Program

•  Representations of Lie algebras
  Prof. Stéphane Launois, Université de Caen Normandie, France
  Abstract: This course introduces finite-dimensional complex Lie algebras and their representations. It covers the classification of simple Lie algebras, the construction of simple Lie algebras associated with irreducible root systems, and potentially explores Verma modules and universal enveloping algebras. The course is example-driven.
  
  

• Representations of bound quiver algebras
  Dr. Amit Kuber, Indian Institute of Technology, Kanpur
  Abstract: Quivers are directed (finite) multi-graphs with the possibility of loops. Despite their simplicity, representation theory of (bound) quivers is a synonym for representation theory of finite-dimensional associative algebras. The course introduces two techniques: Gabriel's quiver-theoretic approach and the Auslander-Reiten theory, with examples and visualizations.
  
  

• Introduction to finite Coxeter groups and their representations
  Dr. Pooja Singla, Indian Institute of Technology, Kanpur
  Abstract: This course explores finite Coxeter groups, their properties, and classifications. It includes a review of fundamental topics in finite group representation theory, focusing on Coxeter groups, and covers character theory, the structure, and computation of irreducible representations.
  
  

• Tree Module: Hom-Sets and Indecomposability
  Annoy Sengupta, Indian Institute of Technology, Kanpur
  Abstract: A finite-dimensional monomial algebra is a path algebra over a quiver Q quotiented out by an admissible ideal generated by a set of paths in Q of length ≥ 2. Over such algebras, tree modules are shown to be indecomposable using universal covering techniques introduced by Peter Gabriel in 1981, in The universal cover of a representation-finite algebra, published in Representations of algebras (Puebla, 1980), Lecture Notes in Mathematics, volume 903, pages 68–105. A description of a basis for the space of homomorphisms Hom(M₁, M₂), where M₁ and M₂ are tree modules, was given by William W. Crawley-Boevey in 1989, in Maps between representations of zero-relation algebras, Journal of Algebra, volume 126, issue 2, pages 259–263, using graph maps. In this talk, we will see an alternative short proof of these two results. 

• Horrocks’ theorem for odd orthogonal group
  Sugilesh H ,  Cochin University of Science and Technology
  Abstract: Horrocks’ theorem provides a criterion for determining when a vector bundle over the affine line A¹ is trivial. The algebraic formulation of Horrocks’ theorem states: “Let R be a commutative local ring, and let P be a finitely generated projective module over the polynomial ring R[X]. If the localization Pₓ is free over R[X], then P itself must be free over R[X].” A. A. Suslin and V. I. Kopeiko proved Horrocks’ theorem for even-dimensional orthogonal groups in 1977. This result gives the splitting of an even-dimensional orthogonal matrix with entries from the polynomial rings R[X] or R[X⁻¹] into a product of an even orthogonal matrix with entries from R and an even elementary orthogonal matrix with entries from the polynomial rings R[X] or R[X⁻¹], respectively. We constructed some subgroups of the orthogonal group and obtained related splitting theorems. We proved Horrocks’ theorem for the odd elementary orthogonal group. 

• Modules with Descending Chain Conditions on Endosubmodules
  Theophilus Gera,  Sardar Vallabhbhai National Institute of Technology
  Abstract: We investigate endoartinian modules, which satisfy the descending chain condition on endoimages, and establish new characterizations that unify classical and generalized chain conditions. Over commutative rings, endoartinianity coincides with the strongly ACCR* and DCCR* conditions. For right principally injective rings, endoartinian and endonoetherian notions are equivalent. Addressing a question of Facchini and Nazemian, we show that endoartinian and isoartinian modules coincide over simple rings, and classify semiprime endoartinian rings as finite products of matrix rings over principal right ideal domains. We further show that endoartinianity is equivalent to the Köthe property in rings with central idempotents, and characterize such rings as finite products of artinian uniserial rings.

• On Distinguished p-Modular Representations of the Group SL₂(Fₚ)
  Flor de May C. Lanohan,  Mindanao State University-Iligan Institute of Technology
  Abstract: A representation of a finite group G is a pair (π, V), where V is a finite-dimensional vector space over a field K, and π : G × V → V defines the group action. If the field K has positive characteristic ℓ > 0, then the representation (π, V) is called an ℓ-modular representation. The representation theory of p-adic groups over algebraically closed fields of characteristic p, such as the general linear group GLₙ(F) or the special linear group SLₙ(F), where the entries vary in a finite extension F of the field ℚₚ of p-adic numbers, is important for the p-modular Langlands correspondences, a set of profound conjectures that bridge number theory and linear algebra. A significant concept in this context is the notion of distinguished representations. Given a subgroup H of G and a character χ of H (i.e., a group homomorphism from H to K*), a representation (π, V) of G is said to be (H, χ)-distinguished when Hom_H(π, χ) ≠ 0, where Hom_H(π, χ) consists of all linear maps ℓ : V → χ such that for all h ∈ H and v ∈ V, ℓ(π(h)v) = χ(h)ℓ(v). When χ is trivial (i.e., χ = 1), we simply say that (π, V) is H-distinguished. In this talk, we present some preliminary results on the determination of some p-modular distinguished irreducible representations of SL₂(Fₚ), where p is prime and Fₚ is the field with p elements. 

    *Joint work with Ramla Abdellatif (University of Picardie Jules Verne) and Jocelyn P. Vilela (Mindanao State University-Iligan Institute of Technology

• Algebraic interpretation of Quantum Computing
  Dr. Tahir Manzoor,  J.B. Institute of Engineering and Technology
  Abstract: I would like to emphasize on how the unified language Clifford algebra or geometric algebra in connection with the Lie algebra can be used to interpret quantum units in quantum computing. How quantum computing looks without the concept of algebra, how algebraic interpretation of quantum units gives rise to the fiber bundles, and other things will be discussed

• Schur's Lemma
  Romita Biswas, Pondicherry University
  Abstract: Representation theory (which makes life a little more linear) studies abstract algebraic structures through linear approximations. Schur's Lemma, here show how matrices rarely commute, it provides insights into the structure of representations. It provides a nice way to check irreducible representations by showing that any linear transformation commuting with all operators in the representation must be a scalar multiple of the identity