### Research Interests

The braid group, Bv, has a well- known representation due to Artin in the group of automorphisms of the free group with n generators. Applying the Fox derivatives and Magnus representation, we get the classical Burau representation which is of degree n. D.D. Long gave a new derivative of the Burau representation by using an action of the automorphism group of the group of a free group with n generators on the representation variety of the free group and the group G, where G is a compact semi simple Lie group . We say that a group is linear if it admits a faithful representation into the general linear group for some natural number n and some field K. The question of faithfulness of the braid group has been the subject of research for a long time. It has been shown that the Burau representation is faithful for values of n smaller than 4 and non faithful for values on larger than 4. However, the faithfulness of the Burau representation in the case n=4 is still unknown. Our target then is to construct possible linear irreducible representations of the braid group and hope we could find one that is proved to be faithful. We need also to mention that the concept of faithfulness plays an important role in the field of cryptography, where there is an undergoing significant research in that field.

### Publications

1. M.N. Abdulrahim, A Faithfulness Criterion of the Gassner representation of the Pure Braid Group, Proceedings of the American Mathematical Society, V. 125, no. 5, May 1997, 1249-1257.

2. M.N.Abdulrahim, Complex Specializations of the Reduced Gassner Representation of the Pure Braid Group, Proceedings of the American Mathematical Society, V. 125, no. 6, June 1997, 1617-1624.

3. M.N.Abdulrahim, The reduced Gassner Representation restricted to a normal free subgroup of the Pure Braid Group. Archive Der Mathematik, V. 69, 1997, 8-12.

4. M.N. Abdulrahim, On the Composition of the Burau Representation and the Natural Map B_n \to B_{nk}. Journal of Algebra and its Applications, V. 2, no. 2, June 2003, 169-175.

5. M.N. Abdulrahim & Woo Lee, The Hecke algebra representations of the braid groups B_2 and B_3. Int. J. Appl. Math, V. 14, no. 4, 2003, 325-331.

6. M.N. Abdulrahim, A braid element with eigenvalues as distinct powers of a complex number. Int. J. Appl. Math, V. 15, no. 1, 2004, 17-28.

7. M.N. Abdulrahim, The Number of Descending paths from a given Young Diagram to a smaller one. Mathematical Sciences Research Journal, V.8, no. 10, 2004, 312-317

8. M.N. Abdulrahim, On Embeddings of Pure Braid Groups P_n \to P_{n+k}, Int. J. Appl. Math, V. 16, no. 1, 2004, 1-12

9. M.N. Abdulrahim, Pure Braids as Automorphisms of free Groups. Journal of Algebra and its Applications, V. 4, no. 4, 2005, 435-440.

10. M.N. Abdulrahim & E. Formanek, Tensor Products of Specializations of the Burau Representation, Journal of Pure and applied Algebra, V. 203, no. 1-3, 2005, 104-112.

11. M.N. Abdulrahim, Wada’s Representation and the natural map B_n \to B_{2n}, Int. J. Pure and Appl. Math, V.18, no.2, 2005, 137-142.

12. M.N. Abdulrahim, Generalizations of the standard Artin representation are unitary, Int. J. Math. Math. Sci., no. 8, 2005, 1321-1326.

13. M.N. Abdulrahim, Wada’s Representations of the Pure Braid Group, International Journal of Mathematics, V.16, no. 7, 2005, 757-763.

14. M.N. Abdulrahim, Wada's representation is of Burau type. International Journal of Algebra & Computation, V. 17 , no.3, 2007, 557-565.

15. M.N. Abdulrahim & N. Kassem, The interplay between linear representations of the braid group. Int. J. Math. Math. Sci. 2007, Art. ID 16186, 9 pp.

16. M.N. Abdulrahim, The Kernel of Wada's Representation restricted to a normal free subgroup of the Pure braid group, International journal of Applied Mathematics, V. 21, no. 2, 2008, 239-245.

17. M.N. Abdulrahim & D. Santina, The Hecke Algebra Representation of the braid group $\;B_3$ is conjugate to a unitary representation, International journal of Applied Mathematics, V. 21, no. 3, 2008, 369-374

18. M.N. Abdulrahim, On the Composition of Wada's representation and the natural map $\; P_n \to P_{n+k}$}, Advances in Algebra, V. 1, no. 1, 2008, 47-55.

19 . M.N. Abdulrahim & M.Al-Tahan, Complex Specializations of Krammer’s Representation of the Braid Group, B_3, J. Math. Stat. , V.4, no. 4, 2008, 213-216.

20 .M. N. Abdulrahim, Tensor Products of the Gassner representation of the pure braid group, Open Math J., V. 2, 2009, 12-15

21. M.N. Abdulrahim, Tensoring Three irreducible linear representations of the Braid Group, Linear and Multi Linear Algebra ,V.57, no. 8, 2009, 749-758.

22. M. N. Abdulrahim, Gassner Representation of the Pure Braid Group, P_4,Int. J. Algebra, V. 3, no. 16, 2009, 793-798.

23. M.N. Abdulrahim, On the irreducibility of linear representations of the pure braid group, Tamkang Journal of Mathematics, V. 41, no.3, 2010, 283-292.

24. M.N. Abdulrahim & M.Al-Tahan, On a class of irreducible representations of the braid group, B_n, International journal of Applied Mathematics, V. 23, no. 4, 2010, 681-691.

25. M.N. Abdulrahim & M.Al-Tahan, Krammer's Representation of the Pure Braid Group, P_3, Int. J. Math. Math. Sci. 2010, Art. ID 806502, 10 pp.

26. M.N. Abdulrahim & G. H. Abboud, Pure Braid group representations of low degree, Int. J. Algebra, V. 5, no. 21-24, 2011, 1011-1020.

27. W.M. Zeid & M.N. Abdulrahim, Irreducibility of the Tensor product of specializations of the Burau representation of the braid groups, ISRN Algebra, V. 2011, Article ID 104263, 9 pages.

28. H.A.Tarraf & & M.N. Abdulrahim, Higher Dimensional Irreducible Representations of the Pure Braid group, Journal of Mathematics Research, V.3, no.4, 2011, 141-151

29. M.N. Abdulrahim & Hariri, Mariam, Another proof of the faithfulness of the Lawrence-Krammer representation of the braid group $B\sb 3$. ISRN Algebra 2012, Art. ID 858959, 13 pp .

30. M.Chreif & M.N. Abdulrahim, Irreducible representations of the pure braid group P_n, International journal of Applied Mathematics, V. 25, no. 5, 2012, 723-728

31 G. H. Abboud & M. N.Abdulrahim, On the irreducibility of Wada’s representation of the pure braid group P_4, , Open Math J., V. 5, 2012, 15-18

32. W.M. Zeid & M.N. Abdulrahim, Tensor product of Wada’s representations of the braid group, B_3, Int. Math. Forum, V. 7, no. 29-32, 2012, 1413-1418

33. H.A.Tarraf & & M.N. Abdulrahim, Tensor Product of Krammer’s representations of the Pure braid group, P_3, Advances in Pure Math, 2012,2, 190- 194

34. M.N. Abdulrahim & I.Daakur, Wada’s representations of the pure braid group of high degree, Theoretical Mathematics & Applications, V.2, no. 1, 2012, 117-125.

35. M.Al-Tahan, M.N.Abdulrahim & S.S. Habre, Multi-parameter Burau representations, Tamkang Journal of Mathematics, V. 44, no.1, 2013, 91-98.

36. M.Al-Tahan, M.N. Abdulrahim, A new six Dimensional representation of the braid group on three strands and its irreducibility and unitarizability, British Journal of Mathematics & Computer Science, V.3, no.3, 2013, 275-280

37. M.Al-Tahan, M.N. Abdulrahim, An Overview of Wada’s representations, Journal of Mathematical & Computational Science, V.3,no.6, 2013, 1430-1443

38. M.Al-Tahan, M.N. Abdulrahim & S.S. Habre, Spin Representations with negative indices, Tamkang Journal of Mathematics, V.45, no. 4, 2014, 367-374.

39. N.Maanna, M.N. Abdulrahim, Tuba’s Representation of the pure braid group on three strands, British Journal of Mathematics & Computer Science V.4, no. 16, 2014,2381-2402.

40.M.Dally, M.N. Abdulrahim, On the irreducibility of Artin’s group of graphs, Journal of Mathematics Research, V.7, no.2, 2015, 117-124.

41 M.Chreif & M.N. Abdulrahim, Attacks on the Faithfulness of the Burau Representation of the Braid Group $B_4$ Journal of Mathematics Research, V.8, no.1, 2016, 5-9.

42 M. . Chreif & M. N. Abdulrahim, On the Irreducibility of the complex specialization of the representation of the Hecke algebra of the complex reflection group G , Journal of Linear & Topological Algebra, V.5, No.4, 2016, 263-270

43 M. Chreif & M. N. Abdulrahim , The Hecke Algebra representation of the complex reflection group G is unitary, JP Journal of Algebra, Number Theory and Applications, V.39, No.2, 2017, 151-164

44 M.M..Dally, M.N. Abdulrahim, On the irreducibility of Perron representations of
degrees 4 and 5, Eur.J.Pure Appl.Math, V.11, No. 1, 2018, 215-237.

45 H.A..Haidar, M.N.Abdulrahim, On the irreducibility of fourth dimensional Tuba’s
representation of the pure braid group on three strands, Eur.J.Pure Appl.Math,
V.11, No. 3, 2018, 682-701

1. Colloquium Lecture at The American University of Beirut, Lebanon, 2009 . Title of the talk: Tensor Product of Specializations of the Burau Representation.

2. Colloquium Lecture at The American University of Beirut at the conference of Lie Groups and Related Topics, April 23rd, 2014.

3. Organized a workshop in mathematics in March 2009 (Workshop on Representation Theory and Number Theory ) as a partner with American University of Beirut in the LSMS conference in April 26th, 2012.

4. Organized a workshop on Partial Differential Equations in April 22nd, 2015

5. Organized A workshop on Mathematical Analysis on March 15th, 2016.