Quiver representations and their dense orbits

Smooth embeddings of SL(2) and PGL(2)

An affine algebraic group

Stable representations of quivers

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Let [Formula: see text] be a [Formula: see text]-equivariant algebraic principal [Formula: see text]-bundle over a normal complex affine variety [Formula: see text] equipped with an action of [Formula: see text], where [Formula: see text] and [Formula: see text] are complex linear algebraic groups. Suppose [Formula: see text] is contractible as a topological [Formula: see text]-space with a dense orbit, and [Formula: see text] is a [Formula: see text]-fixed point. We show that if [Formula: see text] is reductive, then [Formula: see text] admits a [Formula: see text]-equivariant isomorphism with the product principal [Formula: see text]-bundle [Formula: see text], where [Formula: see text] is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal [Formula: see text]-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal [Formula: see text]-bundles over any complex toric variety, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math. 27(14) (2016)].

Given a variety $$\mathbb{V}$$ of bounded residuated lattices satisfying the Stone identity $$\neg x \lor \neg\neg x = \top$$ , the free algebras in $$\mathbb{V}$$ over a set X of cardinality |X| are represented as weak Boolean products over the Cantor space 2|X| of a family of free algebras in an associated variety of (not necessarily bounded) residuated lattices with a bottom added.

In this paper, we give correspondences between unary algebras, semigroups and congruences on free semigroups. We establish isomorphisms between the complete lattice of varieties of semigroups and the complete lattices of families of varieties of unary algebras, and families of filters of congruences on free semigroups. Similar correspondences between generalized varieties and pseudovarieties of semigroups and corresponding families of algebras and congruences are also established.

This paper provides a complete classification of four-dimensional complex Poisson algebras, extending previous work on lower-dimensional cases. We develop a systematic approach that combines the classification of four-dimensional complex commutative associative algebras with the classification of four-dimensional complex Lie algebras, allowing us to determine all compatible Poisson structures up to isomorphism. Our main result presents 71 distinct types: 60 isolated non-isomorphic Poisson algebras with non-trivial commutative product and 11 one-parameter families of non-isomorphic Poisson algebras. In addition to these, we include the known four-dimensional complex Lie algebras, viewed as Poisson algebras with trivial commutative product. This purely algebraic classification provides a foundation for future studies, including the geometric classification of Poisson algebras, their degenerations, and the analysis of orbit closures in the corresponding algebraic variety.

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure richer than the usual Kashiwara crystals. Along with the geometric study of Nakajima quiver varieties, in the same context, this yields a notion of generalized crystals, coming with a tensor product. As an application, we define the semicanonical basis of the Hopf algebra generalizing quantum groups, which was already equipped with a canonical basis. The irreducible components of the Nakajima varieties provide the family of highest weight crystals associated to dominant weights, as in the classical case.

If \({\mathcal{A}}\) is an infinite primal algebra, then we shall represent any algebra in the variety \({V\,(\mathcal{A}}\)) generated by \({\mathcal{A}}\) as a limit reduced power of \({\mathcal{A}}\). Furthermore, we show that any homomorphism between algebras in \({V\,(\mathcal{A}}\)) can be induced by mappings between underlying sets of the limit reduced powers. With this representation of the morphisms between algebras in \({V\,(\mathcal{A}}\)) at hand, we will construct a category equivalent to the category \({V\,(\mathcal{A}}\)).

We verify computationally a conjecture on the field independence of tree representations of Euclidean quivers, with dimension vector bounded by the minimal radical vector of the quiver. This includes a large class of exceptional representations, in particular all the regular non-homogeneous exceptionals. In addition we also present some thought-provoking findings, which further confirms the combinatorial nature of the category of representations of tame quivers.

Jet schemes, quantum dilogarithm and Feigin-Stoyanovsky's principal subspaces

Given a finitely generated group H, the set Hom(H, SL2C) inherits the structure of an algebraic variety R(H) called the representation variety of H. This algebraic variety is an invariant of H. Let [Formula: see text] where p, t are integers greater than one. In this paper, a formula is produced yielding the number of four dimensional irreducible components of the affine algebraic variety R(Gpt).

The object of study of this paper is the following multi-determinantal algebraic variety, $\text{SING}_{n, m}$ , which captures the symbolic determinant identity testing (SDIT) problem (a canonical version of the polynomial identity testing (PIT) problem), and plays a central role in algebra, algebraic geometry and computational complexity theory. $\text{SING}_{n, m}$ is the set of all $m$ -tuples of $n\times n$ complex matrices which span only singular matrices. In other words, the determinant of any linear combination of the matrices in such a tuple vanishes. The algorithmic complexity of testing membership in $\text{SING}_{n, m}$ is a central question in computational complexity. Having almost a trivial probabilistic algorithm, finding an efficient deterministic algorithm is a holy grail of derandomization, and to top it, will imply super-polynomial circuit lower bounds! A sequence of recent works suggests efficient deterministic “geodesic descent” algorithms for memberships in a general class of algebraic varieties, namely the null cones of (reductive) linear group actions. Can such algorithms be used for the problem above? Our main result is negative: $\text{SING}_{n, m}$ is not the null cone of any such group action! This stands in stark contrast to a non-commutative analog of this variety (for which such algorithms work), and points to an inherent structural difficulty of $\text{SING}_{n, m}$ . In other words, we provide a barrier for the attempts of derandomizing SDIT via these algorithms. To prove this result we identify precisely the group of symmetries of $\text{SING}_{n, m}$ . We find this characterization, and the tools we introduce to prove it, of independent interest. Our characterization significantly generalizes a result of Frobenius for the special case $m=1$ (namely, computing the symmetries of the determinant). Our proof suggests a general method for determining the symmetries of general algebraic varieties, an algorithmic problem that was hardly studied and we believe is central to algebraic complexity.