Quivers with loops and generalized crystals

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.

Tensor products on quiver representations

The decomposition of certain tensor products of representations in the principal series with representations in the supplementary series of the group SL (n, C) into irreducible components

Semicanonical Bases Arising From Enveloping Algebras

This is an introductory text on the theory of modules over rings or, equivalently, over algebras, addressed to beginning graduate students. It stresses the importance of the categorical perspective and the use of homological tools. Modules are studied both from a classical point of view and a categorical one. For this purpose, rudiments of category theory, homological algebra and representations of quivers are introduced and applied in the study of module categories. After introducing such fundamental tools as the Hom functor and the tensor product comes the study of projective, injective and flat modules. The classical results, such as the classification of modules over principal ideal domains, the Jordan-Hölder, Wedderburn-Artin, and Morita theorems, are proven, as well as the Eilenberg-Watts theorems. Radicals (of modules, algebras and categories) are thoroughly studied. Homological invariants, such as the extension and torsion modules or the homological dimensions are introduced using derived functors and are applied in module theory. The book stresses the construction and analysis of examples.

Canonical bases for Fock spaces and tensor products

We give an analogue of the classical exponential map on Lie groups for Hopf\n$*$-algebras with differential calculus. The major difference with the\nclassical case is the interpretation of the value of the exponential map,\nclassically an element of the Lie group. We give interpretations as states on\nthe Hopf algebra, elements of a Hilbert $C^{*} $-bimodule of $\\frac{1}{2}$\ndensities and elements of the dual Hopf algebra. We give examples for complex\nvalued functions on the groups $S_{3}$ and $\\mathbb{Z}$, Woronowicz's matrix\nquantum group $\\mathbb{C}_{q}[SU_2] $ and the Sweedler-Taft algebra.\n

We discuss some conjectures for bases of the GL(n) × ℂ* — equivariant K-theory of the closure of the conjugacy class of a nilpotent n × n matrix. One basis is a canonical basis coming from Lusztig’s theory of canonical bases of affine Hecke algebras. The other (conjectural) basis has graded characters which, when expanded in terms of irreducible characters, has coefficient polynomials that are q-analogues of Littlewood-Richardson coefficients.

We consider the problem of gambling on a quantum experiment and enforce rational behaviour by a few rules. These rules yield, in the classical case, the Bayesian theory of probability via duality theorems. In our quantum setting, they yield the Bayesian theory generalised to the space of Hermitian matrices. This very theory is quantum mechanics: in fact, we derive all its four postulates from the generalised Bayesian theory. This implies that quantum mechanics is self-consistent. It also leads us to reinterpret the main operations in quantum mechanics as probability rules: Bayes' rule (measurement), marginalisation (partial tracing), independence (tensor product). To say it with a slogan, we obtain that quantum mechanics is the Bayesian theory in the complex numbers.

Consider any representation ϕ \boldsymbol {\phi } of a finite-dimensional Lie algebra g \mathfrak {g} by derivations of the completed symmetric algebra S ^ ( g ∗ ) \hat {S}(\mathfrak {g}^*) of its dual. Consider the tensor product of S ^ ( g ∗ ) \hat {S}(\mathfrak {g}^*) and the exterior algebra Λ ( g ) \Lambda (\mathfrak {g}) . We show that the representation ϕ \boldsymbol {\phi } extends canonically to the representation ϕ ~ \tilde {\boldsymbol {\phi }} of that tensor product algebra. We construct an exterior derivative on that algebra, giving rise to a twisted version of the exterior differential calculus with the universal enveloping algebra in the role of the coordinate algebra. In this twisted version, the commutators between the noncommutative differentials and coordinates are formal power series in partial derivatives. The square of the corresponding exterior derivative is zero like in the classical case, but the graded Leibniz rule is deformed.

Twisted argyle quivers and Higgs bundles

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$-canonical bases for the highest weight integrable $\textbf U$-modules and their tensor products regarded as $\textbf {U}^\imath$-modules, as well as an $\imath$-canonical basis for the modified form of the $\imath$-quantum group $\textbf {U}^\imath$. A key new ingredient is a family of explicit elements called $\imath$-divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$. We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi-$K$-matrix and the constructions of $\imath$-canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.

<p>We can view quiver representations of a fixed dimension vector as an algebraic variety over an algebraically closed field $K$. There is an action of the product of general linear groups on each of these varieties where the orbits of the action correspond to isomorphism classes of quiver representation. A $K$-algebra $A$ is said to have the dense orbit property if for each dimension vector, the product of the general linear group acts on each irreducible component of the module variety with a dense orbit. Under certain conditions, a $K$ algebra $A$ is representation finite if and only if it $A$ has the dense orbit property. The implication representation finite implies the dense orbit property is always true. The converse is not true in general, as shown by Chindris, Kinser, and Weyman in \cite{ryan}. Our main theorem of this thesis builds on their work to give a family of representation infinite algebras with the dense orbit property. We also give a conjectured classification of indecomposables with dense orbits. \par</p> <p>In the future, we hope the work presented here can be used to find even more examples of representation infinite algebra with the dense orbit property to then develop deeper theory to classify algebras with the dense orbit property that are representation infinite.</p>

In this paper, we construct free commutative integro-differential algebras by applying the method of Gröbner–Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential Rota–Baxter (DRB) algebras of order n. We also obtain a weakly monomial order on these algebras, allowing us to obtain Gröbner–Shirshov bases for free commutative integro-differential algebras on a set. We finally generalize the concept of functional monomials to free differential algebras with arbitrary weight and generating sets from which to construct a canonical linear basis for free commutative integro-differential algebras.

The Knutson Index of the representation ring