## MRI-GQT Masterclass

Virasoro algebras and positive energy representations.
Lecturers: Gert Heckman, Johan van de Leur.

Book
V.G. Kac & A.K. Raina,
Bombay Lectures On Highest Weight Representations of Infinite Dimensional Lie Algebras,
Advanced Series in Mathematical Physics Vol. 2,
World Scientific Publishing Co., Inc., Teaneck, NJ, 1987. xii+145 pp. ISBN: 9971-50-395-6; 9971-50-396-4

Location and Time of the Lectures
Every Friday, 13.45 -17.00 h. (Starting February 15) in Minnaert building , room 018.

Organization
Each meeting consists of 2 lectures of 45 minutes followed by a session in which students explain homework exercises on the blackboard (for which they receive a grade).

Schedule

February 15:
Lecture: Gert Heckman: The Knizhnik-Zamolodchikov equation (Literature: E. Looijenga, Arrangements, KZ systems and Lie algebra homology )
Exercises:
Show that both the Cherednik-Dunkl-connection and the Knizhnik-Zamolodchikov-connection are flat.

February 22:
Lecture: Gert Heckman: Why central extensions? (Literature: Sternberg, Group Theory and Physics)
Exercises: 1 show that the a) <Lu,Lv>=<u,v> implies L is linear and that b) <Lu,Lv>=<v,u> implies L is anti-linear.
2 Prove: Let [v_1]\in P(V) , then there exists a unique [v_2]\in P(V), with ([v_1], [v_2])=0 and P(V)={[v]; ([v], [v_1])+([v], [v_2])=1}.
3 Check that G = g\oplus Cc (vector space) a(x,y)=l([x,y]) for some l\in g* defines a 2-cocycle. Prove that G is isomorphic to G = g\oplus Cc as a direct sum of Lie algebras.

February 29:
Lecture: Johan van de Leur: Lecture 1 from the book by Kac and Raina
Exercises:
1 Show that w(d_n)=-d_n defines an anti-linear anti-involution and calculate the corresponding real Lie algebra.
2 Prove Proposition 1.2.
3 Calculate all central extensions of the semi-direct product of the Witt algebra and the abelian Lie algebra C[ z, z^{-1} ].
Possibly extra: Show that the Vandermonde determinant is equal up to a sign to \prod_{i<j} ( \lambda_i - \lambda_j )

March 7:
Lecture: Gert Heckman: Lecture 2 and part of Lecture 3 from the book by Kac and Raina
Exercises:
1 Calculate the central term in the proof of Proposition 2.3.
2 Show that B is an irreducible representation of the oscillator algebra.
3 Check that \omega(L_n)=L_{-n}. \omega (c)=\overline c.
4 Verify formula (3.25) for all m and n \in Z, where \tilde L_0 and \tilde L_k given by (3.24).

March 14:
Lecture: Gert Heckman: Rest of Lecture 3 from the book by Kac and Raina, Johan van de Leur: Part of Lecture 4
Exercises: 1 Show that the \Lambda^k V is an irreducible gl_n module and calculate its highest weight.
2 Prove Proposition 3.7

March 28:
Lecture: Johan van de Leur: Lecture 4
Exercises: 1 Show that the elements (4.14) form an orthonormal basis of F^{(o)} with respect to the Hermitian form defined in (3.34-35)
2 Show the formula's (4.59-63)

April 4:
Lecture: Gert Heckman: Lecture 8
Exercises: 1 Compute det_2(c,h) explicitly.
2. Check that omega'(L_n)=(-1)^n L_{-n}, is an anti-linear anti-involution. What are the corresponding unitary L(c,h) modules?

April 11:
Lecture: Johan van de Leur : Lecture 9
Exercises: 1 Sow that \alph(t^k a,t^l b)=k \delta_{k,-l} tr (ab), where a, b are in gl_n.
2 Let g be a Lie algebra with real form g_0, i.e. g=g_0+ig_0, and let \omega be the corresponding anti-linear anti involution. Show that \sigma(x)= - \overline{\omega(x)} is a linear involution of g.
3 Show that (9.33b) holds.
4. For su_{1,1} we have the antilinear involution \omega_1 defined by \omega_1(x)=-y, \omega_1(y)=-x, \omega_1(h)=h.
For su_{2} we have the antilinear involution \omega_2 defined by \omega_2(x)=y, \omega_2(y)=x, \omega_2(h)=h.
What are the unitary highest weight representations for su_{1,1} and su_{2} ? See also Humphreys Section 7.2.

April 18:
Lecture: Gert Heckman : Lecture 10
Exercises: 1 The anti-linear anti-involution on the laffine Lie algebra is given by \omega(x(n))=\omega(x)(-n). Using that \tilde L(\Lambda), with Lambda dominant integral, is unitary for \tilde g, show that tilde L(\Lambda) is also unitary for Vir.
2 Consider the finite dimensional Lie algebras of type E_6, E_7 and E_8. For which k do we have
k/(k+g)-(k+1)/(k+1+g) < (1-n)/((g+1)n), here n is the rank of the Lie algebra, so in this case 6, 7 and 8.

April 25:
Lecture:
Johan van de Leur: Part of Lecture 11.
Exercises: 1 Show that the affine Weyl group of type A_1^{(1)} consists of { t_k, t_k r_1 | k\in Z }..
2. Deduce formula (11.16).

May 2: NO LECTURE

May 9: LAST LECTURE
Lecture: Johan van de Leur: Rest of lecture 11. + Gert Heckman part of lecture 12

exercises oral exam

Oral exams are on June 12 and 13.