Algebra Minicourses

Lecturer: Johan Öinert (Blekinge Institute of Technology, Sweden)

Title: Epsilon-strongly graded rings - theory and applications

Abstract: In this lecture series, we will introduce the audience to the theory of group-graded rings and more precisely epsilon-strongly graded rings. Some key results will be dissected and explained, and applications  e.g. partial crossed prudcts and Leavitt path algebras will be put on display.

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Lecturer: Sujit Kumar Sardar (Jadavpur University, India)

Title: Understanding groupoid approach in the study of inverse semigroups and some

combinatorial algebras

Abstract: A groupoid is a small category consisting of isomorphisms only. Groupoids generalize groups and have close relationship with inverse semigroups which capture partial symmetries on objects. In this talk we will understand groupoid and its components from a purely algebraic point of view. The study of topological groupoids is quite classical in literature. It appears in several areas of mathematics including operator algebras, theory of inverse semigroups, ergodic theory etc. Among the interesting connections between groupoids and inverse semigroups one can talk about the non-commutative generalization of Stone duality due to Lawson which interrelates ample groupoids (or Stone groupoids) with Boolean inverse semigroups. We will try to explore this. One of the interesting and perhaps the most striking fact about topological groupoids is that they can be used to model algebras constructed from combinatorial objects. Once this is done the structural properties of the algebra can be characterized via the algebraic and topological properties of the modeling groupoid. This idea was pioneered for graph C∗-algebras by Paterson and for discrete inverse semigroup algebras by Steinberg. In this talk we will witness groupoid approaches in the study of (i) Leavitt path algebras of directed graphs over commutative unital rings, commutative semirings and Clifford semifields, (ii) Kumjian-Pask algebras of higher-rank graphs over commutative unital rings.

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Lecturer: Ardeline Mary Buhphang (North-Easter Hill University, India)

Title: A Functorial Approach to Extension Problems and Annihilating Conditions in Noncommutative Algebras

Abstract: This lecture series develops a functorial viewpoint on two themes of noncommutative algebra: the extension problem and annihilating conditions. The emphasis is on studying classical constructions such as the socle and the nil radical as endofunctors on appropriate module or ring categories, and on analysing how these functors interact with homomorphism extension, injectivity, and annihilation. Within the framework of monomial and Leavitt path algebras, we examine when homomorphisms defined on the socle extend to endomorphisms of the algebra or the ring, and how annihilator, nil, and Jacobson radical behave under these functorial operations. The lectures present the equivalence between socle-injectivity and self-injectivity for finite monomial algebras (Buhphang–Goswami–Kuber), the structure of socle-injective semiprime rings and their connections with quotient and endomorphism rings (Buhphang–Das–Gonz´alez–Siles Molina), and the behaviour of nil ideals and Jacobson radicals in Leavitt path algebras over commutative rings (Dutta–Buhphang). The series attempts to describe how functorial constructions namely, the socle and radical endofunctors provide a categorical framework that unifies extension problems, annihilating conditions, and structural properties of noncommutative algebras.