In algebra, the Nichols algebra of a
braided vector space (with the braiding often induced by a finite group) is a
braided Hopf algebra In mathematics, a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf algebra ''H'', particularly the Nichols algebra of a braided vec ...
which is denoted by
and named after the mathematician Warren Nichols. It takes the role of quantum Borel part of a pointed
Hopf algebra[ such as a ]quantum groups
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 ...
and their well known finite-dimensional truncations. Nichols algebras can immediately be used to write down new such quantum groups by using the Radford biproduct.
The classification of all such Nichols algebras and even all associated quantum groups
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 ...
(see Application) has been progressing rapidly, although still much is open: The case of an abelian group was solved in 2005,[ but otherwise this phenomenon seems to be very rare, with a handful examples known and powerful negation criteria established (see below). See also this List of finite-dimensional Nichols algebras.
The finite-dimensional theory is greatly governed by a theory of ]root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
s and 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, strikingly similar to those of 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.[ A comprehensive introduction is found in the lecture of Heckenberger.][
]
Definition
Consider a Yetter–Drinfeld module ''V'' in the Yetter–Drinfeld category In mathematics a Yetter–Drinfeld category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms.
Definition
Let ''H'' be a Hopf algebra over a field ''k''. Let \Delt ...
. This is especially a braided vectorspace, see Braided monoidal category 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 ...
.
The tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
of a Yetter–Drinfeld module is always a Braided Hopf algebra In mathematics, a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf algebra ''H'', particularly the Nichols algebra of a braided vec ...
. The coproduct and counit of is defined in such a way that the elements of are primitive, that is
for all
::
::
The Nichols algebra can be uniquely defined by several equivalent characterizations, some of which focus on the Hopf algebra structure and some are more combinatorial. Regardless, determining the Nichols algebra explicitly (even decide if it's finite-dimensional) can be very difficult and is open in several concrete instances (see below).
Definition I: Combinatorical formula
Let be a braided vector space, this means there is an action of the braid group on for any , where the transposition acts as . Clearly there is a homomorphism to the symmetric group but neither does this admit a section, nor does the action on in general factorize over this.
Consider nevertheless a set-theoretic section sending transposition to transposition and arbitrary elements via any reduced expression. This is not a group homomorphism, but Matsumoto's theorem (group theory) In group theory, Matsumoto's theorem, proved by , gives conditions for two reduced words of a Coxeter group to represent the same element.
Statement
If two reduced words represent the same element of a Coxeter group, then Matsumoto's theorem stat ...
tells us that the action of any on is well-defined independently of the choice of a reduced expression. Finally the Nichols algebra is then
::
::
This definition was later (but independently) given by Woronowicz. It has the disadvantage of being rarely useful in algebraic proofs but it represents an intuition in its own right and it has the didactical advantage of being very explicit and independent of the notation of a Hopf algebra.
Definition II: Prescribed primitives
The Nichols algebra is the unique Hopf algebra in the braided category generated by the given , such that are the only primitive elements.
This is the original definition due to Nichols and it makes very transparent the role of the Nichols algebra as a fundamental notion in the classification of Hopf algebras.
Definition III: Universal quotient
Let . There exists a largest ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
with the following properties:
::
:: (this is automatic)
The Nichols algebra is
::
Definition IV: Nondegenerate Pairing
The unique Hopf pairing factorizes to a nondegenerate Hopf pairing between and this fact characterizes the Nichols algebra uniquely. This theoretically very helpful characterization is due to Lusztig.
Definition V: Skew derivatives
This is a somewhat explicit form of the previous definition: Chosen a homogeneous basis (i.e. coaction/graduation ) one may define skew derivations , using the universal property of the tensor algebra:
::
::
Then the Nichols algebra is the quotient of by the largest homogeneous ideal which contains no constants and is invariant under all derivations . Roughly spoken, one may look in for elements in the kernel of all skew-derivations and divide these out; then look again for all elements that are now in the kernel of all skew-derivatives and divide them out as well etc.
Examples
We give examples of finite-dimensional Nichols algebras. Over characteristic ''p'', this effect already may appear in the non-braided situation, namely the truncated universal envelopings of p-restricted Lie algebras. In characteristic zero and with a braiding coming from an abelian group, this seems to be a similarly frequent occurrence (however more involved, see Classification). For ''G'' nonabelian on the other side, only very few examples are known so far, and powerful negation criteria exclude many groups at all (see Classification).
1-dimensional examples
As a first example, consider the 1-dimensional Yetter–Drinfeld module over the Group Hopf algebra
In mathematics, the group Hopf algebra of a given group (mathematics), group is a certain construct related to the symmetries of Group action (mathematics), group actions. Deformations of group Hopf algebras are foundational in the theory of quantu ...
''H'' = ''k'' ''Z/2Zwith the Cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
multiplicatively denoted (as usual in algebra) and generated by some ''g''.
* Take as ''H''-coaction (resp. Z/2Z-graduation) on :
* Take as ''H''-action (resp. Z/2Z-action) on :
* Thus the braiding is
Then, depending on the sign choice, the Nichols algebras are:
::
Note that the first is as expected (the non-braided case), while the second has been truncated to the point that it's finite-dimensional! Similarly, ''Vq'' over a higher cyclic group with ''g'' acting by some ''q'' in ''k'' has Nichols algebra if ''q'' ≠ 1 is a primitive ''n''-th root of unity, and