Research

Tensor products of irreducible highest weight representations of Kac-Moody algebras and their generalizations:

• The tensor product is an essential construction in representation theory. Many fascinating conjectures are made to understand the tensor products of various representations, which stimulates a lot of research in representation theory. In my thesis, I tried to understand when a given tensor product of irreducible highest weight representations of Kac-Moody algebras is isomorphic to another such tensor product. This was first studied by C. S. Rajan (Ann. of Math (2) 2004) for finite-dimensional simple Lie algebras. We found a much more straightforward proof in R. Venkatesh et al. [Adv. Math. 231 (2012), no. 6, 3162–3171] for C. S. Rajan’s theorem and generalized his result to indecomposable Kac-Moody Lie algebras.

• We can ask a similar question for Borcherds-Kac-Moody algebras as well. Since BKM algebras are built from sl_2 and Heisenberg copies, the problem of determining when a given two tensor products of irreducible highest weight representations of these algebras are isomorphic to another becomes a much more complicated problem than for Kac-Moody algebras. We need a more careful study of characters in this case. We settled this problem positively and gave the complete characterization in S. Reif et al. [J. Algebra 592 (2022), 402–423].

Affine Demazure modules and their fusion product decomposition:

• Demazure modules of affineKac-Moody algebras have been the subject of intensive study due to their connection with the representation theory of quantum affine algebras, crystal bases, mathematical physics, Macdonald polynomials, and more. One of the key structural results concerning the g-stable Demazure modules is their decomposition into a fusion product of smaller Demazure modules of the same level. This fact plays a vital role in understanding their graded structure and has many applications. We proved this decomposition theorem in a series of papers for all affine Lie algebras. A particular case was considered in R. Venkatesh [Algebr. Represent. Theory 18 (2015), no. 2, 307–321]. The untwisted case was first settled in V. Chari et al. [J. Algebra 455 (2016), 314–346] and we established this result for twisted affine Kac-Moody Lie algebras in D. Kus et al. [Represent. Theory 20 (2016), 94–127]. A key combinatorial fact that was used in Chari et al. (op. cit.) to prove this decomposition theorem was settled for all cases uniformly in R. Venkatesh et al. [J. of Lie Theory 32 (2022), No. 1, 261-266].

• We also introduced a family of graded current algebra representations in V. Chari et al. [Comm. Math. Phys. 333 (2015), no. 2, 799–830], referred as the Chari-Venkatesh representations in the literature. These representations include many exciting families of graded current algebra representations that arise q=1 limit of irreducible representations of quantum affine algebras. We proved that Chari-Venkatesh modules include g-stable affine Demazure modules, and using this, we greatly simplified Mathieu’s presentation of these modules. This simplified presentation was used to prove the decomposition theorem for g-stable affine Demazure modules. We also generalized the notion of Q-systems in simply-laced types using this simplified presentation of g-stable Demazure modules.

• Recently in a joint work Deniz Kus, for untwisted affine Kac-Moody algebra, we proved an embedding of higher-level Demazure module into a tensorproduct of lower level Demazure modules (e.g., level one in type A), which becomes in the limit (for anti-dominant weights) the well-known embedding of finite-dimensional irreducible modules of the underlying simple Lie algebra into the tensor product of fundamental modules. To achieve this goal, we first simplified the presentation of these modules extending the results of V. Chari et al. [Comm. Math. Phys. 333 (2015)] for the 𝔤-stable case. As an application, we also proposed a crystal theoretic way to find classical decompositions with respect to a maximal semi-simple Lie subalgebra by identifying the Demazure crystal as a connected component in the corresponding tensor product of crystals. This is a preprint available here [arXiv:2112.14830]. 

Classification of Regular subalgebras:

• Classifying closed subroot systems of a given root system is a central problem in Lie theory. This needs to be done to understand the structure of various objects studied in Lie theory. For example, this has connections with the classification of regular subalgebras of semi-simple Lie algebras and their generalizations, the classification of reflection subgroups of (affine) Weyl groups, the classification of closed connected subgroups of a compact Lie group that has a maximal rank (Borel-de Siebenthal theory), and more. We are mainly interested in classifying regular subalgebras of affine Kac- Moody Lie algebras and their various generalizations. Regular subalgebras naturally appear in both mathematical and physical contexts. For example, the Lie algebra E10 contains a regular subalgebra isomorphic to AE3 reflects the inclusion of Einstein gravity into 11-dimensional supergravity. By improving some earlier work of Felikson, Retakh, and Tumarkin, we obtained a classification of all regular subalgebras of affine Kac-Moody Lie algebras in K. Roy et al. [Transform. Groups 24 (2019), no. 4, 1261-1308.]. In this paper, we developed an algorithm to get all the closed subroot systems affine root systems and also established a one-to-one correspondence between regular subalgebras and closed subroot systems.

• We can ask a similar question for various generalizations of affine Kac-Moody Lie algebras. The essential generalizations are:

1. Kac-Moody algebras (particularly Hyperbolic Kac-Moody algebras)

2. Borcherds-Kac-Moody algebras

3. Extended affine Lie algebras (EALA)

4. affine Kac-Moody Liesuper algebras

We studied this classification problem for EALA. The nullity one EALA are nothing but affine Kac-Moody Lie algebras, and the nullity two EALA naturally has connections with Double affine Hecke algebras via double affine Weyl groups. So, nullity two is the most crucial class of EALA next to nullity one. We settled the combinatorial problem on classifying the maximal closed subroot systems of affine reflection systems in D. Kus et al. [Mosc. Math. J. 21 (2021), no. 1, 99–127]. In particular, this work helps us to understand what happens in the nullity two case. We expect a one-to-one correspondence between the closed subroot systems of nullity two EALA and their regular subalgebras. This is a work in progress.

• Closed subsets are also natural objects of interest, because they come from Cartan invariant subalgebras. It is not hard to see that symmetric closed subsets of a finite crystallographic root system must be a closed subroot system. This simple fact seems complicated to prove for affine root systems (even for untwisted cases). We recently proved this assertion is indeed true for all affine root systems. As an application, we use this result to establish an algorithm in D. Biswas et al. [in preparation] to classify all closed subsets of the untwisted affine root system in terms of their gradient and null root parts.

Borcherds-Kac-Moody algebras and graph polynomials:

• We found an unexpected connection between the root multiplicities of Borcherds-Kac-Moody algebras and the generalized chromatic polynomials of graphs in D. Kus et al. [J. Algebra 499 (2018), 538–569]. Using the combinatorics of Lyndon's words, we constructed bases for some root spaces

of Borcherds algebras and gave the Lie theoretic proof of Stanley's reciprocity theorem of chromatic polynomials in this paper. This connection was first established in R. Venkatesh et al. [J. Algebraic Combin. 41 (2015), no. 4, 1133–1142] for Kac-Moody algebras.

• This bridge between Borcherds-Kac-Moody algebras and Graph polynomials needs to be explored further. My main interest lies in constructing bases of partially commutative Lie algebras consisting of right-normed Lie words. Theoretically, we know such a basis exists for these algebras. But coming up with a combinatorial model seems to be very difficult even for some simpler graphs. For example, only such a combinatorial model was established for free Lie algebras (complete graph case) by Chibrikov in [J. Algebra 2006].

• C. D. Godsil showed that there is a tree T = T(G) associated with any given simple graph G such that m(G − u, x)/m(G, x) = m(T − r, x)/m(T, x), where m(G, x) denotes the matching polynomial of G. He used his identity to obtain real rootedness of the matching polynomial of graphs. He also used his identity to get some sharp bounds of roots of the matching polynomial of graphs. These root bounds are very important, for example, this was used to show the existence of Ramanujan graphs. Similar identities are available in the literature for the independence polynomial of G. We are particularly interested in the identity that comes from the stable path tree of G (studied by Ferenc Bencs, 2018). Multi-variate generalizations of “these identities” can be viewed as the characters of highest weight representations of Borcherds–Kac–Moody Lie algebras of G. So this suggests that these identities have a much deeper connection at the level of the representations. We proved indeed this is true and the multi-variate version of “these identities” comes from the corresponding isomorphisms of the respective highest weight representations of Borcherds–Kac–Moody Lie algebras.