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"The pigs did not actually work, but directed and supervised the others. With their superior knowledge it was natural that they should assume the leadership."

George Orwell, Animal Farm


Ulrich Krähmer

School of Mathematics and Statistics
University of Glasgow
15 University Gardens
Glasgow G12 8QW

Office: Room 432

Phone: Please use email!

Email: ulrich dot kraehmer at glasgow dot ac dot uk


A main theme of my research is to study algebraic analogues and generalisations of structures from geometry, topology or physics. So although my core subject is algebra, my work is inspired to some extent by these other fields. Some concrete key words that one can look up in order to get a better idea of what I do are homological algebra, noncommutative geometry, Hopf algebras, quantum groups and Dirac operators.

At present the following research and survey articles are available:

"From computer algorithms to quantum field theory: an introduction to operads", Snapshots of Modern Mathematics 17 (2015) .pdf

"Cyclic homology arising from adjunctions", Theory Appl. of Categories, 30/32 (2015), 1067-1095 (with Niels Kowalzig and Paul Slevin) .pdf

"Factorisations of distributive laws", J. Pure Appl. Algebra 220/4 (2015), 1403-1418 (with Paul Slevin) .pdf

"Racks, Leibniz algebras and Yetter-Drinfel'd modules", Georgian J. Math 22/4 (2015), 529-542 (with Friedrich Wagemann) .pdf

"A Lie-Rinehart algebra with no antipode", Communications in Algebra 43/10 (2015), 4049-4053 (with Ana Rovi) .pdf

"On the Dolbeault-Dirac operator of quantized symmetric spaces ", Transactions London Mathematical Society, 2/1 (2015), 33-56 (with Matthew Tucker-Simmons) .pdf

"Untwisting a twisted Calabi-Yau algebra", J. Algebra 406 (2014), 272-289 (with Jake Goodman) .pdf

"Batalin-Vilkovisky structures on Ext and Tor", J. Reine Angew. Math. 697 (2014), 159-219 (with Niels Kowalzig) .pdf

"On the Hochschild (co)homology of quantum homogeneous spaces", Israel J. Math. 189 (2012), 237-266 .pdf

"A residue formula for the fundamental Hochschild class of the Podles sphere" J. K-Theory 12/2 (2013), 257-271 (with Elmar Wagner) .pdf

"A residue formula for the fundamental Hochschild class of the Podles sphere - the SINGULAR code" .sin

"A residue formula for the fundamental Hochschild 3-cocycle for SUq(2)", J. Lie Theory 22/2 (2012), 557-585 (with Adam Rennie and Roger Senior) .pdf

"Cyclic structures in algebraic (co)homology theories", Homology, Homotopy and Applications 13/1 (2011), 297-318 (with Niels Kowalzig) .pdf

"Twisted spectral triples and covariant differential calculi", Banach Center Publ. 93 (2011), 177-188 (with Elmar Wagner) .pdf

"On the principality of piecewise trivial comodule algebras", J. Noncomm. Geometry 5/4 (2011), 591-614 (with Piotr Hajac, Rainer Matthes, Bartek Zielinski) .pdf

"Duality and products in algebraic (co)homology theories", J. Algebra 323/7 (2010), 2063-2081 (with Niels Kowalzig) .pdf

"The Hochschild cohomology ring of the standard Podles quantum sphere", Arab. J. Sci. Eng. Sect. C Theme Issues 33/2 (2009), 325-335 .pdf

"Braided homology for quantum groups", J. K-Theory 4 (2009), 299-332 (with Tom Hadfield) .pdf

"Twisted homology of quantum SL(2) - part II", J. K-Theory 6 /1 (2009), 69-98 (with Tom Hadfield) .pdf

"Twisted homology of quantum SL(2) - the FELIX code" .flx

"On the non-standard Podles spheres", in "C*-algebras and elliptic theory II" (2008), 145-147 .pdf

"Poincare duality in Hochschild (co)homology", in "New techniques in Hopf algebras and graded ring theory" (2007), 117-126 .pdf

"On piecewise trivial Hopf-Galois extensions", Czech. J. Phys. 56/10-11 (2006), 1221-1226 (with Bartek Zielinski) .pdf

"On the Hochschild homology of quantum SL(N)", C.R. Acad. Sci. Paris 343/1 (2006), 9-13 (with Tom Hadfield) .pdf

"Twisted homology of quantum SL(2)", K-Theory 34/4 (2005), 327-360 (with Tom Hadfield) .pdf

"On quantum double groups and quantum flag manifolds", PhD thesis University of Leipzig (2004) .pdf

"Dirac operators on quantum flag manifolds", Lett. Math. Phys. 67/1 (2004), 49-59 .pdf

"FRT-duals as quantum enveloping algebras", J. Algebra 264/1 (2003), 68-81 .pdf

Talk to me!

I reserve

Thursdays 1500-1800

in my calendar as office hours, so whether you are a student or not, you will find me in this slot in my office just waiting for your visit.

If you want to make an individual appointment as you want to talk about something that needs more time or privacy, just drop me an email and request an appointment in a specified time slot between 0815 and 1815 that is not marked as busy in my calendar:


The African Institutes of Mathematical Sciences offer students from all African countries a one year Taught MSc programme that comes with free accommodation, subsistance, tuition and a laptop. There is a very tough selection process and hence the students that one meets there are extremely motivated.

I had already twice the opportunity to teach a course on representations of finite groups at AIMS Ghana and also supervised some end of year essays, and I can only recommend everyone to apply for lecturing a course there as it is a fantastic experience. Feel free if you want to ask any questions about my trip there, but for official information see their web pages:

AIMS Cameroon

AIMS Ghana

AIMS Senegal

AIMS South Africa

Mathematics masterclasses

Continuing a long tradition, some engaged school teachers organise under the patronage of the Royal Institution of Great Britain masterclasses in mathematics, click here for general information. Below I provide the material for three such lectures that I had the pleasure to give in Glasgow and in Stirling. Feel free to reuse this material and do not hesitate to contact me for getting the original latex sources:

"Knots as mathematical objects" .pdf

"Counting to Infinity - The principle of mathematical induction" .pdf

"On the foundations of mathematics" .pdf

"Symmetry" .pdf

At the University of Glasgow the masterclasses run always in autumn on four Saturday mornings. Feel free to contact me for information about future sessions and I will redirect you to the current person in charge.

Further downloads

Slides for a talk about operads for TedX Glasgow:

"From exploration to innovation via abstraction: a brief introduction to operads" .pdf

Notes for a lecture in our functional analysis course proving that c_0 has no complemement in \ell^\infty:

"On the category of Banach spaces" .pdf

Notes on the climax of our undergraduate group theory course:

"Sylow's theorem" .pdf

Slides from a minicourse on Dirac operators given at the 2nd School and Conference on Noncommutative Geometry at IPM Tehran in 2009. The topics covered and their presentation is influenced by the other courses that were given, but still it is a fairly generic introduction to the subject.

Lecture 1: "Projective modules and connections" .pdf

Lecture 2: "Clifford algebras and spinors" .pdf

Lecture 3: "Spin structures and Dirac operators" .pdf

Lecture 4: "From $S^2=\mathbb{C} P^1$ to Borel-Weil" .pdf

Slides of a 90 minutes lecture about the Knizhnik-Zamolodchikov equations and the Drinfeld-Kohno theorem:

"Knizhnik-Zamolodchikov and Drinfeld-Kohno" .pdf

Slides of a 45 minutes survey talk about compact quantum groups:

"Long live compact quantum groups" .pdf

Slides of a one hour survey talk about noncommutative geometry:

"Noncommutative geometry - An elementary introduction" .pdf

A short survey on Koszul algebras:

"Notes on Koszul algebras" .pdf

A brief discussion of the difference between algebraic line bundles on a real or complex affine variety and smooth line bundles on the underlying smooth manifold:

"Picard groups: real and complex, algebraic and continous" .pdf

Some notes from an afternoon on which I learned how to use Mathematica:

"A quick dip into mathematica" .pdf


WalkIt - the UK route planner for pedestrians

FELIX - a computer algebra system good in noncommutative Gröbner bases

The LiE online demo - a computer algebra tool good in simple Lie algebras

Math Journal Price Survey - avoid publishing in greedy journals (based on 2008 data)

Why Linux is better

Why Linux

The universal currency converter

The best small country in the world - some facts about Scotland

The exact time - if you want to be in time

LaTeX Polish - helps typing maths in Warsaw

Last modified 29.02.2016