Propwise.sg) District 18, 999 12 months LH condo<br><br>Discounts are usually given throughout the preview, like 5~10% of the list value. Whatever the xx% reductions (differ from developers to builders), the underside line is the engaging PSF you'll take pleasure in to purchase on the very Preview day. One other benefit is that you've got the precedence to decide on your alternative unit should nobody else choose the same. In a sizzling property market, many units are snapped up at VIP Preview day, whereas some initiatives are absolutely sold even before the public come to know about it.<br><br>Freehold Condominium at Bukit Timah, near to approaching King Albert Park MRT. Total of 536 units from 1BR to 4BR, with 162 luxurious brand [http://www.fcscarousel.info/FCSCarousel_contents/?q=node/92923 new property for sale] and improved flats accessible for sale. Corporations are anticipated to offer jobs within the north that complement labour-intensive industries across the Causeway. With the extreme growth of Iskandar just across the border, more jobs are expected to be created inside the northern part of Singapore. The new coming intergrated CIQ complicated will likely be situated North of Woodlands MRT Station with the Northern terminus of the Thomson MRT Line Along with JB Sentral, they'll act because the link of Rapid Transit System (RTS), adding higher connectivity between Malaysia and Singapore. Private Apartment @ EC Prices! From Only $8xxPSF!<br><br>A particular, tailored strategy that naturally combines market expertise with private consumer service, any new launch apartment or property launch request or pledge required; AtasLaunch.com is the skilled group you can rely on to ship the answer, outcome and support that will surpass your expectations. The Denton Home was constructed within the late 1700s as a farm home. In the mid-1800s, the property was renovated, and it later saw commercial use as a funeral home When McDonald's stepped in to purchase the property in 1985, New Hyde Park residents have been upset at the concept that the franchise was coming to town with plans of tearing the constructing down. for allowing the corporate to drill for natural gasoline on their property. New Condo Waterfront @ Faber MAY<br><br>At AtasLaunch.com our thoughts is about on becoming the main and most admired marketer of new condominium and property launches in Singapore. When making essential property selections that impression you now and in the future, you want an exceptional actual estate accomplice. We perceive that your new property is a crucial part of your life, it brings warm to your loved ones, it is the place recollections are crystallised and it offers to your future. So for those who've ever puzzled why there seem to be so many "developer's crew" sales workers around in such Singapore property sales, nicely, that is why. The Peak @ Cairnhill - is the following upcoming Residential condominium in Singapore District 9. It is a Uncommon Freehold Growth that will outshine the Read Extra New Rental 2014 in Might FREEHOLD |
| | |
| A class of '''Kac–Moody algebras''' called '''[[affine Lie algebra]]s''' is of particular importance in mathematics and [[theoretical physics]], especially [[conformal field theory]] and the theory of [[exactly solvable model]]s. Kac discovered an elegant proof of certain combinatorial identities, the [[Macdonald identities]], which is based on the representation theory of affine Kac–Moody algebras. Howard Garland and [[James Lepowsky]] demonstrated that [[Rogers-Ramanujan identities]] can be derived in a similar fashion.<ref>(?) {{cite journal |first=H. |last=Garland |first2=J. |last2=Lepowsky |title=Lie algebra homology and the Macdonald-Kac formulas |journal=[[Inventiones Mathematicae|Invent. Math.]] |volume=34 |issue=1 |year=1976 |pages=37–76 |doi=10.1007/BF01418970 }}</ref>
| |
| | |
| == History of Kac-Moody algebras ==
| |
| The initial construction by [[Élie Cartan]] and [[Wilhelm Killing]] of finite dimensional [[simple Lie algebra]]s from the [[Cartan integer]]s was type dependent. In 1966 [[Jean-Pierre Serre]] showed that relations of [[Claude Chevalley]] and [[Harish-Chandra]],<ref name="H-C">{{cite journal |last=Harish-Chandra |title=On some applications of the universal enveloping algebra of a semisimple Lie algebra |journal=[[Transactions of the American Mathematical Society|Trans. Amer. Math. Soc.]] |volume=70 |issue=1 |year=1951 |pages=28–28 |doi= 10.1090/S0002-9947-1951-0044515-0|jstor=1990524 }}</ref> with simplifications by [[Nathan Jacobson]],<ref name="Ja">{{cite book |last=Jacobson |first=N. |title=Lie algebras |series=Interscience Tracts in Pure and Applied Mathematics |volume=10 |publisher=Interscience Publishers (a division of John Wiley & Sons) |location=New York-London |year=1962 }}</ref> give a defining presentation for the [[Lie algebra]].<ref name="Se">{{cite book |last=Serre |first=J.-P. |title={{lang|fr|Algèbres de Lie semi-simples complexes}} |language=French |publisher=W. A. Benjamin |location=New York-Amsterdam |year=1966 }}</ref> One could thus describe a simple Lie algebra in terms of generators and relations using data from the matrix of Cartan integers, which is naturally [[positive-definite matrix|positive definite]].
| |
| | |
| In his 1967 thesis, [[Robert Moody]] considered Lie algebras whose [[Cartan matrix]] is no longer positive definite.<ref name="M1">{{cite journal |last=Moody |first=R. V. |title=Lie algebras associated with generalized cartan matrices |journal=Bull. Amer. Math. Soc. |volume=73 |issue= 2|year=1967 |pages=217–222 |url=http://www.ams.org/journals/bull/1967-73-02/S0002-9904-1967-11688-4/S0002-9904-1967-11688-4.pdf |doi=10.1090/S0002-9904-1967-11688-4 }}</ref><ref name="M2">Moody 1968, ''A new class of Lie algebras''</ref> This still gave rise to a Lie algebra, but one which is now infinite dimensional. Simultaneously, '''Z'''-[[graded Lie algebra]]s were being studied in Moscow where [[I. L. Kantor]] introduced and studied a general class of Lie algebras including what eventually became known as '''Kac–Moody algebras'''.<ref name="Kan">{{cite journal |last=Kantor |first=I. L. |title=Graded Lie algebras |language=Russian |journal=Trudy Sem. Vektor. Tenzor. Anal. |volume=15 |issue= |year=1970 |pages=227–266 |doi= }}</ref> [[Victor Kac]] was also studying simple or nearly simple Lie algebras with polynomial growth. A rich mathematical theory of infinite dimensional Lie algebras evolved. An account of the subject, which also includes works of many others is given in (Kac 1990).<ref>Kac, 1990</ref> See also (Seligman 1987).<ref name="Sel">{{cite journal |last=Seligman |first=George B. |title=Book Review: Infinite dimensional Lie algebras |journal=Bull. Amer. Math. Soc. |series=N.S. |volume=16 |year=1987 |issue=1 |pages=144–150 |doi=10.1090/S0273-0979-1987-15492-9 }}</ref>
| |
| | |
| ==Definition==
| |
| A Kac–Moody algebra is given by the following:
| |
| # An ''n''×''n'' [[generalized Cartan matrix]] {{nowrap|1=''C'' = (''c<sub>ij''</sub>)}} of [[rank (linear algebra)|rank]] ''r''.
| |
| # A [[vector space]] <math>\mathfrak{h}</math> over the [[complex number]]s of dimension 2''n'' − ''r''.
| |
| # A set of ''n'' [[linearly independent]] elements <math>\alpha_i^\vee\ </math> of <math>\mathfrak{h}</math> and a set of ''n'' linearly independent elements <math>\alpha_i</math> of the [[dual space]] <math>\mathfrak{h}^*</math>, such that <math>\alpha_i(\alpha_j^\vee) = c_{ji}</math>. The <math>\alpha_i</math> are analogue to the [[Root system of a semi-simple Lie algebra|simple roots]] of a semi-simple Lie algebra, and the <math>\alpha_i^\vee</math> to the simple coroots.
| |
| | |
| The Kac–Moody algebra is the Lie algebra <math>\mathfrak{g}</math> defined by [[generating set|generators]] <math>e_i</math> and <math>f_i</math> (<math>i \in \{1,\ldots,n\}</math>) and the elements of <math>\mathfrak{h}</math> and relations | |
| *<math>[h,h'] = 0\ </math> for <math>h,h' \in \mathfrak{h}</math>;
| |
| *<math>[h,e_i] = \alpha_i(h)e_i</math>, for <math>h \in \mathfrak{h}</math>;
| |
| *<math>[h,f_i] = -\alpha_i(h)f_i</math>, for <math>h \in \mathfrak{h}</math>;
| |
| *<math>[e_i,f_j] = \delta_{ij}\alpha_i^\vee </math>, where <math> \delta_{ij}</math> is the Kronecker delta;
| |
| *<math>\textrm{ad}(e_i)^{1-c_{ij}}(e_j) = 0</math> and <math>\textrm{ad}(f_i)^{1-c_{ij}}(f_j) = 0</math>, where <math>\textrm{ad}: \mathfrak{g}\to\textrm{End}(\mathfrak{g}),\textrm{ad}(x)(y)=[x,y],</math> is the [[Adjoint representation of a Lie algebra|adjoint representation]] of <math>\mathfrak{g}</math>.
| |
| | |
| A [[real number|real]] (possibly infinite-dimensional) [[Lie algebra]] is also considered a Kac–Moody algebra if its [[complexification]] is a Kac–Moody algebra.
| |
| | |
| ==Root-space decomposition of a Kac-Moody algebra==
| |
| <math>\mathfrak{h}</math> is the analogue of a [[Cartan subalgebra]] for the Kac–Moody algebra <math>\mathfrak{g}</math>. | |
| | |
| If <math>x\neq 0</math> is an element of <math>\mathfrak{g}</math> such that
| |
| | |
| :<math>\forall h\in\mathfrak{h}, [h,x]=\lambda(h)x</math>
| |
| | |
| for some <math>\lambda\in\mathfrak{h}^*\backslash\{0\}</math>, then <math>x</math> is called a '''root vector''' and <math>\lambda</math> is a '''root''' of <math>\mathfrak{g}</math>. (The zero functional is not considered a root by convention.) The set of all roots of <math>\mathfrak{g}</math> is often denoted by <math>\Delta</math> and sometimes by <math>R</math>. For a given root <math>\lambda</math> one denotes by <math>\mathfrak{g}_\lambda</math> the '''root space''' of <math>\lambda</math>, that is
| |
| | |
| :<math>\mathfrak{g}_\lambda = \{x\in\mathfrak{g}:\forall h\in\mathfrak{h}, [h,x] = \lambda(h)x\}</math>. | |
| | |
| It follows from the defining relations of <math>\mathfrak{g}</math> that <math>e_i\in\mathfrak{g}_{\alpha_i}</math> and <math>f_i\in\mathfrak{g}_{-\alpha_i}</math>. Also, if <math>x_1\in\mathfrak{g}_{\lambda_1}</math> and <math>x_2\in\mathfrak{g}_{\lambda_2}</math>, then <math>[x_1,x_2]\in\mathfrak{g}_{\lambda_1+\lambda_2}</math> by the [[Jacobi identity]].
| |
| | |
| A fundamental result of the theory is that any Kac–Moody algebra can be decomposed into the [[direct sum]] of <math>\mathfrak{h}</math> and its root spaces, that is
| |
| | |
| :<math> \mathfrak{g} = \mathfrak{h}\oplus\bigoplus_{\lambda\in\Delta} \mathfrak{g}_\lambda</math>,
| |
| | |
| and that every root <math>\lambda</math> can be written as <math>\lambda = \sum_{i=1}^n z_i\alpha_i</math> with all the <math>z_i</math> being [[integers]] of the same [[Sign (mathematics)|sign]].
| |
| | |
| ==Types of Kac–Moody algebras==
| |
| Properties of a Kac–Moody algebra are controlled by the algebraic properties of its generalized Cartan matrix ''C''. In order to classify Kac–Moody algebras, it is enough to consider the case of an ''indecomposable'' matrix''C'', that is, assume that there is no decomposition of the set of indices ''I'' into a disjoint union of non-empty subsets ''I''<sub>1</sub> and ''I''<sub>2</sub> such that ''C''<sub>''ij''</sub> = 0 for all ''i'' in ''I''<sub>1</sub> and ''j'' in ''I''<sub>2</sub>. Any decomposition of the generalized Cartan matrix leads to the direct sum decomposition of the corresponding Kac–Moody algebra:
| |
| | |
| : <math>\mathfrak{g}(C)\simeq\mathfrak{g}(C_1)\oplus\mathfrak{g}(C_2),</math>
| |
| | |
| where the two Kac–Moody algebras in the right hand side are associated with the submatrices of ''C'' corresponding to the index sets ''I''<sub>1</sub> and ''I''<sub>2</sub>.
| |
| | |
| An important subclass of Kac–Moody algebras corresponds to ''[[symmetrizable matrix|symmetrizable]]'' generalized Cartan matrices ''C'', which can be decomposed as ''DS'', where ''D'' is a [[diagonal matrix]] with positive integer entries and ''S'' is a [[symmetric matrix]]. Under the assumptions that ''C'' is symmetrizable and indecomposable, the Kac–Moody algebras are divided into three classes:
| |
| | |
| *A [[positive definite matrix]] ''S'' gives rise to a finite-dimensional [[simple Lie algebra]].
| |
| *A [[positive semidefinite matrix]] ''S'' gives rise to an infinite-dimensional Kac–Moody algebra of '''affine type''', or an [[affine Lie algebra]].
| |
| *An [[indefinite matrix]] ''S'' gives rise to a Kac–Moody algebra of '''indefinite type'''.
| |
| *Since the diagonal entries of ''C'' and ''S'' are positive, ''S'' cannot be [[negative definite matrix|negative definite]] or negative semidefinite.
| |
| | |
| Symmetrizable indecomposable generalized Cartan matrices of finite and affine type have been completely classified. They correspond to [[Dynkin diagram]]s and [[affine Dynkin diagram]]s. Very little is known about the Kac–Moody algebras of indefinite type. Among those, the main focus has been on the (generalized) Kac–Moody algebras of '''hyperbolic type''', for which the matrix ''S'' is indefinite, but for each proper subset of ''I'', the corresponding submatrix is positive definite or positive semidefinite. Such matrices have rank at most 10 and have also been completely determined.<ref>{{cite journal |last=Carbone |first=L. |last2=Chung |first2=S. |last3=Cobbs |first3=C. |last4=McRae |first4=R. |last5=Nandi |first5=D. |last6=Naqvi |first6=Y. |last7=Penta |first7=D. |title=Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits |journal=[[Journal of Physics A|J. Phys. A: Math. Theor.]] |volume=43 |issue=15 |pages=155209 |year=2010 |doi=10.1088/1751-8113/43/15/155209 |arxiv=1003.0564 }}</ref>
| |
| | |
| ==See also==
| |
| *[[Weyl–Kac character formula]]
| |
| *[[Generalized Kac–Moody algebra]]
| |
| <!--
| |
| ==Notes==
| |
| <references />--> | |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *[[Robert Moody|R.V. Moody]], ''A new class of Lie algebras'', [[Journal of Algebra|J. of Algebra]], 10 (1968) pp. 211–230
| |
| *[[Victor Kac|V. Kac]], ''Infinite dimensional Lie algebras'', 3rd edition, Cambridge University Press (1990) ISBN 0-521-46693-8 [http://books.google.com/books?id=kuEjSb9teJwC&lpg=PP1&dq=Victor%20G.%20Kac&pg=PP1#v=onepage&q&f=false]
| |
| *A. J. Wassermann, [http://arxiv.org/abs/1004.1287 Lecture notes on Kac–Moody and Virasoro algebras]
| |
| *{{springer|id=K/k055050|author=|title=Kac–Moody algebra}}
| |
| *V.G. Kac, ''Simple irreducible graded Lie algebras of finite growth'' Math. USSR Izv., 2 (1968) pp. 1271–1311, Izv. Akad. Nauk USSR Ser. Mat., 32 (1968) pp. 1923–1967
| |
| *[[Shrawan Kumar|S. Kumar]], ''Kac-Moody Groups, their Flag Varieties and Representation Theory'', 1st edition, Birkhäuser (2002). ISBN 3-7643-4227-7.
| |
| | |
| ==External links==
| |
| * [http://www.emis.de/journals/SIGMA/Kac-Moody_algebras.html SIGMA: Special Issue on Kac-Moody Algebras and Applications]
| |
| | |
| {{DEFAULTSORT:Kac-Moody Algebra}}
| |
| [[Category:Lie algebras]]
| |
Rogozin's reaction is a dig at the most recent sanctions leveled by the U.S. in response to the Ukraine disaster. The White Home issued a press release that they may " deny export license functions for any high-technology gadgets that would contribute to Russia's army capabilities. These Departments also will revoke any present export licenses that meet these conditions." This immediately impacts the Russian tech sector, particularly satellites and the protection industry.
Prepared-fitted. If you go to a new launch show flat, you'll be awed at how posh the interior designs are. However you may be a little too naive to think that the fittings, furniture, electronics, and so on all come along with your purchase. The good factor is that a lot of the principal objects are included. Just ask the gross sales staff about it and they might make clear what gadgets are included and what aren't. You may even have the luxurious to pick out the type of flooring and tiles to use amongst others. You'll unlikely need to spend a huge sum on renovations. If you happen to do, perhaps you will have bought the wrong property.
The Panorama, positioned along Ang Mo Kio Ave 2, would be the next upcoming new launch with strolling distance to approaching new MRT Station, Mayflower, and likewise surrounded with Training Establishment with only a stone throw away Latest improvement by Hong Leong Holdings Pte Ltd location at Flora Drive. eight-storey condominium improvement comprising 1BR, 2BR, 2+Study, 3BR, 4BR, 4BR Twin Key & 3BR Roof Terrace Flats. Items are effectively-designed & useful layout fully-geared up with free oven, fridge, washer/ dryer (Microwave oven for selected items) Respond to the rental listings of new projects in the same district to check the recognition of properties for lease there. Feb Gross sales Plunge – Has the Storm Arrived? (at Propwise.sg) District 18, 999 12 months LH condo
Discounts are usually given throughout the preview, like 5~10% of the list value. Whatever the xx% reductions (differ from developers to builders), the underside line is the engaging PSF you'll take pleasure in to purchase on the very Preview day. One other benefit is that you've got the precedence to decide on your alternative unit should nobody else choose the same. In a sizzling property market, many units are snapped up at VIP Preview day, whereas some initiatives are absolutely sold even before the public come to know about it.
Freehold Condominium at Bukit Timah, near to approaching King Albert Park MRT. Total of 536 units from 1BR to 4BR, with 162 luxurious brand new property for sale and improved flats accessible for sale. Corporations are anticipated to offer jobs within the north that complement labour-intensive industries across the Causeway. With the extreme growth of Iskandar just across the border, more jobs are expected to be created inside the northern part of Singapore. The new coming intergrated CIQ complicated will likely be situated North of Woodlands MRT Station with the Northern terminus of the Thomson MRT Line Along with JB Sentral, they'll act because the link of Rapid Transit System (RTS), adding higher connectivity between Malaysia and Singapore. Private Apartment @ EC Prices! From Only $8xxPSF!
A particular, tailored strategy that naturally combines market expertise with private consumer service, any new launch apartment or property launch request or pledge required; AtasLaunch.com is the skilled group you can rely on to ship the answer, outcome and support that will surpass your expectations. The Denton Home was constructed within the late 1700s as a farm home. In the mid-1800s, the property was renovated, and it later saw commercial use as a funeral home When McDonald's stepped in to purchase the property in 1985, New Hyde Park residents have been upset at the concept that the franchise was coming to town with plans of tearing the constructing down. for allowing the corporate to drill for natural gasoline on their property. New Condo Waterfront @ Faber MAY
At AtasLaunch.com our thoughts is about on becoming the main and most admired marketer of new condominium and property launches in Singapore. When making essential property selections that impression you now and in the future, you want an exceptional actual estate accomplice. We perceive that your new property is a crucial part of your life, it brings warm to your loved ones, it is the place recollections are crystallised and it offers to your future. So for those who've ever puzzled why there seem to be so many "developer's crew" sales workers around in such Singapore property sales, nicely, that is why. The Peak @ Cairnhill - is the following upcoming Residential condominium in Singapore District 9. It is a Uncommon Freehold Growth that will outshine the Read Extra New Rental 2014 in Might FREEHOLD