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| 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 <math>\mathfrak{B}(V)</math> and named after the mathematician Warren Nichols. It takes the role of quantum Borel part of a pointed Hopf algebra <ref name=AS02/> 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 [[Braided Hopf algebra|Radford biproduct]]<ref name=AS02/>
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| Susan Montgomory asked<ref name=HS08/> to classify which finite groups admit at all finite dimensional indecomposable Nichols algebras over ''k'' = '''C'''. The answering of this question, moreover the classification of all such Nichols algebras and even all associated [[quantum groups]] (see Application) is recently progressing rapidly, although still much is open: The case of an abelian group has been solved 2005,<ref name=H05/> but otherwise this phenomenon seems to be a very rare occasion, with a handful examples known and powerful negation criteria established (see below).
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| The finite dimensional theory is greatly governed by a theory of [[root system]]s and [[Dynkin diagram]]s, strikingly similar to those of [[semisimple Lie algebra]]s.<ref name=HS08/> A comprehensive introduction is found in the lecture of Heckenberger <ref name=H08/>
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| ==Definition==
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| Consider a Yetter–Drinfeld module ''V'' in the [[Yetter–Drinfeld category]] <math> {}^H_H\mathcal{YD}</math>. This is especially a braided vectorspace, see [[Braided monoidal category]].
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| The [[tensor algebra]] ''TV'' of a Yetter–Drinfeld module <math> V\in {}^H_H\mathcal{YD}</math> is always a [[Braided Hopf algebra]]. The coproduct Δ of ''TV'' is defined in such a way that the elements of ''V'' are primitive, that is
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| ::<math>\Delta(v)=1\otimes v+v\otimes 1</math> for all ''v'' in ''V''.
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| ::The counit ε: ''TV'' → ''k'' then satisfies the equation ε(''v'') = 0 for all ''v'' in ''V''.
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| As an example, all [[Universal enveloping algebra|universal envelopings]] of Lie algebras are (trivially braided) quotients thereof for <math>V=\mathfrak{g}</math> by [[Semisimple Lie algebra|Serre relations]]: The basis consists of derivational elements (=primitives), the higher powers usually solemnly of higher derivations and there are no further quotients still preserving ''V'' ("universal"). In the braided context, the "true" enveloping turns out to be often much smaller:
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| ''(note that in the further development of the theory, see below, the Nichols algebra takes rather the role of the quantum Borel part of the enveloping!)''
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| ::There are '''three equivalent characterizations''', as proven by W. Nichols around 1978 <ref name=MS00 /> ''(better source ? see discussion)'' They essentially means, that the "superfluous elements" in the braided Hopf algebra ''TV'' are exactly corresponding to "extra-primitives" arrising in higher degree, and are exactly detected by all partial skew-derivations being zero "as if it were a constant".
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| ===Definition I: Universal quotient===
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| Let <math>V\in {}^H_H\mathcal{YD}</math>. There exists a largest [[ideal (ring theory)|ideal]] of ''TV'' with the following properties:
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| :: <math> I\subset \bigoplus _{n=2}^\infty T^nV,</math>
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| ::<math> \Delta (I)\subset I\otimes TV+TV\otimes I.</math>
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| :(hence ''I'' is also a [[Coalgebra|coideal]]) One has <math>I\in {}^H_H\mathcal{YD}</math>, and the quotient ''TV/I'' is a braided Hopf algebra in <math> {}^H_H\mathcal{YD}</math>, the '''Nichols algebra'''. It is the "smallest" braided Hopf algebra containing ''V'' as a primitive elements (''a-priori'' among others, but see below!)
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| ===Definition II: Prescribed primitives===
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| There's a '''unique''' braided Hopf algebra, generated by ''V'' being primitives, which contains no other primitive elements. It is as well the Nichols algebra.
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| ===Definition III: Skew derivatives===
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| Chosen a homogeneous basis <math> v_i\in V .</math> (i.e. coaction/graduation <math> v_i\mapsto g_i\otimes v_i</math>) one may define '''skew derivations''' <math>\partial_i</math>, using the universal property of the tensor algebra:
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| ::<math>\partial_i(1)=0 \quad \partial_i(v_j)=\delta_{ij} </math>
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| ::<math>\partial_i(ab)=a\partial_i(b)+\partial_i(a)(g_i.b) </math>
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| Then the Nichols algebra is the quotient of ''TV'' by the largest homogeneous ideal which contains no constants and is invariant under all derivations <math>\partial_i</math>.
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| ==Examples==
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| 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).
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| ===1-dimensional examples===
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| As a first example, consider the 1-dimensional Yetter–Drinfeld module <math>V_\pm=kx</math> over the [[Group Hopf algebra]] ''H'' = ''k''['''Z'''/2'''Z'''] with the [[Cyclic group]] multiplicatively denoted (as usual in algebra) and generated by some ''g''.
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| * Take as ''H''-coaction (resp. '''Z'''/2'''Z'''-graduation) on <math>V_\pm</math>: <math> x\mapsto g\otimes x</math>
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| * Take as ''H''-action (resp. '''Z'''/2'''Z'''-action) on <math>V_\pm</math>: <math> g\otimes x\mapsto \pm x</math>
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| * Thus the braiding is <math>x\otimes x\rightarrow \pm x\otimes x</math>
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| Then, depending on the sign choice, the Nichols algebras are:
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| ::<math>\mathfrak{B}(V_+)=k[x]\qquad \mathfrak{B}(V_-)=k[x]/(x^2)</math>
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| 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<sub>q</sub>'' over a higher cyclic group with ''g'' acting by some ''q'' in ''k'' has Nichols algebra <math>\mathfrak{B}(V_q)=k[x]/(x^n)</math> if ''q'' ≠ 1 is a primitive ''n''-th root of unity, and <math>\mathfrak{B}(V_q)=k[x]</math> otherwise.
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| ''(from a physical perspective, the ''V''<sub>+</sub> corresponds to a boson, while ''V''<sub>–</sub> 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)''
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| ===Higher-rank examples over ''G'' abelian: braided commutators===
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| The next examples show the interaction of two basis elements: Consider the two-dimensional Yetter–Drinfeld module ''V''<sub>0,1</sub> = ''kx'' ⊕ ''ky'' over the [[group Hopf algebra]] ''H'' = ''k''['''Z'''/2'''Z''' × '''Z'''/2'''Z'''] with the [[Klein four group]] multiplicatively denoted and generated by some ''g,h''.
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| * Take as ''H''-coaction/graduation on ''V''<sub>0,1</sub>: <math> x\mapsto g\otimes x</math> and <math> x\mapsto g\otimes x</math>
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| * Take as ''H''-action (resp. '''Z'''/2'''Z'''-action) on ''V''<sub>0,1</sub>:
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| ** <math> g\otimes x\mapsto -x</math>
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| ** <math> g\otimes y\mapsto +y</math>
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| ** <math> h\otimes y\mapsto -y</math>
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| ** <math> h\otimes x\mapsto \pm x</math> with ''"+"'' for ''V''<sub>0</sub> (symmetric) and ''"–"'' for ''V''<sub>1</sub> (asymmetric)
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| * Thus the braiding is
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| ** <math>x\otimes x\rightarrow -x\otimes x</math>
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| ** <math>y\otimes y\rightarrow -y\otimes y</math>
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| ** <math>x\otimes y\rightarrow y\otimes x</math>
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| ** <math>y\otimes x\rightarrow \pm x\otimes y</math>
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| Then, depending on the sign choice, the Nichols algebras are of dimension 4 and 8 (they appear in the classification under <math>q_{12}q_{21}=\pm 1</math>):
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| ::<math>\mathfrak{B}(V_0)=k[x,y]/(x^2,y^2,xy+yx),</math>
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| ::<math>\mathfrak{B}(V_1)=k[x]/(x^2,y^2,xyxy+yxyx)</math>
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| There one can see the striking resemblance to [[Semisimple Lie algebra]]s: In the first case, the [[braided vector space|braided commutator]] [''x'', ''y''] (here: anticommutator) is zero, while in the second, the [[Root system|root string]] is longer [''x'', [''x'', ''y'']] = 0. Hence these two belong to [[Dynkin diagram]]s <math>A_1\cup A_1</math> and A<sub>2</sub>.
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| <table> | |
| <tr> | |
| <td width="50"></td>
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| <td>[[File:A1A1.png|180px]]</td>
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| <td width="180"></td>
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| <td>[[File:Dynkin diagram A2.png|180px]]</td>
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| <td width="50"></td>
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| </tr>
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| </table>
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| One also constructs examples with even longer root strings ''V''<sub>2</sub>, ''V''<sub>3</sub> corresponding to [[Dynkin diagram]]s B<sub>2</sub>, G<sub>2</sub> (but as well no higher ones).
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| <table>
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| <tr>
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| <td width="45"></td>
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| <td>[[File:Dynkin diagram B2.png|180px]]</td>
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| <td width="180"></td>
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| <td>[[File:Dynkin diagram G2b.png|180px]]</td>
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| <td width="50"></td>
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| </tr>
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| </table>
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| ===Known examples over ''G'' nonabelian===
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| 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 <math>\mathcal{O}_{[g]}^\chi</math> corresponds to [[Conjugacy class]] of the group (together with an irreducible representation of the [[Centralizer and normalizer|centralizer]] of ''g''). An arbitrary Yetter–Drinfeld module is a [[direct sum]] of such <math>\mathcal{O}_{[g]}^\chi</math>, 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 coinncide!
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|
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| <table border ="1" cellpadding="10" align="center">
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| <tr> <th> Group ''G'' </th><th> Conjugacy class </th><th> Dimension of ''V'' </th><th> Dimension of <math>\mathfrak{B}(V)</math> </th><th> Source </th></tr>
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| <tr><td> [[Symmetric group]]   <math>\mathbb{S}_3\;</math> </td><td> <math>\mathcal{O}_{[(12)]}\;</math> </td>
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| <td> 3 </td><td> 12 </td><td> <ref name=MS00/> </td></tr>
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| <tr><td> [[Symmetric group]]   <math>\mathbb{S}_4\;</math> </td><td> <math>\qquad\mathcal{O}_{[(12)]}\;</math> </td>
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| <td> 6 </td><td> 576 </td><td> <ref name=MS00/></td></tr>
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| <tr><td>[[Symmetric group]]   <math>\mathbb{S}_4\;</math> </td><td> <math>\qquad\mathcal{O}_{[(1234)]}\;</math> </td>
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| <td> 6 </td><td> 576 </td><td> <ref name=AG03/></td></tr>
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| <tr><td>[[Symmetric group]]   <math>\mathbb{S}_5\;</math> </td><td> <math>\qquad\mathcal{O}_{[(12)]}\;</math> </td>
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| <td> 10 </td><td> 8294400 </td><td> <ref name=MS00/><ref name=FK99/></td></tr>
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| <tr><td>[[Dihedral group]]   <math>\mathbb{D}_4\;</math> </td><td> <math>\qquad\mathcal{O}_{[ab]}\oplus \mathcal{O}_{[b]}</math> </td>
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| <td> 4 </td><td> 64 </td><td> <ref name=MS00/></td></tr>
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| </table>
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| Some more examples are found e.g. in the ''"zoo"'' on M. Grana's Webpace page<ref name=GranaZoo/> ''(please add more information, see discussion)''.
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| Recently, a family of A<sub>2</sub> example of rank 2 over extensions of <math>\mathbb{D}_4\;</math> has been constructed<ref name=HS10/>
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| ==Classification==
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| ''the following section could use more details. Please add to it, see discussion''
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| ===Over abelian groups===
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| The Nichols algebras of finite dimension over '''abelian groups''' in ''k'' = '''C''' were classified by Istvan Heckenberger<ref name=H05/> in the years 2004–2005 by classifying arithmetic [[root system]]s and generalized [[Dynkin diagram]]s; where already Kharchenko had proven them to posess a [[Poincaré–Birkhoff–Witt theorem|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) | |
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| ::<math>x_i\otimes x_j \mapsto q_{ij}x_j\otimes x_i</math>
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| While mostly only the classical ''Cartan-cases'' appear, there are several exotic diagrams possible for small primes, such as a triangle
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| [[File:Dynkin Diagram Triangle.jpg|thumb|A rank 3 Dynkin diagram associated to a finite-dimensional Nichols algebra]]
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| In these cases the [[Weyl group|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 ''(picture?)''.
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| ===Negative criteria: abelian subracks===
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| 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<ref name=AFGV10a/> ''(more details!)'':
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| :: <TT>Consider a subrack that is abelian, check which representation my be inherited from the larger rack, and looked up in Heckenbegers List <ref name=H05/></TT> | |
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| 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'''
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| ::<math>x\otimes x\;\stackrel{\tau}{\longmapsto}\;q(x\otimes x)\;\;\Longleftrightarrow\;\; g.x=qx</math>
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| It is usually used to excludes ''g'' e.g. of being of odd order and/or χ of high dimension:<ref name=AFGV10b/>
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| * If ''g'' is '''real''' (i.e. conjugated to its inverse) then ''q'' = –1 (especially ''g'' has to be of even order)
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| * If ''g'' is '''quasi-real''' (i.e. conugated to some ''j''-th power) then
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| **either ''q'' = –1 as above
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| **or <math>g^{(j^2)}=g</math> and the representation χ is one-dimensional with ''q'' = ζ<sub>3</sub> a [[Root of unity|primitive 3rd root of unity]] (especially the order of ''g'' is divisible by 3)
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| * If contrary ''g'' is an [[Involution (mathematics)#Group theory|involution]] and some centralizing ''h'' = ''tgt'' then the [[eigenvalue]]s of the ''h'' (viewed as matrix) acting on <math>\mathcal{O}_{[g]}^\chi</math> is strongly restricted.
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| ===Root systems over nonabelian groups===
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| On the other hand, Schneider and Heckenberger established the existence an arithmetic root system over nonabelian groups<ref name=HS08/> (as in the abelian case), including again a [[Weyl group]]oid and a [[Poincaré–Birkhoff–Witt theorem|Poincaré–Birkhoff–Witt basis]] of iterated (braided) commutators
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| Immediate consequences are implied for ''rank 2'' Nichols algebras <math>\mathfrak{B}\left(\mathcal{O}_{[g]}\oplus\mathcal{O}_{[h]}\right)</math> which ''g, h'' '''discommuting'''; then:
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| * The braided commutators [''x'', ''y''] of elements <math>x\in\mathcal{O}_{[g]}\; y\in\mathcal{O}_{[h]}</math> are '''not all zero'''.
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| * The space of braided commutators <math>ad_{\mathcal{O}_{[g]}}\mathcal{O}_{[h]}=[\mathcal{O}_{[g]},\mathcal{O}_{[h]}]</math> form an '''irreducible''' sub-Yetter–Drinfeld module <math>\mathcal{O}_{[gh]}</math> (i.e. the root is unique as in the Lie algebra case)
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| * They're '''"close to commuting"'' <math>\;(gh)^2=(hg)^2</math>
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| This implies roughly, that finite dimensional Nichols algebras over nonabelian groups have to be (if at all) of very low rank.
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| ===Negative criteria: nonabelian subracks (type D)===
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| 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.
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| A specific case<ref name=AFGV10b/> where this is highly successful is '''type D''', i.e. for <math>r,s\in [g]\;</math>
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| * ''r'', ''s'' not conjugate in the generated subgroup <math>\langle r,s\rangle\;</math>
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| * <math>(rs)^2\neq(sr)^2\;</math>
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| in this case the Nichols algebra of the subrack is '''infinite dimensional''' and so is the entire Nichols algebra
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| ===Known groups not admitting finite dimensional Nichols algebras===
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| Both negation techniques above have been very fruitful to '''negate''' (indecomposable) finite-dimensional Nichols algebras:<ref name=AFGV10b/>
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| * for [[Alternating groups]]s <math>\mathbb{A}_{n\geq 5}</math> <ref name=AFGV10c/>
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| * for [[Symmetric group]]s <math>\mathbb{S}_{n\geq 6}</math> except a short list of examples<ref name=AFGV10c/>
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| * some [[group of Lie type]] ''(sources, complete list?)''
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| * all [[Sporadic group]]s except a short list of possibilities (resp. conjugacy classes in ATLAS notation) that are all real or ''j'' = 3-quasireal:
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| **...for the [[Fisher group]] <math>Fi_{22}\;</math> the classes <math>22A,22B\;</math>
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| **...for the [[baby monster group]] ''B'' the classes <math>16C,\;16D,\;32A,\;32B,\;32C,\;32D,\;34A,\;46A,\;46B\;</math>
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| **...for the [[monster group]] ''M'' the classes <math>32A,\;32B,\;46A,\;46B,\;92A,\;92B,\;94A,\;94B\;</math>
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| Usually a large amount of conjugacy classes ae of type D ("not commutative enough"), while the others tend to posess 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-group]]s of order 2048 resp. 128 and currently no restrictions on χ.
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| ==Applications==
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| The Nichols algebra appears as '''quantum Borel part''' in the classification of finite-dimensional pointed Hopf algebras<ref name=AS02/> (without small primes) by Nicolas Andruskiewitsch and Hans-Jürgen Schneider, especially [[Quantum groups]]. For example <math>U_q(\mathfrak{g})</math> and their well known truncations for ''q'' a root of unity decompose just like an ordinary [[Semisimple Lie algebra]] into ''E''´s (Borel part), dual ''F''´s and ''K''´s (Cartan algebra):
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| ::<math>U_q(\mathfrak{g})\cong \left(\mathfrak{B}(V)\otimes k[\mathbb{Z}^n]\otimes\mathfrak{B}(V^*)\right)^\sigma</math>
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| Here, as in the classical theory ''V'' is a vectorspace of dimension ''n'' (the '''rank''' of <math>\mathfrak{g}</math>) spanned by the ''E''´s, and σ (a so-called cocylce twist) creates the nontrivial '''linking''' between ''E''´s and ''F''´s. Note that in contrast to classical theory, more than two linked components may appear. See ''cit. loc.'' for an exotic example with 4 parts of type A<sub>3</sub>.[[File:Dynkin4A3lift.png|thumb|generalized Dynkin diagram for a pointed Hopf algebra linking four A3 copies]]
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| The classification roughly reduces a given hypothetical example to a [[Braided Hopf algebra|Radford biproduct]] of the (coradical-) group and the (connected-) part, which contains the Nichols algebra, by taking the corresponding "graded object" (killing all linkings). With the knowledge from the classification of finite dimensional Nichols algebras above, the authors prove no additional elements to appear in the connected part (generation in degree 1), and finally describe all possible liftings as "dotted lines" in generalized [[Dynkin diagrams]].
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| Recently, this correspondence has been greatly extended to identify certain so-called '''coideal subalgebras''' to be in 1:1 correspondence<ref name=HS09/> to the [[Weyl group]], which has been conjectued as "numercal coincidence" earlier and proven in certain cases by-hand.
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| ==References==
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| <ref name=AS02>Andruskiewitsch, Schneider: ''Pointed Hopf algebras'', New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.</ref>
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| <ref name=HS08>Heckenberger, Schneider: ''Root system and Weyl gruppoid for Nichols algebras'', 2008.</ref>
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| <ref name=H05>Heckenberger: ''Nichols algebras of diagonal type and arithmetic root systems'', Habilitation thesis 2005.</ref> | |
| <ref name=H08>Heckenberger: ''Nichols Algebras'' (Lecture Notes), 2008 http://www.mi.uni-koeln.de/~iheckenb/na.pdf</ref>
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| <ref name=MS00>Schneider, Milinski: ''Nichols algebras over Coxeter groups'', 2000.</ref>
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| <ref name=AG03>Andruskiewisch, Grana: ''From racks to pointed Hopf algebras'', 2003.</ref>
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| <ref name=FK99>Fomin,Kirilov: ''Quadratic algebras, Dunkl elements and Schubert calculus'', 1999.</ref>
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| <ref name=GranaZoo>Grana: http://mate.dm.uba.ar/~matiasg/zoo.html</ref>
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| <ref name=HS10>Heckenberger, Schneider: ''Nichols algebras over groups with finite root system of rank 2 I'', 2010.</ref>
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| <ref name=AFGV10a>Andruskiewitsch, Fantino, Grana, Vendramin: ''On Nichols algebras associated to simple racks'', 2010.</ref>
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| <ref name=AFGV10b>Andruskiewitsch, Fantino, Grana, Vendramin: ''Pointed Hopf algebras over the sporadic simple groups'', 2010.</ref>
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| <ref name=AFGV10c>Andruskiewitsch, Fantino, Grana, Vendramin: ''Finite-dimensional pointed Hopf algebras with alternating groups are trivial'', 2010.</ref>
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| <ref name=HS09>Heckenberger, Schneider: ''Right coideal subalgebras of Nichols algebras and the Duflo order of the Weyl grupoid'', 2009.</ref>
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| <references />
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| [[Category:Hopf algebras]]
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| [[Category:Quantum groups]]
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