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 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 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 (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 systems and Dynkin diagrams, strikingly similar to those of semisimple Lie algebras. A comprehensive introduction is found in the lecture of Heckenberger.
Definition Consider a Yetter–Drinfeld module V in the Yetter–Drinfeld category . This is especially a braided vectorspace, see Braided monoidal category.
The tensor algebra of a Yetter–Drinfeld module is always a Braided Hopf algebra. 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) 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 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 H = k[Z/2Z] with the Cyclic group 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 otherwise.
(from a physical perspective, the V+ corresponds to a boson, while V– represents a fermion restricted by Pauli exclusion principle; an analogy that repeats when considering braided commutators, being (anti)commutators in these cases, see also Supersymmetry as a quantum group and discussion)
Higher-rank examples over G abelian: braided commutators The next examples show the interaction of two basis elements: Consider the two-dimensional Yetter–Drinfeld module V0,1 = kx ⊕ ky over the group Hopf algebra H = k[Z/2Z × Z/2Z] with the Klein four group multiplicatively denoted and generated by some g,h. Take as H-coaction/graduation on V0,1: and Take as H-action (resp. Z/2Z-action) on V0,1:
with "+" for V0 (symmetric) and "–" for V1 (asymmetric) Thus the braiding is
Then, depending on the sign choice, the Nichols algebras are of dimension 4 and 8 (they appear in the classification under ):
There one can see the striking resemblance to Semisimple Lie algebras: In the first case, the braided commutator [x, y] (here: anticommutator) is zero, while in the second, the root string is longer [x, [x, y]] = 0. Hence these two belong to Dynkin diagrams and A2.
One also constructs examples with even longer root strings V2, V3 corresponding to Dynkin diagrams B2, G2 (but as well no higher ones).
Universal enveloping of Lie algebras, Quantum groups
Nichols algebras are probably best known for being the Borel part of the quantum groups and their generalizations. More precisely let
be the diagonal Yetter-Drinfel'd module over an abelian group with braiding
where is the Killing form of a semisimple (finite-dimensional) Lie algebra , then the Nichols algebra is the positive part of Lusztig's small quantum group
Includes Super-Lie algebras
There are more diagonal Nichols algebras than Lie algebras in Heckenbergers list, and the root system theory is systematic, but more complicated (see below). In particular is contains also the classification of Super-Lie-Algebras (example below) as well as certain Lie algebras and Super-Lie-Algebras that only appear in a specific finite characteristic.
Thus Nichols algebra theory and root system theory provides a unified framework for these concepts.
Nondiagonal braidings, Nonabelian groups Only a handful of finite-dimensional Nichols algebras over k = C are known so far. It is known that in this case each irreducible Yetter–Drinfeld module corresponds to Conjugacy class of the group (together with an irreducible representation of the centralizer of g). An arbitrary Yetter–Drinfeld module is a direct sum of such , the number of summands is called rank; each summand corresponds to anode in the Dynkin diagram (see below). Note that for the abelian groups as above, the irreducible summands are 1-dimensional, hence rank and dimension coincide.
Particular examples include the Nichols algebra associated to the conjugacy class(es) of reflections in a Coxeter group, they are related to the Fomin Kirilov algebras. It is known these Nichols algebras are finite dimensional for but already the case is open since 2000. Another class of examples can be constructed from abelian case by a folding through diagram automorphisms.
See here for a list List of finite-dimensional Nichols algebras to the extent of our knowledge.
Root system
A very remarkable feature is that for every Nichols algebra (under sufficient finiteness conditions) there exists a generalized root system with a set of roots , which controls the Nichols algebra. This has been discovered in for diagonal Nichols algebras in terms of the bicharacter and in for general semisimple Nichols algebras. In contrast to ordinary crystallographic root systems known from Lie algebras, the same generalized root system may possess several be different Weyl chambers, corresponding to non-equivalent choices of sets of positive roots and simple positive roots , having different Cartan matrices and different Dynkin diagrams.
The different Weyl chambers correspond in fact to different non-isomorphic Nichols algebras which are called Weyl-equivalent. Quantum groups are very special with respect to the fact that here all Borel parts are isomorphic; nevertheless even in this case Lusztig's reflection operator is again not a Hopf algebra isomorphism!
Definition of Weyl groupoid and generalized roots system
Let where is the rank, with formal basis .
We first discuss generalized Cartan graphs as in: A generalized Cartan matrix is an integral matrix such that
A Cartan graph is a set of such Cartan matrices parametrized by a set of objects/chambers , together with (object change) morphism such that
Define maps
(note that Lie algebra literature has also the transpose convention for , e.g. in Humphrey's book) The Weyl groupoid is the category with objects and morphisms formally the groups generated by the The set of real roots is the set Define , Then a root system of type is a set with
For with finite
Equivalence to Crystallographic Hyperplane Arrangements
In it was shown that Weyl groupoids are in 1:1 correspondence to crystallographic hyperplane arrangements. These are a set of hyperplanes in through the origin and choices of normal vectors such that for every simplicial chamber bounded by hyperplanes with normal vectors all other chosen normal vector can be expressed as integral linear combination of the .
In the set of all finite crystallographic hyperplane arrangements (and hence finite Weyl groupoids or finite generalized root systems) have been classified. Apart from the reflection arrangements there is one more infinite family and altogether 74 exceptionswith rank up to .
Example of rank 3 (also a super Lie algebra)
The smallest crystallographic hyperplane arrangement, Weyl groupoid, generalized root system, which is not of ordinary Lie type, is as follows. It appears for a diagonal Nichols algebra, even a super Lie algebra. The hyperplane arrangement can be constructed from a cuboctahedron (a platonic solid):
It has roots ( resp. hyperplanes, in the pictures bounding equilateral triangle resp. diagonals in squares, in the super Lie algebra odd resp. even roots). It visibly has different types of Weyl chambers (equilateral triangles resp. right triangles) with different Cartan matrices in which the roots in terms of simple roots are as follows:
In the picture the white chamber, e.g with basis . Clearly, the Dynkin diagram of this type of chamber is a simply-laced triangle, Reflection on brings us to the second type of chamber
In the picture the gray chamber, e.g with basis . The Dynkin diagram of this type of chamber is just (but one more root).