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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
Oral exams are on June 12
and 13.