In mathematics, a braided vectorspace
is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
together with an additional structure map
symbolizing interchanging of two vector
tensor copies:
::
such that the
Yang–Baxter equation
In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve the ...
is fulfilled. Hence drawing
tensor diagrams with
an overcrossing the corresponding composed morphism is unchanged when a
Reidemeister move
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
Life
He was a brother of Marie Neurath.
Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttinge ...
is applied to the tensor diagram and thus they present a representation of the
braid group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
.
As first example, every vector space is braided via the trivial braiding (simply flipping). A
superspace
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numb ...
has a braiding with negative sign in braiding two odd vectors. More generally, a diagonal braiding means that for a
-base
we have
::
A good source for braided vector spaces entire
braided monoidal categories In mathematics, a ''commutativity constraint'' \gamma on a monoidal category ''\mathcal'' is a choice of isomorphism \gamma_ : A\otimes B \rightarrow B\otimes A for each pair of objects ''A'' and ''B'' which form a "natural family." In particu ...
with braidings between any objects
, most importantly the modules over
quasitriangular Hopf algebra
In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of H \otimes H such that
:*R \ \Delta(x)R^ = (T \circ \Delta)(x) for all x \in H, where \Delta is the cop ...
s and
Yetter–Drinfeld modules over finite groups (such as
above)
If
additionally possesses an
algebra structure inside the braided category ("braided algebra") one has a braided commutator (e.g. for a
superspace
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numb ...
the
anticommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
):
::
Examples of such braided algebras (and even
Hopf algebras Hopf is a German surname. Notable people with the surname include:
* Eberhard Hopf (1902–1983), Austrian mathematician
* Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
*Heinz Hopf (actor) (1934–2001), Swed ...
) are the
Nichols algebra In algebra, the Nichols algebra of a braided vector space (with the braiding often induced by a finite group) is a braided Hopf algebra which is denoted by \mathfrak(V) and named after the mathematician Warren Nichols. It takes the role of quantum ...
s, that are by definition generated by a given braided vectorspace. They appear as quantum Borel part of
quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...
s and often (e.g. when finite or over an abelian group) possess an
arithmetic root system, multiple
Dynkin diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
s and a
PBW-basis made up of braided commutators just like the ones in
semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
s.
[Andruskiewitsch, Schneider: ''Pointed Hopf algebras'', New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.]
Hopf algebras
Quantum groups
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