Research projects

    Here are some topics on which I and my group members have made some research contributions. The research articles are available here. 

Lie Maps on Lie Algebras

Let L be a Lie algebra. A Lie polynomial is an element of a free Lie algebra, say on k-generators. Such an element defines a map on L by evaluation, called a Lie map. The subject studies whether such maps are surjective and, if not, whether we can add the image finitely many times to get L. In one of the projects I am studying, whether an analog of Ore’s conjecture and Thompson conjecture can be proved for a simple Lie algebra. In another project, we want to classify if a subset containing 0 and closed under automorphisms is always an image of a Lie polynomial. Further, if one can specifically get such a polynomial which gives the given subset as image.

Polynomial Maps on Central Simple Algebras 

  The Waring problem is naturally considered for an algebra 𝓐 (not necessarily commutative) over a field 𝕂, which is a generalisation of the classical problem in number theory for integers. A longstanding problem is the Matrix Waring Problem, where one asks for the smallest number m so that every element of the matrix algebra 𝓐 = M(n, 𝕂) is a sum of m many k-th powers. A lot of interesting results are known. We have solved the asymptotic version of this problem over a finite field. Briefly speaking, if the finite field is large enough, we can write every element as a sum of TWO k-th powers. This is the best possible result. In recent work, we have also extended this work to triangular matrix algebra.

  More generally, we can consider polynomial maps on 𝓐. Let f be a polynomial over 𝕂 in n variables x1 ,…, xn and consider the map w on 𝓐n to 𝓐 given by evaluation under f. Some classic examples are the sum of powers, multilinear polynomials, commutators, etc. The main problem is understanding its image and, in particular, determining if such a map is surjective. This problem generalises the Matrix Waring problem, where the algebra 𝓐=M(n, 𝕂) and the polynomial f is a sum of k-powers.  In this direction, we have studied images of diagonal polynomials over real, complex, and finite fields. These results generalise the earlier work on the Matrix Waring problem to this polynomial and prove an analogous result to that of Richman about the subjectivity of the diagonal map on M(m, R). Another general problem is the Lovov-Kaplansky conjecture, which asks if the image of a multilinear polynomial is always a vector space over M(m, 𝕂) when 𝕂 is infinite. This conjecture is solved for small m, and some partial results for other cases are known. Along with this, we are also looking into more general problems as well, e.g., to determine the image of a polynomial with coefficients in the algebra itself and have obtained results in the case of M(2, C), some of it is in the line of L'vov-Kaplansky conjecture.   

Word Maps on Groups

  The subject of word maps on groups has been one of the most active research areas in group theory over last two decades. The rapid developments started with the breakthrough results of Larsen, Liebeck, O’Brien, Shalev, and Tiep, who solved the Waring problem for finite simple groups and quasi-simple groups. They proved that for a large enough size of group G, any non-trivial word w, every element is a product of two (and resp. three) elements of w(G) when G is a finite simple (resp. quasisimple) group. We have been using probabilistic methods for this problem, especially for finite groups of Lie type. Through the work of Wall, Macdonald, Kung, Stong, Fulman, Praeger, etc, the probability of finding semisimple, regular, regular semi-simple elements in such groups is known. They have developed a method called the cycle index, which essentially is a Generating Function for these counts, and use this to get asymptotic results. We used these techniques to find such elements in the image of a power map on GL(n, q), O(n, q), Sp(2n, q), and U(n, q) and get an explicit formula for the proportions. We write their generating functions explicitly.  We have been able to get a precise formula for any finite group of Lie type when q tends to infinity, using the structure of maximal tori in these groups.

  Over the last five years, we have tried to solve some of Shalev’s conjectures, too. We solved a particular conjecture of Shalev for power maps, which asked for bounds on images of certain words for groups of type An and 2An. We also have such results for triangular groups. The groups beyond simple groups are less explored, and we want to focus on some of them, too. 

Differential Central Simple Algebra

  Splitting of a central simple algebra is a well-studied phenomenon that usually requires a finite extension of the base field. The idea is to determine what field extension after base change will make a CSA into a matrix algebra. However, splitting a `CSA with a derivation' requires a finitely generated field extension, which might involve a transcendental extension. This is called differential splitting. We studied this explicitly for quaternion and, more generally, symbol algebras. 

  Let 𝕜 be a field and δ be a derivation of 𝕜. For m ≥ 2, assume that 𝕜 contains a primitive m-th root of unity ω. For α, β in 𝕜*, the symbol algebra 𝓐 = (α, β)𝕜,ω  is an m2-dimensional 𝕜-algebra generated by u, v in 𝓐 satisfying the relations um = α, vm = β, and vu  = ω uv. An additive map d:  R → R on a ring R is said to be a derivation on R if d(xy) = xd(y) + d(x)y for all x, y in R. For m ≥ 2, we study derivations on symbol algebras of degree m over fields with characteristic not dividing m. A differential central simple algebra over a field 𝕜 is split by a finitely generated extension of 𝕜. For certain derivations on symbol algebras, we provide an explicit construction of differential splitting fields and give bounds on their algebraic and transcendence degrees. We further analyze maximal subfields that split certain differential symbol algebras. In further work, we explicitly construct examples of derivations for quaternion algebra, which achieve various possible splitting degrees. The maximum possible degree is 4, and we give examples of derivations for which splitting field degrees are 1, 2,3 and 4.

Breadth Type of Nilpotent Lie Algebras

  For a natural number m, a Lie algebra L (always assumed to be of finite dimension) over a field 𝕜 is said to be of breadth type (0, m) if the co-dimension of the centralizer of every non-central element is of dimension m. This project is motivated by the classification of p-groups of small sizes. The notion of conjugate type plays an important role there. The analogous notion in the case of a Lie algebra could be the co-dimension of various centralizers. We have classified finite dimensional nilpotent Lie algebras of breadth type (0, 3) over 𝔽q of odd characteristic up to isomorphism. Encouraged by these results, we decided to explore the subject further and try to get results over more general fields. Naturally, as opposed to simple Lie algebras, the nilpotent Lie algebras are not very well understood. We have got a partial classification of the nilpotent Lie algebras of a finite dimension of breadth type (0, m) over finite fields of even characteristic, ℂ, and  ℝ.

Commuting Probability and Branching Matrix

  In this work, we looked at how to get the commuting probability for certain classical groups. The commuting probability cpn(G) measures the probability of finding an n-tuple in G that commutes pairwise. We proved that this quantity is related to computing branching, which we have computed for some classical groups in an earlier work. This topic is well-studied in topology as well by considering the space Hom(ℤn, G) for a Lie group G. It is a computationally very difficult problem, and often we have to resort to computers for guessing the correct solutions.

Z-classes in groups

Let G be a group. We say two elements in G are z-conjugate if their centralisers in G are conjugate. Since two conjugate elements are always z-conjugate, z-conjgacy is a weaker notion than conjugacy and often used to indicate the orbit behaviour. Indeed, when a compact group acts on a compact manifold the number of z-equivalent orbits are finitely many. From the work of Steinberg it follows that for simple algebraic groups even thoug the number of conjugacy classes are infinitely many, there are finitely many z-classes. In one of the results we have proved that over fields of type (F), certain cohomoligical property, reductive groups have finitely many z-classes. In further work we have established the number of such classes for various classical groups over finite fields. 

Real and Strongly Real elements in groups

An element is a group G is said to be real (or reversible) if it is conjugate to its own inverse. It is said to be strongly real if it is a product of two involutions. For finite groups, the number of real conjugacy classes is equal to the number of real irreducible characters. Such characters are of two kind: orthogonal and symplectic. The question has been to figure out if the number of orthogonal characters are equal to the number of strongly real elements. I studied real elements for algebraic groups and various classical groups, G_2 etc over the years starting from my PhD thesis.