F q [M 2], F q [GL 2], and F q [SL 2] as Quantized Hyperalgebras

A ring R is called right Johns if R is right noetherian and every right ideal of R is a right annihilator. R is called strongly right Johns if the matrix ring M _n (R) is right Johns for each integer n ≥ 1. The Faith–Menal conjecture is an open conjecture on QF rings. It says that every strongly right Johns ring is QF. It is proved that the conjecture is true if every closed left ideal of the ring R is finitely generated. This result improves the known result that the conjecture is true if R is a left CS ring.     

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ?S_n and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.     

Finite GK-dimensional pre-Nichols algebras of super and standard type

We prove that finite GK-dimensional pre-Nichols algebras of super and standard type are quotients of the corresponding distinguished pre-Nichols algebras, except when the braiding matrix is of type super A and the dimension of the braided vector space is three. For these two exceptions we explicitly construct substitutes as braided central extensions of the corresponding pre-Nichols algebras by a polynomial ring in one variable. Via bosonization this gives new examples of finite GK-dimensional Hopf algebras.     

Composition algebras and outer automorphisms of algebraic groups via triality

In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form n over a field k of characteristic not two, and a category arising from an action of the projective similarity group of n on certain pairs of automorphisms of the group scheme PGO⁺(n) defined over k. This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known results on composition algebras from our equivalence.     

Weyl equivalence for rank 2 Nichols algebras of diagonal type

Yetter-Drinfel'd modules of diagonal type admit an equivalence relation which preserves dimension and Gel'fand-Kirillov dimension of the corresponding Nichols algebras. This relation is determined explicity for all rank 2 Yetter-Drinfel'd modules where the Gel'fand-Kirillov dimension is known to be finite. Supported by the European Community under a Marie Curie Intra-European Fellowship.     

Construction of cocycles for bicrossproduct of Lie groups

We obtain an explicit formula for finding cocycles on a matched pair of Lie groups by using cocycles on the corresponding pair of Lie algebras. This formula for cocycles allows one to construct examples of locally compact quantum groups via bicrossproduct of Lie groups.     

Higher Indicators for the Doubles of Some Totally Orthogonal Groups

We investigate the indicators for certain groups of the form ? _k  ? D _l and their doubles, where D _l is the dihedral group of order 2l. We subsequently obtain an infinite family of totally orthogonal, completely real groups which are generated by involutions, and whose doubles admit modules with second indicator of ?1. This provides us with answers to several questions concerning the doubles of totally orthogonal finite groups.     

Factorization of simple modules for certain restricted two-parameter quantum groups

We study the representations of the restricted two-parameter quantum groups of types B and G. For these restricted two-parameter quantum groups, we give some explicit conditions which guarantee that a simple module can be factored as the tensor product of a one-dimensional module with a module that is naturally a module for the quotient by central group-like elements. That is, given θ a primitive lth root of unity, the factorization of simple ?θy,θz,( )-modules is possible, if and only if (2(y - z), l) = 1 for =??2n+1; (3(y - z), l) = 1 for g= G₂.     

Vertex representation of quantum N-toroidal algebra for type F4

Abstract

Recently Gao-Jing-Xia-Zhang defined the structures of quantum N-toroidal algebras uniformally, which are a kind of natural generalizations of the classical quantum toroidal algebras, just like the relation between 2-toroidal Lie algebras and N-toroidal Lie algebras. Based on this work, we construct a level-one vertex representation of quantum N-toroidal algebra for type F₄. In particular, we can also obtain a level-one vertex representation of quantum toroidal algebra for type F₄ as our special cases.     

The geometry of root subgroups in Ree groups of type 2 F 4

For G, a member of this family of twisted groups, an incidence structure is described whose points are the elementary abelian root subgroups of G. A correspondence is established between the lines of this structure which satisfy a certain regularity condition and its planes, and it is shown that this correspondence induces a polarity on a metasymplectic space in which the structure can be embedded.     

Finite Presentability and the Homological Type FP m for a Class of Hopf Algebras

We define and study the property finite presentability in the category  of Hopf algebras that are smash product of universal enveloping algebra of a Lie algebra by a group algebra. We show that for such Hopf algebras finite presentability is equivalent with finite presentability as an associative k-algebra.     

The Yoneda Algebra of a 𝒦2 Algebra Need not be Another 𝒦2 Algebra

The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. 𝒦₂ algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a 𝒦₂ algebra would be another 𝒦₂ algebra. We show that this is not necessarily the case by constructing a monomial 𝒦₂ algebra for which the corresponding Yoneda algebra is not 𝒦₂.     

The J-invariant and Tits algebras for groups of inner type E 6

A connection between the indices of the Tits algebras of a split linear algebraic group G and the degree one parameters of its motivic J-invariant was introduced by Quéguiner-Mathieu, Semenov and Zainoulline through use of the second Chern class map in the Riemann-Roch theorem without denominators. In this paper we extend their result to higher Chern class maps and provide applications to groups of inner type E ₆.     

Reflection groupoids of rank two and cluster algebras of type A

We extend the classification of finite Weyl groupoids of rank two. Then we generalize these Weyl groupoids to ‘reflection groupoids’ by admitting non-integral entries of the Cartan matrices. This leads to the unexpected observation that the spectrum of the cluster algebra of type An−3 completely describes the set of finite reflection groupoids of rank two with 2n objects.     

The peak algebra and the descent algebras of types B and D

We show the existence of a unital subalgebra of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that is the image of the descent algebra of type B under the map to the descent algebra of type A which forgets the signs, and also the image of the descent algebra of type D. The algebra contains a two-sided ideal which is defined in terms of interior peaks. This object was introduced in previous work by Nyman (2003); we find that it is the image of certain ideals of the descent algebras of types B and D. We derive an exact sequence of the form . We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions and by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces and over and explain the connection with previous work of Stembridge (1997); we also obtain new properties of his descents-to-peaks map and construct a type B analog.

     

On the compact real forms of the lie algebras of type E 6 and F 4

We give a construction of the compact real form of the Lie algebra of type E ₆, using the finite irreducible subgroup of shape 3³⁺³: SL₃(3), which is isomorphic to a maximal subgroup of the orthogonal group Ω₇(3). In particular we show that the algebra is uniquely determined by this subgroup. Conversely, we prove from first principles that the algebra satisfies the Jacobi identity, and thus give an elementary proof of existence of a Lie algebra of type E ₆. The compact real form of F ₄ is exhibited as a subalgebra.