Definition I: Combinatorical formula

Let ${\displaystyle V}$ be a braided vector space, this means there is an action of the braid group ${\displaystyle \mathbb {B} _{n}}$ on ${\displaystyle V^{\otimes n}}$ for any ${\displaystyle n\in \mathbb {N} }$, where the transposition ${\displaystyle (i,i+1)}$ acts as ${\displaystyle id\otimes \cdots \otimes \tau \otimes id\cdots id}$. Clearly there is a homomorphism to the symmetric group ${\displaystyle \pi$ :\mathbb {B} _{n}\to \mathbb {S} _{n}} but neither does this admit a section, nor does the action on ${\displaystyle V^{\otimes n}}$ in general factorize over this.

Consider nevertheless a set-theoretic section ${\displaystyle s:\mathbb {S} _{n}\to \mathbb {B} _{n}}$ 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 ${\displaystyle s(\sigma )}$ on ${\displaystyle V^{\otimes n}}$ is well-defined independently of the choice of a reduced expression. Finally the Nichols algebra is then

${\displaystyle {\mathfrak {W}}_{n}:=\sum _{\sigma \in \mathbb {S} _{n}}s(\sigma ):\;\;V^{\otimes n}\to V^{\otimes n}\qquad {\text{(quantum symmetrizer or quantum shuffle map)}}}$

${\displaystyle {\mathfrak {B}}(V):=\bigoplus _{n\geq 0}V^{\otimes n}/Ker({\mathfrak {W}}_{n})}$

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 ${\displaystyle {\mathfrak {B}}(V)}$ is the unique Hopf algebra in the braided category ${\displaystyle {}_{H}^{H}{\mathcal {YD}}}$ generated by the given ${\displaystyle V\in {}_{H}^{H}{\mathcal {YD}}}$, such that ${\displaystyle V\subset {\mathfrak {B}}(V)}$ 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 ${\displaystyle V\in {}_{H}^{H}{\mathcal {YD}}}$. There exists a largest ideal ${\displaystyle {\mathfrak {I}}\subset TV}$ with the following properties:

${\displaystyle {\mathfrak {I}}\subset \bigoplus _{n=2}^{\infty }T^{n}V,}$

${\displaystyle \Delta ({\mathfrak {I}})\subset {\mathfrak {I}}\otimes TV+TV\otimes {\mathfrak {I}}}$ (this is automatic)

The Nichols algebra is

${\displaystyle {\mathfrak {B}}(V):=TV/{\mathfrak {I}}}$

Definition IV: Nondegenerate Pairing

The unique Hopf pairing ${\displaystyle V\otimes TV^{*}\to k}$ factorizes to a nondegenerate Hopf pairing between ${\displaystyle {\mathfrak {B}}(V)\otimes {\mathfrak {B}}(V^{*})\to k}$ 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 ${\displaystyle v_{i}\in V}$ (i.e. coaction/graduation ${\displaystyle v_{i}\mapsto g_{i}\otimes v_{i}}$) one may define skew derivations ${\displaystyle \partial _{i}}$, using the universal property of the tensor algebra:

${\displaystyle \partial _{i}(1)=0\quad \partial _{i}(v_{j})=\delta _{ij}}$

${\displaystyle \partial _{i}(ab)=a\partial _{i}(b)+\partial _{i}(a)(g_{i}.b)}$

Then the Nichols algebra ${\displaystyle {\mathfrak {B}}(V)}$ is the quotient of ${\displaystyle TV}$ by the largest homogeneous ideal which contains no constants and is invariant under all derivations ${\displaystyle \partial _{i}}$. Roughly spoken, one may look in ${\displaystyle TV}$ 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.

1-dimensional examples

As a first example, consider the 1-dimensional Yetter–Drinfeld module ${\displaystyle V_{\pm }=kx}$ 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 ${\displaystyle V_{\pm }}$: ${\displaystyle x\mapsto g\otimes x}$
• Take as H-action (resp. Z/2Z-action) on ${\displaystyle V_{\pm }}$: ${\displaystyle g\otimes x\mapsto \pm x}$
• Thus the braiding is ${\displaystyle x\otimes x\rightarrow \pm x\otimes x}$

Then, depending on the sign choice, the Nichols algebras are:

${\displaystyle {\mathfrak {B}}(V_{+})=k[x]\qquad {\mathfrak {B}}(V_{-})=k[x]/(x^{2})}$

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, V_q over a higher cyclic group with g acting by some q in k has Nichols algebra ${\displaystyle {\mathfrak {B}}(V_{q})=k[x]/(x^{n})}$ if q ≠ 1 is a primitive n-th root of unity, and ${\displaystyle {\mathfrak {B}}(V_{q})=k[x]}$ 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 V_0,1 = 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 V_0,1: ${\displaystyle x\mapsto g\otimes x}$ and ${\displaystyle x\mapsto g\otimes x}$
• Take as H-action (resp. Z/2Z-action) on V_0,1:
  • ${\displaystyle g\otimes x\mapsto -x}$
  • ${\displaystyle g\otimes y\mapsto +y}$
  • ${\displaystyle h\otimes y\mapsto -y}$
  • ${\displaystyle h\otimes x\mapsto \pm x}$ with "+" for V₀ (symmetric) and "–" for V₁ (asymmetric)
• Thus the braiding is
  • ${\displaystyle x\otimes x\rightarrow -x\otimes x}$
  • ${\displaystyle y\otimes y\rightarrow -y\otimes y}$
  • ${\displaystyle x\otimes y\rightarrow y\otimes x}$
  • ${\displaystyle y\otimes x\rightarrow \pm x\otimes y}$

Then, depending on the sign choice, the Nichols algebras are of dimension 4 and 8 (they appear in the classification under ${\displaystyle q_{12}q_{21}=\pm 1}$):

${\displaystyle {\mathfrak {B}}(V_{0})=k[x,y]/(x^{2},y^{2},xy+yx),}$

${\displaystyle {\mathfrak {B}}(V_{1})=k[x]/(x^{2},y^{2},xyxy+yxyx)}$

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 ${\displaystyle A_{1}\cup A_{1}}$ 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

${\displaystyle V:=x_{1}\mathbb {C} \oplus x_{2}\mathbb {C} \oplus \cdots \oplus x_{n}\mathbb {C} }$

be the diagonal Yetter-Drinfel'd module over an abelian group ${\displaystyle \Lambda =\mathbb {Z} ^{n}=\langle K_{1},\ldots ,K_{n}\rangle }$ with braiding

${\displaystyle x_{i}\otimes x_{j}\mapsto q_{ij}x_{j}\otimes x_{i}\qquad q_{ij}:=q^{(\alpha _{i},\alpha _{j})}}$

where ${\displaystyle (\alpha _{i},\alpha _{j})}$ is the Killing form of a semisimple (finite-dimensional) Lie algebra ${\displaystyle {\mathfrak {g}}}$, then the Nichols algebra is the positive part of Lusztig's small quantum group

${\displaystyle {\mathfrak {B}}(V)=u_{q}({\mathfrak {g}})^{+}}$

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 ${\displaystyle {\mathcal {O}}_{[g]}^{\chi }}$ 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 ${\displaystyle {\mathcal {O}}_{[g]}^{\chi }}$, 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 ${\displaystyle \mathbb {S} _{3},\mathbb {S} _{4},\mathbb {S} _{5},\mathbb {D} _{4}}$ but already the case ${\displaystyle \mathbb {S} _{6}}$ 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.

Definition of Weyl groupoid and generalized roots system

Let ${\displaystyle I=\{1,\ldots n\}}$ where ${\displaystyle n}$ is the rank, with formal basis ${\displaystyle \alpha _{1},\ldots \alpha _{n}}$.

We first discuss generalized Cartan graphs as in:^[6]

• A generalized Cartan matrix ${\displaystyle c_{ij},\;\;i,j\in I}$ is an integral matrix such that
  • ${\displaystyle c_{ii}=2,\quad c_{ij}<0}$
  • ${\displaystyle c_{ij}=0\Rightarrow c_{ji}=0}$
• A Cartan graph is a set of such Cartan matrices ${\displaystyle c_{ij}^{a},\;\forall a\in A}$ parametrized by a set of objects/chambers ${\displaystyle A}$, together with (object change) morphism ${\displaystyle r_{i}:A\to A}$ such that
  • ${\displaystyle r_{i}^{2}=id}$
  • ${\displaystyle c_{ij}^{a}=c_{ij}^{r_{i}(a)}}$
• Define maps

${\displaystyle s_{i}^{a}:\mathbb {Z} ^{I}\to \mathbb {Z} ^{I}}$

${\displaystyle s_{i}^{a}:\;\alpha _{j}\mapsto \alpha _{j}-c_{ij}^{a}\alpha _{i}}$

(note that Lie algebra literature has also the transpose convention for ${\displaystyle c_{ij}}$, e.g. in Humphrey's book)

• The Weyl groupoid is the category with objects ${\displaystyle A}$ and morphisms ${\displaystyle Hom(a,b)}$ formally the groups generated by the ${\displaystyle s_{i}^{a}:a\to r_{i}(a)}$
• The set of real roots ${\displaystyle \Phi _{a}^{+}}$ is the set ${\displaystyle \{id^{a}s_{i_{1}}\cdots s_{i_{k}}(\alpha _{j})\,|\,k\in \mathbb {N} _{0},\,i_{1},\dots ,i_{k},j\in I\}\subseteq \mathbb {N} ^{I}}$
• Define ${\displaystyle m_{i,j}^{a}=|\Phi ^{a}\cap (\mathbb {N} _{0}\alpha _{i}+\mathbb {N} _{0}\alpha _{j})|}$,
• Then a root system ${\displaystyle \Phi }$ of type ${\displaystyle (c_{i,j}^{a})_{a\in A}}$ is a set
  • ${\displaystyle \Phi ^{a}=\Phi _{+}^{a}\cup -\Phi _{+}^{a}}$ with ${\displaystyle \Phi _{+}^{a}=\Phi ^{a}\cap \mathbb {N} _{0}^{I}}$
  • ${\displaystyle \Phi ^{a}\cap \mathbb {Z} \alpha _{i}=\{\alpha _{i},-\alpha _{i}\}}$
  • ${\displaystyle s_{i}^{a}(\Phi ^{a})=\Phi ^{r_{i}(a)}}$
  • For ${\displaystyle i\neq j}$ with ${\displaystyle m_{ij}}$ finite ${\displaystyle (r_{i}r_{j})^{m_{i,j}^{a}}(a)=a}$

Equivalence to Crystallographic Hyperplane Arrangements

In ^[7] it was shown that Weyl groupoids are in 1:1 correspondence to crystallographic hyperplane arrangements. These are a set of hyperplanes in ${\displaystyle \mathbb {R} ^{n}}$ through the origin and choices of normal vectors such that for every simplicial chamber bounded by ${\displaystyle n}$ hyperplanes with normal vectors ${\displaystyle \alpha _{1}^{\perp },\ldots \alpha _{n}^{\perp }}$ all other chosen normal vector ${\displaystyle \alpha ^{\perp }}$ can be expressed as integral linear combination of the ${\displaystyle \alpha _{i}^{\perp }}$.

In ^[8] 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 ${\displaystyle A_{n},B_{n},C_{n},D_{n},E_{6},E_{7},E_{8},F_{4},G_{2}}$ there is one more infinite family and altogether 74 exceptionswith rank up to ${\displaystyle 8}$.

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 ${\displaystyle 7}$ roots (${\displaystyle 4}$ resp. ${\displaystyle 3}$ hyperplanes, in the pictures bounding equilateral triangle resp. diagonals in squares, in the super Lie algebra odd resp. even roots). It visibly has ${\displaystyle 2}$ 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:

${\displaystyle \Phi _{+}^{a}=\{(1,0,0),(0,1,0),(0,0,1),(1,1,0),(1,0,1),(0,1,1),(1,1,1)\}}$

In the picture the white chamber, e.g with basis ${\displaystyle \alpha _{1},\alpha _{2},\alpha _{3}}$. Clearly, the Dynkin diagram of this type of chamber ${\displaystyle a\in A}$ is a simply-laced triangle,

Reflection on ${\displaystyle \alpha _{2}}$ brings us to the second type of chamber

${\displaystyle \Phi _{+}^{a'}=\{(1,0,0),(0,1,0),(0,0,1),(1,1,0),(0,1,1),(1,1,1),(1,2,1)\}}$

In the picture the gray chamber, e.g with basis ${\displaystyle \alpha _{1}',\alpha _{2}',\alpha _{3}'=\alpha _{12},-\alpha _{2},\alpha _{23}}$. The Dynkin diagram of this type of chamber ${\displaystyle a'\in A}$ is just ${\displaystyle A_{2}}$ (but one more root).

This root system is the smallest member of an infinite series. The pictures are from,^[9] where the example is also discussed thoroughly.

Over abelian groups

The Nichols algebras of finite dimension over abelian groups in k = C were classified by Istvan Heckenberger^[2] in the years 2004–2005 by classifying arithmetic root systems and generalized Dynkin diagrams; where already Kharchenko had proven them to possess a Poincaré–Birkhoff–Witt basis of iterated (braided) commutators. The only information one requires is the braiding matrix, which is diagonal in this setting (see examples above)

${\displaystyle x_{i}\otimes x_{j}\mapsto q_{ij}x_{j}\otimes x_{i}}$

While mostly only the classical Cartan-cases appear, there are several exotic diagrams possible for small primes, such as a triangle

A rank 3 Dynkin diagram associated to a finite-dimensional Nichols algebra

In these cases the Weyl reflections of one diagram may not land in the "same" diagram, but a so-called Weyl equivalent. This is also the exact reason, that these exotic cases possess a Weyl-groupoid instead of a usual group.

The generators and relations of a Nichols algebra are not readily available from the root system. Rather, one has to perform tedious work with the Lynond words. This has been completely done in ^[10]

Negative criteria: abelian subracks

Especially for irreducible V there are no submodules; however one may use the more abstract notion of subrack only reflecting the braiding of two contained elements. In several papers, Nicolas Andruskiewitsch et al. gave negative criteria excluding groups at all from possessing (indecomposable) Nichols algebras. Their techniques can be roughly summarized^[11] (more details!):

Consider a subrack that is abelian, check which representation may be inherited from the larger rack, and looked up in Heckenbegers List ^[2]

This ansatz puts sometimes strong conditions especially on the braiding of any g-graded element x with itself (e.g. the first example above shows q ≠ 1). Note that because g is central in the centralizer, it acts on the irreducible representation by a scalar as a consequence of the Schur lemma; hence this selfbraiding resp. 1-dim sub-Yetter-Drinfeld module / braided vectorspace / 1-dim subrack is diagonal

${\displaystyle x\otimes x\;{\stackrel {\tau }{\longmapsto }}\;q(x\otimes x)\;\;\Longleftrightarrow \;\;g.x=qx}$

It is usually used to excludes g e.g. of being of odd order and/or χ of high dimension:^[12]

• If g is real (i.e. conjugated to its inverse) then q = –1 (especially g has to be of even order)
• If g is quasi-real (i.e. conjugated to some j-th power) then
  • either q = –1 as above
  • or ${\displaystyle g^{(j^{2})}=g}$ and the representation χ is one-dimensional with q = ζ₃ a primitive 3rd root of unity (especially the order of g is divisible by 3)
• If contrary g is an involution and some centralizing h = tgt then the eigenvalues of the h (viewed as matrix) acting on ${\displaystyle {\mathcal {O}}_{[g]}^{\chi }}$ is strongly restricted.

Root systems over nonabelian groups

The existence of a root system also in the nonabelian case ^[3] implies rather immediately the following very strong implications:

Immediate consequences are implied for rank 2 Nichols algebras ${\displaystyle {\mathfrak {B}}\left({\mathcal {O}}_{[g]}\oplus {\mathcal {O}}_{[h]}\right)}$ which g, h discommuting; then:

• The braided commutators [x, y] of elements ${\displaystyle x\in {\mathcal {O}}_{[g]}\;y\in {\mathcal {O}}_{[h]}}$ are not all zero.
• The space of braided commutators ${\displaystyle ad_{{\mathcal {O}}_{[g]}}{\mathcal {O}}_{[h]}=[{\mathcal {O}}_{[g]},{\mathcal {O}}_{[h]}]}$ form an irreducible sub-Yetter–Drinfeld module ${\displaystyle {\mathcal {O}}_{[gh]}}$ (i.e. the root is unique as in the Lie algebra case)
• They're '"close to commuting" ${\displaystyle \;(gh)^{2}=(hg)^{2}}$

This implies roughly, that finite-dimensional Nichols algebras over nonabelian groups have to be (if at all) of very low rank or the group has to be close-to-abelian.

Negative criteria: nonabelian subracks (type D)

As the abelian subracks use the structural classification of Heckenberger for Nichols algebras over abelian groups (see above) one can also consider nonabelian subracks. If such a subrack decomposes into several pieces (because now less element are present to conjugate), then the above results on root systems apply.

A specific case^[12] where this is highly successful is type D, i.e. for ${\displaystyle r,s\in [g]\;}$

• r, s not conjugate in the generated subgroup ${\displaystyle \langle r,s\rangle \;}$
• ${\displaystyle (rs)^{2}\neq (sr)^{2}\;}$

in this case the Nichols algebra of the subrack is infinite-dimensional and so is the entire Nichols algebra

Known groups not admitting finite-dimensional Nichols algebras

Both negation techniques above have been very fruitful to negate (indecomposable) finite-dimensional Nichols algebras:^[12]

• for Alternating groups ${\displaystyle \mathbb {A} _{n\geq 5}}$ ^[13]
• for Symmetric groups ${\displaystyle \mathbb {S} _{n\geq 6}}$ except a short list of examples^[13]
• some group of Lie type (sources, complete list?)
• all Sporadic groups except a short list of possibilities (resp. conjugacy classes in ATLAS notation) that are all real or j = 3-quasireal:
  • ...for the Fischer group ${\displaystyle Fi_{22}\;}$ the classes ${\displaystyle 22A,22B\;}$
  • ...for the baby monster group B the classes ${\displaystyle 16C,\;16D,\;32A,\;32B,\;32C,\;32D,\;34A,\;46A,\;46B\;}$
  • ...for the monster group M the classes ${\displaystyle 32A,\;32B,\;46A,\;46B,\;92A,\;92B,\;94A,\;94B\;}$

Usually a large amount of conjugacy classes ae of type D ("not commutative enough"), while the others tend to possess sufficient abelian subracks and can be excluded by their consideration. Several cases have to be done by-hand. Note that the open cases tend to have very small centralizers (usually cyclic) and representations χ (usually the 1-dimensional sign representation). Significant exceptions are the conjugacy classes of order 16, 32 having as centralizers p-groups of order 2048 resp. 128 and currently no restrictions on χ.