List Of Finite-dimensional Nichols Algebras
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In mathematics, a Nichols algebra is a
Hopf algebra 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), Swedis ...
in a braided category assigned to an object ''V'' in this category (e.g. a
braided vector space In mathematics, a braided vectorspace \;V is a vector space together with an additional structure map \tau symbolizing interchanging of two vector tensor copies: ::\tau:\; V\otimes V\longrightarrow V\otimes V such that the Yang–Baxter equation ...
). The Nichols algebra is a quotient of the tensor algebra of ''V'' enjoying a certain universal property and is typically infinite-dimensional. Nichols algebras appear naturally in any pointed Hopf algebra and enabled their classification in important cases.Andruskiewitsch, Schneider: ''Pointed Hopf algebras'', New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002. The most well known examples for Nichols algebras are the Borel parts U_q(\mathfrak)^+ of the infinite-dimensional quantum groups when ''q'' is no root of unity, and the first examples of finite-dimensional Nichols algebras are the Borel parts u_q(\mathfrak)^+ of the Frobenius–Lusztig kernel (''small quantum group'') when ''q'' is a root of unity. The following article lists all known finite-dimensional Nichols algebras \mathfrak(V) where V is a Yetter–Drinfel'd module over a finite group G, where the group is generated by the support of V. For more details on Nichols algebras see Nichols algebra. * There are two major cases: ** G abelian, which implies V is diagonally braided x_i\otimes x_j\mapsto q_x_j\otimes x_i. ** G nonabelian. * The rank is the number of irreducible summands V=\bigoplus_ V_i in the semisimple Yetter–Drinfel'd module V. * The irreducible summands V_i=\mathcal_^\chi are each associated to a conjugacy class subset G and an irreducible representation \chi of the centralizer \operatorname(g). * To any Nichols algebra there is by attached ** a generalized root system and a Weyl groupoid. These are classified in. ** In particular several Dynkin diagrams (for inequivalent types of Weyl chambers). Each Dynkin diagram has one vertex per irreducible V_i and edges depending on their braided commutators in the Nichols algebra. * The
Hilbert series In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homoge ...
of the graded algebra \mathfrak(V) is given. An observation is that it factorizes in each case into polynomials (n)_t:=1+t+t^2+\cdots + t^. We only give the Hilbert series and dimension of the Nichols algebra in characteristic 0. Note that a Nichols algebra only depends on the braided vector space V and can therefore be realized over many different groups. Sometimes there are two or three Nichols algebras with different V and non-isomorphic Nichols algebra, which are closely related (e.g. cocycle twists of each other). These are given by different conjugacy classes in the same column.


State of classification

(as of 2015)


Established classification results

* Finite-dimensional diagonal Nichols algebras over the complex numbers were classified by Heckenberger in. The case of arbitrary characteristic is ongoing work of Heckenberger, Wang. * Finite-dimensional Nichols algebras of semisimple Yetter–Drinfel'd modules of rank >1 over finite nonabelian groups (generated by the support) were classified by Heckenberger and Vendramin in.


Negative criteria

The case of rank 1 (irreducible Yetter–Drinfel'd module) over a nonabelian group is still largely open, with few examples known. Much progress has been made by Andruskiewitsch and others by finding subracks (for example diagonal ones) that would lead to infinite-dimensional Nichols algebras. As of 2015, known groups not admitting finite-dimensional Nichols algebras are * for alternating groups \mathbb_ * for symmetric groups \mathbb_ except a short list of examples * some group of Lie type such as most PSL_n(\mathbb_q) and most unipotent classes in Sp_(\mathbb_q) * 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 Fisher group Fi_\; the classes 22A,22B\; **...for the baby monster group ''B'' the classes 16C,\;16D,\;32A,\;32B,\;32C,\;32D,\;34A,\;46A,\;46B\; **...for the monster group ''M'' the classes 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.


Over abelian groups

Finite-dimensional diagonal Nichols algebras over the complex numbers were classified by Heckenberger in in terms of the braiding matrix q_, more precisely the data q_,q_q_. The small quantum groups u_q(\mathfrak)^+ are a special case q_=q^, but there are several exceptional examples involving the primes 2,3,4,5,7. Recently there has been progress understanding the other examples as exceptional Lie algebras and super-Lie algebras in finite characteristic.


Over nonabelian group, rank > 1


Nichols algebras from Coxeter groups

For every finite coxeter system (W,S) the Nichols algebra over the conjugacy class(es) of reflections was studied in (reflections on roots of different length are not conjugate, see fourth example fellow). They discovered in this way the following first Nichols algebras over nonabelian groups : The case \mathbb_2\cong \mathbb_2 is the rank 1 diagonal Nichols algebra u_i(A_1)^+ of dimension 2.


Other Nichols algebras of rank 1


Nichols algebras of rank 2, type Gamma-3

These Nichols algebras were discovered during the classification of Heckenberger and Vendramin.


The Nichols algebra of rank 2 type Gamma-4

This Nichols algebra was discovered during the classification of Heckenberger and Vendramin.


The Nichols algebra of rank 2, type T

This Nichols algebra was discovered during the classification of Heckenberger and Vendramin.


The Nichols algebra of rank 3 involving Gamma-3

This Nichols algebra was the last Nichols algebra discovered during the classification of Heckenberger and Vendramin.


Nichols algebras from diagram folding

The following families Nichols algebras were constructed by Lentner using diagram folding, the fourth example appearing only in characteristic 3 was discovered during the classification of Heckenberger and Vendramin. The construction start with a known Nichols algebra (here diagonal ones related to quantum groups) and an additional automorphism of the Dynkin diagram. Hence the two major cases are whether this automorphism exchanges two disconnected copies or is a proper diagram automorphism of a connected Dynkin diagram. The resulting root system is folding / restriction of the original root system. By construction, generators and relations are known from the diagonal case. The following two are obtained by proper automorphisms of the connected Dynkin diagrams A_,E_6 Note that there are several more foldings, such as D_4,D_n and also some not of Lie type, but these violate the condition that the support generates the group.


Poster with all Nichols algebras known so far

(Simon Lentner, University Hamburg, please feel free to write comments/corrections/wishes in this matter: simon.lentner at uni-hamburg.de)


References

Schneider, Milinski: ''Nichols algebras over Coxeter groups'', 2000. {{DEFAULTSORT:Finite-dimensional Nichols algebras Hopf algebras Quantum groups Mathematics-related lists