Rolf Farnsteiner

Representation Theory

In Physics one is often interested in matrix solutions to certain systems of algebraic equations such as 

x2 = 0  ;   y2 = 0  ;  (xy)2 - (yx)2 = 0

Here the variables x and y are not supposed to commute with each other (since matrices usually do not commute). One problem is to find solutions that are truly different from each other. Certainly, pairs of square matrices of different sizes would be considered different. For matrices of a given size, one has the notion of similarity which one knows from the theory of canonical forms. Note that if X,Y are (n x n)-matrices that satisfy the system above and S is an arbitrary invertible (n x n)-matrix, then the pair

(SXS-1, SYS-1)

also is a solution. Such solutions will not be considered different.

The proper mathematical framework to accommodate the foregoing features is the representation theory of algebras over a field k (the ground field providing the entries for our matrices). In this theory, one first defines an abstract object A in which the system of equations holds. In our case this would be the free algebra k{x,y}on the generators x,y modulo the ideal I generated by the set

{x2 , y2 ,  (xy)2 - (yx)2}

 

Thus, the algebra A := k{x,y}/I  has the requisite property with respect to the residue classes of x and y, which we will also denote by x and y. Now let M be a finite-dimensional vector space over k and consider a homomorphism

f : A ---> Endk(M)

from A into the k-algebra of linear transformations on M. Since f respects addition and multiplication, the transformations f(x) and f(y) also satisfy our system of equations. By choosing a basis of  M, we produce a solution consisting of square matrices.

The map f is called a representation of the associative algebra A, and M is referred to as an A-module. For ease of notation one writes

a.m := f(a)(m)        for all a in A and m in M.

In this setting the similarity of matrices corresponds to the notion of an isomorphism between two A-modules M and N. Such an isomorphism is given by a bijective linear u : M --> N transformation such that

u(a.m) = a.u(m)       for all a in A and m in M.

The main goal of the representation theory of algebras is the classification (up to isomorphism) of the representations of an associative k-algebra A.

The theory of algebras and their representations is generally considered to originate with Hamilton's definition of Hypercomplex numbers and his famous construction of the quaternions. By the end of the 19-th century algebras where successfully employed in the representation theory of finite groups. Maschke's 1898 Theorem states that the representations of the group algebras of finite groups over fields of characteristic zero are semisimple, that is, every representation is a direct sum of simple subrepresentations, of which there are only finitely many. The structure of finite-dimensional algebras with the latter property was completely determined by Wedderburn in 1908. His result was generalized by E. Artin in 1925 to algebras satisfying a somewhat weaker finiteness condition, the so-called Artin algebras. At around the same time the final version of the theorem of Krull-Remak-Schmidt was proved. This important result states that any finite-dimensional A-module M can be written in an essentially unique way as a direct sum

M = M1 +  M2 + .........+ Mr

of submodules, which in turn cannot be written as direct sums of proper submodules. This reduces the classification problem to the determination of these so-called indecomposable modules.

If the algebra A satisfies Maschke's Theorem it is called semisimple. For such algebras, all indecomposable modules are simple, so that there are only finitely many indecomposable modules. There are other algebras with this property, and we say that a k-algebra A has finite representation type if it admits only finitely many non-isomorphic indecomposable modules. For instance, the theorem of finitely generated modules over principal ideal domains implies that the truncated polynomial algebra  k[X]/(Xn) has finite representation type.

Unfortunately, most algebras are not of finite representation type. An early result by Higman asserts that a group algebra over a field of characteristic p>0 has finite representation type if and only if its p-Sylow subgroups are cyclic. If  p=2, then our algebra above is the group algebra of the dihedral group of order 8. Thus, it is its own 2-Sylow subgroup, yet it is not cyclic, so it has infinite representation type.

According to a famous result due to Drozd, an algebra of infinite representation type is either tame or wild. In the former case, all but finitely many indecomposable A-modules of a given dimension can be parametrized by essentially one parameter of the base field. It turns out that for p=2 the algebra defined above enjoys this property. If A is wild, then the classification of the indecomposables is harder than bringing two matrices simultaneously into Jordan form, a problem which is generally considered hopeless.

Since most algebras are of wild representation type, one has to find a new way of describing indecomposable modules. This is accomplished by the so-called Auslander-Reiten theory, which was initiated by M. Auslander and I. Reiten in the early seventies of the last century.

 

Publications

1. Lie algebras with faithful completely reducible representations.
      Abh. Math. Sem. Univ. Hamburg  51 (1981), 244-251  (with Helmut Strade)

  2. Periodicity and representation type of modular Lie algebras.
      J. reine angew. Math. 464 (1995), 47-65

  3. Representations of blocks associated to induced modules of restricted Lie algebras.
      Math. Nachr. 179 (1996), 57-88

  4. On the distribution of AR-components of restricted Lie algebras.
      Contemp. Math. 229 (1998), 139-157

  5. Auslander-Reiten theory for restricted Lie algebras.
      Ohio State Univ. Math. Res. Inst. Publ. 7 (1998), 165-186

  6. On Auslander-Reiten quivers of enveloping algebras of restricted Lie algebras.
      Math. Nachr. 202 (1999), 43-66

  7. Representations of nilpotent Lie algebras.
      Arch. Math. 72 (1999), 28-39

  8. On support varieties of Auslander-Reiten components.
      Indag. Math. 10 (1999), 221-234

  9. Modules of solvable infinitesimal groups and the structure of representation-finite cocommutative Hopf algebras.
      Math. Proc. Camb. Phil. Soc. 127 (1999), 441-459  (with Detlef Voigt)

10. On cocommutative Hopf algebras of finite representation type.
      Adv. in. Math. 155 (2000), 1-22  (with Detlef Voigt)

11. Auslander-Reiten components for Lie algebras of reductive groups.
      Adv. in. Math. 155 (2000), 49-83 

12. On the Auslander-Reiten quiver of an infinitesimal group.
      Nagoya Math. J. 160 (2000), 103-121 

13. Schemes of tori and the structure of tame restricted Lie algebras.
       J. London Math. Soc. 63 (2001), 553-570  (with Detlef Voigt)

14. Infinitesimal unipotent group schemes of complexity 1.
       Colloq. Math. 89 (2001), 179-192  (with Gerhard Röhrle and Detlef Voigt)

15. Representations of infinitesimal groups of characteristic 2.
      Arch. Math. 78 (2002), 184-188  (with Detlef Voigt)

16. Almost split sequences of Verma modules.
      Math. Ann. 322 (2002), 701-743  (with Gerhard Röhrle)

17. Classification of restricted Lie algebras with tame principal block.
      J. reine angew. Math. 546 (2002), 1-45  (with Andrzej Skowronski)

18. On infinitesimal groups of tame representation type.
      Math. Z. 244 (2003), 479-513  (with Detlef Voigt)

19. Block representation type of Frobenius kernels of smooth groups.
      J. reine angew. Math. 586 (2005), 45-69

20. Auslander-Reiten Components for G1T-modules. 
      J. Algebra Appl. 4 (2005), 739-759

21. The tame infinitesimal groups of odd characteristic. (with Andrzej Skowronski)  
      Adv. in Math. 205 (2006), 229-274  

22. Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks.
      Invent. math. 166 (2006), 27-94

23. Tameness and complexity of finite group schemes.
      Bull. London Math. Soc. 39 (2007), 63-70

24. Support Spaces and Auslander-Reiten components.
      Contemp. Math. 442 (2007), 61-87

25. Galois actions and blocks of tame infinitesimal group schemes.
      Trans. Amer. Math. Soc. 359 (2007), 5867-5898