Supergroup (physics)

The concept of supergroup is a generalization of that of group. In other words, every supergroup carries a natural group structure, but there may be more than one way to structure a given group as a supergroup. A supergroup is like a Lie group in that there is a well defined notion of smooth function defined on them. However the functions may have even and odd parts. Moreover, a supergroup has a super Lie algebra which plays a role similar to that of a Lie algebra for Lie groups in that they determine most of the representation theory and which is the starting point for classification.

Details

More formally, a Lie supergroup is a supermanifold G together with a multiplication morphism $\mu :G\times G\rightarrow G$ , an inversion morphism $i:G\rightarrow G$ and a unit morphism $e:1\rightarrow G$ which makes G a group object in the category of supermanifolds. This means that, formulated as commutative diagrams, the usual associativity and inversion axioms of a group continue to hold. Since every manifold is a super manifold, a Lie supergroup generalises the notion of a Lie group.

There are many possible supergroups. The ones of most interest in theoretical physics are the ones which extend the Poincaré group or the conformal group. Of particular interest are the orthosymplectic groups Osp(M|N)^[1] and the superunitary groups SU(M|N).

An equivalent algebraic approach starts from the observation that a super manifold is determined by its ring of supercommutative smooth functions, and that a morphism of super manifolds corresponds one to one with an algebra homomorphism between their functions in the opposite direction, i.e. that the category of supermanifolds is opposite to the category of algebras of smooth graded commutative functions. Reversing all the arrows in the commutative diagrams that define a Lie supergroup then shows that functions over the supergroup have the structure of a Z₂-graded Hopf algebra. Likewise the representations of this Hopf algebra turn out to be Z₂-graded comodules. This Hopf algebra gives the global properties of the supergroup.

There is another related Hopf algebra which is the dual of the previous Hopf algebra. It can be identified with the Hopf algebra of graded differential operators at the origin. It only gives the local properties of the symmetries i.e., it only gives information about infinitesimal supersymmetry transformations. The representations of this Hopf algebra are modules. Like in the non-graded case, this Hopf algebra can be described purely algebraically as the universal enveloping algebra of the Lie superalgebra.

In a similar way one can define an affine algebraic supergroup as a group object in the category of superalgebraic affine varieties. An affine algebraic supergroup has a similar one to one relation to its Hopf algebra of superpolynomials. Using the language of schemes, which combines the geometric and algebraic point of view, algebraic supergroup schemes can be defined including super Abelian varieties.

Notes

↑ (M|N) is pronounced "M vertical bar N." M denotes the bosonic dimensions and N denotes the Grassmann dimensions (cf. Larus Thorlacius, Thordur Jonsson (eds.), M-Theory and Quantum Geometry, Springer, 2012, p. 263).

References

supergroup in nLab

String theory
Background	Strings History of string theory First superstring revolution Second superstring revolution String theory landscape
Theory	Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string Type IIA string theory Type IIB string theory Heterotic string N=2 superstring F-theory String field theory Matrix string theory Non-critical string theory Nonlinear sigma model Tachyon condensation RNS formalism GS formalism
String Duality	T-duality S-duality U-duality Montonen–Olive duality Mysterious duality
Particles and Fields	Graviton Dilaton Tachyon Ramond–Ramond field Kalb–Ramond field Magnetic monopole Dual graviton Dual photon
Branes	D-brane NS5-brane M2-brane M5-brane S-brane Black brane Black holes Black string Brane cosmology Quiver diagram Hanany–Witten transition
Conformal Field Theory	Virasoro algebra Mirror symmetry Conformal anomaly Conformal algebra Superconformal algebra Vertex operator algebra Loop algebra Kac–Moody algebra Wess–Zumino–Witten model
Gauge Theory	Anomalies Instantons Chern–Simons form Bogomol'nyi–Prasad–Sommerfield bound Exceptional Lie groups (G₂, F₄, E₆, E₇, E₈) ADE classification Dirac string p-form electrodynamics
Geometry	Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G₂ manifold Spin(7) manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten domain wall K-theory (physics) Twisted K-theory
Supersymmetry	Supergravity Superspace Lie superalgebra Lie supergroup
Holography	Holographic principle AdS/CFT correspondence
M-theory	Matrix theory Introduction to M-theory
String theorists	Nima Arkani-Hamed Michael Francis Atiyah Tom Banks David Berenstein Raphael Bousso Gerald Cleaver Thomas Curtright Robbert Dijkgraaf Jacques Distler Michael Douglas Michael Duff Sergio Ferrara Willy Fischler Daniel Friedan Sylvester James Gates Ferdinando Gliozzi Rajesh Gopakumar Michael Green Brian Greene David Gross Steven Gubser Sergei Gukov Jeffrey Harvey Petr Hořava Gary Gibbons Shamit Kachru Michio Kaku Renata Kallosh Theodor Kaluza Anton Kapustin Igor Klebanov Vadim Knizhnik Maxim Kontsevich Oskar Klein Andrei Linde Juan Maldacena Stanley Mandelstam Donald Marolf Emil Martinec Shiraz Minwalla Gregory Moore Lubos Motl Sunil Mukhi Robert Myers Dimitri Nanopoulos Asad Naqvi Horatiu Nastase Nikita Nekrasov André Neveu Holger Bech Nielsen Peter van Nieuwenhuizen Sergei Novikov David Olive Hirosi Ooguri Joseph Polchinski Alexander Polyakov Arvind Rajaraman Pierre Ramond Lisa Randall Seifallah Randjbar-Daemi Martin Roček Ryan Rohm Joël Scherk John H. Schwarz Nathan Seiberg Ashoke Sen Steve Shenker Warren Siegel Eva Silverstein Đàm Thanh Sơn Matthias Staudacher Andrew Strominger Raman Sundrum Leonard Susskind Gerard 't Hooft Paul Townsend Sandip Trivedi Cumrun Vafa Gabriele Veneziano Erik Verlinde Herman Verlinde Julius Wess Edward Witten Shing-Tung Yau Tamiaki Yoneya Alexander Zamolodchikov Alexei Zamolodchikov Eric Zaslow Bruno Zumino Barton Zwiebach

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.