What is "supergroup"

n. 1 Any group composed of other groups (in any of several contexts) 2 (context music English) A group of musicians, each a star of another band (especially in rock music) 3 (context physics English) A generalization of groups, used in the study of supersymmetry. 4 (cx telecommunications English) In the (w: L-carrier) system, a multiplexed group of channel groups.

A supergroup is a music group whose members are already successful as solo artists or as part of other groups or well known in other musical professions. Usually used in the context of rock and pop music, the term has been applied to other musical genres such as The Three Tenors in opera.

The term is sometimes applied retrospectively when several members from a group later achieve notable success in their own right. Supergroups are sometimes formed as side projects and thus not intended to be permanent, while other times can become the primary project of the members' careers. Charity supergroups, where prominent musicians perform or record together in support of a particular cause, have been common since the 1980s.

SuperGroup plc is a British international branded clothing company, and owner of the Superdry label. Superdry products combine vintage Americana styling with Japanese inspired graphics. It is listed on the London Stock Exchange and is a constituent of the FTSE 250 Index.

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.

More formally, a Lie supergroup is a supermanifold G together with a multiplication morphism μ : G × G → G, an inversion morphism i : G → G and a unit morphism e : 1 → 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(N/M) and the superunitary groups SU(N/M).

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 Z-graded Hopf algebra. Likewise the representations of this Hopf algebra turn out to be Z-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.

Category:Supersymmetry Category:Hopf algebras

SuperGroup was a 2006 reality show on VH1 that followed five well-known hard rock and heavy metal musicians over a 12-day period during which they lived together in a Las Vegas mansion in order to create, plan and perform a live show together. The show, which aired in seven segments, starred band members Sebastian Bach, Jason Bonham, Scott Ian, Ted Nugent and Evan Seinfeld. Doc McGhee, who had previously worked with Bach and his band Skid Row, appeared as the band's manager.

Journalist Lonn Friend and photographer Ross Halfin, who were long-time contributors for other VH1 features such as Behind the Music, appeared on the show.