KLR algebras of type A have been a revolution in the representation theory of Hecke algebras of a type A flavour, thanks to the the Brundan-Kleshchev-Rouquier isomorphism relating them explicitly to the affine Hecke algebra of type A. KLR algebras of other types exist but are not related to affine Hecke algebras of other types. In this talk I will present a generalisation of the KLR presentation for the affine Hecke algebra of type B and I will discuss some applications.
The Hecke algebra is in general not quasi-hereditary, meaning that its module category is not a highest weight category; while it admits a quasi-hereditary cover via category O for certain rational Cherednik algebras due to Ginzburg-Guay-Opdam-Rouquier. It was proved in type A that this category O can be realized using q-Schur algebra, but this realization problem remains open beyond types A/B/C. An essential step for type D is to study Hu's Hecke subalgebra, which deforms from a wreath product that is not a Coxeter group. In this talk, I'll talk about a new theory allowing us to take the 'quantum wreath product' of an algebra by a Hecke algebra. Our wreath product produces the Ariki-Koike algebra as a special case, as well as new 'Hecke algebras' of wreath products between symmetric groups. We expect them to play a role in answering the realization problem for complex reflection groups.
Noetherianity is a fundamental property of modules, rings, and topological spaces that underlies much of commutative algebra and algebraic geometry. This talk concerns algebraic structures such as the infinite-dimensional polynomial ring K[x1,x2,....] that are not Noetherian as such, but become Noetherian when we regard them up to the action of a large symmetry group.
Murray Gerstenhaber constructed a graded Lie structure (Gerstenhaber bracket) on Hochschild cohomology, which makes Hochschild cohomology a Lie algebra. However, it is not easy to calculate bracket structure with the original definition. There is an alternative technique to compute Gerstenhaber bracket on Hochschild cohomology, introduced by Chris Negron and Sarah Witherspoon. It is also known that Hopf algebra cohomology has a bracket and the bracket is trivial when a Hopf algebra is quasi-triangular. We use a similar technique to the technique given by Negron and Witherspoon to calculate the Lie structure on Hochschild cohomology of the Taft algebra Tp for any integer p>2 which is a nonquasi-triangular Hopf algebra. Then, we find the corresponding bracket on Hopf algebra cohomology of Tp. We show that the bracket is indeed zero on Hopf algebra cohomology of Tp, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi triangular algebra.
We first give some background on the Poincaré series of the algebra of generic matrices and its associated trace ring, and then focus on some recent work, including a conjecture for the denominator of the one variable series for the trace rings. Time permitting we will also say a bit about traces of direct sums.
Murray Gerstenhaber constructed Lie structure (Gerstenhaber bracket) on Hochschild cohomology, which makes Hochschild cohomology a graded Lie algebra. Later, it is shown that Hopf algebra cohomology also has a Lie structure. We will introduce a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber’s original formula for Hochschild cohomology.
After reviewing the basic definitions of tensor categories and the notion of semisimplification of symmetric tensor categories, it will be shown how the semisimplification of the category of representations of the cyclic group of order 3 over a field of characteristic 3 is naturally equivalent to the category of vector superspaces over this field. This allows to define a superalgebra starting with any algebra endowed with an order 3 automorphism. As a noteworthy example, the exceptional composition superalgebras will be obtained, in a systematic way, from the split octonion algebra.
Dixmier and Moeglin showed that if L is a finite-dimensional complex Lie algebra then the primitive ideals of the enveloping algebra U(L) are the prime ideals of Spec(U(L)) that are locally closed in the Zariski topology. In addition, they proved that a prime ideal P of U(L) is primitive if and only if the Goldie ring of quotients of U(L)/P has the property that its centre is just the base field of the complex numbers. Algebras that share this characterization of primitive ideals are said to satisfy the Dixmier-Moeglin equivalence. We give an overview of this property and mention some recent work on proving this equivalence holds for certain classes of twisted homogenous coordinate rings and classes of Hopf algebras of small Gelfand-Kirillov dimension.
In 1901 Young gave an explicit construction of the ordinary irreducible representations of the symmetric groups. In doing this, he introduced content functions for partitions, which are now a key statistic in the semisimple representation theory of the symmetric groups. In this talk I will describe a generalization of Young's ideas to the cyclotomic KLR algebras of affine types A and C. This is quite surprising because Young's seminormal forms are creatures from the semisimple world whereas the cyclotomic KLR algebras are rarely semisimple. As an application, we show that these algebras are cellular and construct their irreducible representations. A special case of these results gives new information about the symmetric groups in characteristic p > 0. If time permits, I will describe how these results lead to an explicit categorification of the corresponding integrable highest weight modules.
The Rouquier blocks, also known as the RoCK blocks, are important blocks of the symmetric groups algebras and the Hecke algebras of type A, with the partitions labelling the Specht modules that belong to these blocks having a particular abacus configuration. We generalize the definition of Rouquier blocks to the Ariki-Koike algebras, where the Specht modules are indexed by multipartitions, and explore the properties of these blocks.
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