Israel Gelfand

How to use this course

Israel Gelfand’s name is attached to more mathematical objects than almost anyone’s of his century: the Gelfand transform, the Gelfand–Naimark theorems, the GNS construction, Gelfand pairs, Gelfand–Tsetlin patterns, the Gelfand–Levitan equation, Gelfand–Fuchs cohomology, Gelfand–Kirillov dimension, the BGG resolution and GKZ hypergeometric systems. Behind the list is a single habit. He looked for the algebraic structure inside analysis, and for the simplest example that shows the whole phenomenon.

Each technical lesson opens with a paragraph for the general reader in italics, then gives precise statements, proof sketches and exercises with folded solutions. The order is roughly chronological: Banach algebras (1938–1943), representations (1943–1950), spectral theory (1951), generalized functions and integral geometry (1950s–1960s), and the algebraic work of his later decades.

Timeline

Okny, 1913–1930

Israel Moiseevich Gelfand was born on 2 September 1913 in Okny, a small town in the Kherson Governorate of the Russian Empire, now in the Odesa region of Ukraine. His family was Jewish, and his father ran a mill. He was largely self-taught in mathematics.

After the Revolution, the son of a mill owner counted as a child of the class enemy. Gelfand was expelled from the technical school he attended and could not complete his secondary education. He never received a school certificate and never studied as an undergraduate.

Moscow without a diploma, 1930–1935

In 1930, aged 16, Gelfand went to Moscow. He took whatever work he could find. One job was as a doorkeeper at the Lenin Library. Soon he was teaching mathematics in the evening institutes that the Soviet state was opening for adult workers, and teaching taught him the subject in turn. He also attended lectures and seminars at Moscow University, among them Mikhail Lavrentiev’s course on complex variables.

In 1932, at 19, he was admitted as a research student at Moscow State University with Andrey Kolmogorov as supervisor. The university took him on the strength of his mathematics alone. Kolmogorov set him to work in the new field of functional analysis, then being built from the work of Stefan Banach, Frigyes Riesz and John von Neumann. Gelfand’s candidate thesis of 1935, Abstract functions and linear operators, studied integration and differentiation of functions with values in Banach spaces. The Gelfand–Pettis integral comes from this work, as does a lemma on the boundedness of convex functionals.

Normed rings, 1935–1943

From 1935 Gelfand held a research post at the Academy of Sciences, and in 1939 he became an associate professor at Moscow University. His central idea of these years was to treat a space of functions or operators not only as a vector space but as a ring. Multiplication, he found, carries the topology: the norm of an element can be computed from its powers, and the points of a space can be recovered as the maximal ideals of its algebra of functions.

He presented this theory of commutative normed rings for the higher doctorate, which he received in 1940, and published it in 1941 as Normierte Ringe. He became a full professor that year. Two applications made the theory famous. A three-line proof of Norbert Wiener’s lemma on absolutely convergent Fourier series showed what the abstraction could do. And with Mark Naimark in 1943 he proved that the algebras now called C*-algebras are exactly the norm-closed algebras of operators on Hilbert space. In the same year he and Dmitrii Raikov proved that every locally compact group has enough irreducible unitary representations to separate its points. Representation theory of infinite groups had a foundation, and Gelfand spent the next decade building on it.

The war years also brought hardship and discrimination. As a Jew, he was later pushed out of his position at the Steklov Institute and out of a full-time professorship at the university. He continued to teach at Moscow University and found a lasting research home at the Institute of Applied Mathematics of the Academy, directed by Mstislav Keldysh, where he did extensive computational work.

Commutative Banach algebras

For the general reader. Continuous functions on a space can be added and multiplied. Gelfand showed that the space itself can be recovered from this algebra alone: its points correspond to the ways of “evaluating” the algebra. The same idea gives a short proof of a hard theorem of Wiener about Fourier series, and it became a model for modern geometry, where spaces are studied through their algebras of functions.

A Banach algebra \(A\) is a complex Banach space with an associative bilinear product satisfying \(\|ab\|\le\|a\|\|b\|\). Assume \(A\) has a unit with \(\|1\|=1\). The spectrum of \(a\) is \(\sigma(a)=\{\lambda\in\mathbb C: a-\lambda\ \text{not invertible}\}\).

Proof sketch. If \(\sigma(a)\) were empty, \(\lambda\mapsto\varphi\big((a-\lambda)^{-1}\big)\) would be an entire function tending to 0 at infinity for every \(\varphi\in A^*\); by Liouville it vanishes, and Hahn–Banach forces \((a-\lambda)^{-1}=0\), impossible. For Gelfand–Mazur, pick \(\lambda\in\sigma(a)\); then \(a-\lambda\) is not invertible, so it is 0. For the radius formula, the resolvent has the Laurent expansion \(-\sum_n a^n\lambda^{-n-1}\) for large \(|\lambda|\), which by analyticity converges on \(|\lambda|>r(a)\); uniform boundedness then gives \(\limsup\|a^n\|^{1/n}\le r(a)\), and \(\lambda^n\in\sigma(a^n)\) gives the reverse inequality. Stanisław Mazur had announced the division-algebra result in 1938 without published proof.

The Gelfand transform

Now let \(A\) be commutative. A character is a non-zero algebra homomorphism \(\chi:A\to\mathbb C\). Its kernel is a maximal ideal, and conversely, by Gelfand–Mazur, every maximal ideal \(M\) gives \(A/M\cong\mathbb C\). The set \(\Delta(A)\) of characters, with the weak-* topology, is a compact Hausdorff space.

Wiener’s lemma

Let \(A=\ell^1(\mathbb Z)\) with convolution, identified with the absolutely convergent Fourier series \(f(\theta)=\sum c_ne^{in\theta}\), \(\|f\|=\sum|c_n|\). A character is determined by \(z=\chi(\delta_1)\); since \(\|\delta_{\pm1}\|=1\), \(|z|\le1\) and \(|z^{-1}|\le1\), so \(|z|=1\). Thus \(\Delta(A)\) is the circle and the Gelfand transform is the Fourier series. The theorem above gives:

C*-algebras: Gelfand–Naimark

For the general reader. Quantum mechanics replaces numbers by operators on a Hilbert space. Gelfand and Naimark found a short list of algebraic axioms that characterize exactly the algebras of such operators. When the algebra is commutative, it is an algebra of ordinary functions on a space; when it is not, one can think of it as functions on a “noncommutative space”, the starting point of Alain Connes’ noncommutative geometry.

A C*-algebra is a Banach algebra with an involution \(a\mapsto a^*\) (conjugate-linear, \((ab)^*=b^*a^*\), \(a^{**}=a\)) satisfying the C*-identity

The 1943 paper used two further axioms, that \(\|a^*a\|=\|a^*\|\|a\|\) and that \(1+a^*a\) is always invertible. Both were later shown to follow from the C*-identity, the second by Irving Kaplansky and others around 1952–53.

The GNS construction

A state is a linear functional \(\omega\) with \(\omega(a^*a)\ge0\) and \(\omega(1)=1\). Define \(\langle a,b\rangle=\omega(b^*a)\), a semi-definite inner product on \(A\). Quotient by the null space \(N=\{a:\omega(a^*a)=0\}\), which is a left ideal, and complete to get \(H_\omega\). Then \(\pi_\omega(a)[b]=[ab]\) defines a representation with cyclic vector \(\xi=[1]\) and \(\omega(a)=\langle\pi_\omega(a)\xi,\xi\rangle\). Taking the direct sum over enough states (supplied by Hahn–Banach and Krein–Milman) gives the faithful representation of the theorem. Irving Segal sharpened the construction in 1947, hence the name Gelfand–Naimark–Segal.

Unitary representations and Gelfand pairs

For the general reader. Fourier analysis decomposes a function on a circle into waves. Symmetry groups such as the rotations of a sphere or the Lorentz group of relativity allow a similar decomposition, but for non-compact groups the building blocks are infinite-dimensional. Gelfand and his collaborators showed that such representations always exist in sufficient number, classified them for the Lorentz group, and found the condition under which the decomposition is especially clean.

The proof applies the GNS construction to positive-definite functions on \(G\) and uses Krein–Milman: extreme points of the convex set of normalized positive-definite functions give irreducible representations, and they separate points. In 1947 Gelfand and Naimark classified the irreducible unitary representations of the Lorentz group \(\mathrm{SL}(2,\mathbb C)\), in parallel with Valentine Bargmann’s work on \(\mathrm{SL}(2,\mathbb R)\). Their book Unitary Representations of the Classical Groups (1950) began the program later completed by Harish-Chandra.

Gelfand pairs

Let \(K\) be a compact subgroup of \(G\). The pair \((G,K)\) is a Gelfand pair if the convolution algebra \(L^1(K\backslash G/K)\) of bi-\(K\)-invariant functions is commutative. Equivalently, for each irreducible unitary representation \(\pi\), the space of \(K\)-fixed vectors has dimension at most 1. The matrix coefficient \(\varphi(g)=\langle\pi(g)v,v\rangle\) of a \(K\)-fixed unit vector is then a spherical function, and these play the role of exponentials in harmonic analysis on \(G/K\).

Proof. For bi-invariant \(f\), \(f\circ\theta=f\). Since \(\theta\) reverses products and preserves Haar measure, \((f_1*f_2)\circ\theta=(f_2\circ\theta)*(f_1\circ\theta)\). Hence \(f_1*f_2=(f_1*f_2)\circ\theta=f_2*f_1\).

Examples: \((\mathrm{SO}(n+1),\mathrm{SO}(n))\) with spherical functions given by Gegenbauer polynomials; \((G,K)\) for every Riemannian symmetric space, with Harish-Chandra’s spherical functions; and \((G\times G,\Delta G)\) for compact \(G\), where harmonic analysis is the Peter–Weyl theorem.

Gelfand–Tsetlin bases

For the general reader. An irreducible representation of a symmetry group is a space with no preferred basis. In 1950 Gelfand and Mikhail Tsetlin found a canonical one for the unitary groups by restricting step by step to smaller and smaller subgroups. The basis vectors are labelled by triangular arrays of integers, which turn hard algebra into counting.

Irreducible representations of \(\mathrm U(n)\) are labelled by highest weights \(\lambda=(\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n)\in\mathbb Z^n\).

Iterate the restriction down the chain \(\mathrm U(n)\supset\mathrm U(n-1)\supset\cdots\supset\mathrm U(1)\). Because each step is multiplicity-free, the decomposition ends in one-dimensional pieces, each determined by a sequence of interlacing rows.

Counting patterns gives \(\dim V_\lambda\), which must equal Weyl’s formula \(\prod_{i<j}\frac{\lambda_i-\lambda_j+j-i}{j-i}\). Gelfand–Tsetlin patterns are in bijection with semistandard Young tableaux, and the polytope of real interlacing arrays is the Gelfand–Tsetlin polytope, now central in the geometry of flag varieties and in integrable systems.

Inverse spectral problems

For the general reader. Can you hear the shape of a string? If you know every frequency at which a vibrating system resonates, can you reconstruct the system? In 1951 Gelfand and Boris Levitan gave an exact procedure for one-dimensional problems: from the spectral data, solve a linear integral equation, and read off the potential. Twenty years later the same equation became the engine of the inverse scattering method for solitons.

Consider the Sturm–Liouville problem on the half-line,

with real continuous \(q\). Let \(\varphi(x,\lambda)\) be the solution with these initial values. There is a non-decreasing spectral function \(\rho(\lambda)\) such that \(f\mapsto\int_0^\infty f(x)\varphi(x,\lambda)\,dx\) is an isometry from \(L^2(0,\infty)\) onto \(L^2(d\rho)\). For \(q=0\), \(h=0\), one has \(\varphi=\cos(\sqrt\lambda\,x)\) and \(\rho_0(\lambda)=\frac2\pi\sqrt\lambda\) for \(\lambda>0\).

For each fixed \(x\) the equation is a Fredholm equation in \(t\), so the reconstruction is linear even though the map \(q\mapsto\rho\) is not. Vladimir Marchenko gave the analogous equation for scattering data, and in 1967 Clifford Gardner, John Greene, Martin Kruskal and Robert Miura used it to solve the Korteweg–de Vries equation: the scattering data of the Schrödinger operator with potential \(u(x,t)\) evolve linearly when \(u\) solves KdV. In 1975 Gelfand and Leonid Dikii built the KdV hierarchy from the asymptotic expansion of the resolvent of this operator.

Generalized functions and integral geometry

For the general reader. Physicists use objects like the Dirac delta, a “function” that is zero everywhere except at one point. Laurent Schwartz made such objects rigorous around 1950. Gelfand and his coauthors wrote the five-volume treatise that made the theory a working tool, and asked a related question: how can a function be reconstructed from its integrals over lines or planes? The answer underlies the CT scanner.

Gelfand triples

The operator \(-i\,d/dx\) on \(L^2(\mathbb R)\) has “eigenfunctions” \(e^{ikx}\), none of which is square-integrable. Gelfand’s answer was to place the Hilbert space between a nuclear space of test functions and its dual, \(\Phi\subset H\subset\Phi'\), for example \(\mathcal S(\mathbb R)\subset L^2(\mathbb R)\subset\mathcal S'(\mathbb R)\).

This “rigged Hilbert space” gives Dirac’s bra–ket formalism a rigorous meaning. It appears in volume 4 of Generalized Functions (1961), with Naum Vilenkin. The series ran from 1958 to 1966, with Shilov, Vilenkin and Mark Graev as coauthors.

The Radon transform

For \(f\) on \(\mathbb R^2\), \(\theta\in S^1\) and \(s\in\mathbb R\), let \(Rf(\theta,s)=\int_{x\cdot\theta=s}f\), the integral over a line, and let \(R^\#g(x)=\int_{S^1}g(\theta,x\cdot\theta)\,d\theta\) be back-projection. Johann Radon inverted \(R\) in 1917. The Gelfand school, especially with Graev and Vilenkin in volume 5 (1962) and later with Simon Gindikin, built integral geometry into a general theory linked to representation theory.

Proof sketch. By the projection-slice theorem the 1D Fourier transform of \(Rf(\theta,\cdot)\) is \(\hat f(\sigma\theta)\). Fourier inversion in polar coordinates gives \(f(x)=\frac{1}{(2\pi)^2}\cdot\frac12\int_{S^1}\int_{\mathbb R}|\sigma|\,\hat f(\sigma\theta)e^{i\sigma x\cdot\theta}\,d\sigma\,d\theta\), and \(|\sigma|\) is the multiplier of \(\mathcal H\partial_s\).

Cohomology, dimension, hypergeometry

For the general reader. From the late 1960s Gelfand’s seminar turned increasingly algebraic. Three of its products are sampled here: a computation of the cohomology of the Lie algebra of vector fields on a circle, which produced the symmetry algebra of string theory; a way to measure the size of noncommutative algebras; and a sweeping generalization of the hypergeometric functions of Gauss.

Gelfand–Fuchs cohomology (1968–70)

With Dmitry Fuchs, Gelfand computed the continuous cohomology of Lie algebras of smooth vector fields, a project that founded a new branch of characteristic-class theory.

The corresponding central extension is the Virasoro algebra.

Gelfand–Kirillov dimension (1966)

For a finitely generated algebra \(A\) with finite-dimensional generating subspace \(V\ni1\),

independent of \(V\). Gelfand and Alexandre Kirillov introduced it to show that the Weyl algebras \(A_n\), \(A_m\) (\(n\ne m\)) have non-isomorphic skew fields of fractions. Their conjecture that the fraction field of \(U(\mathfrak g)\) is always a Weyl skew field over a purely transcendental extension holds for \(\mathfrak{gl}_n\) and nilpotent \(\mathfrak g\), and fails in general.

BGG and GKZ

With Joseph Bernstein and his son Sergei Gelfand, Gelfand constructed in the 1970s a resolution of each finite-dimensional irreducible representation \(L_\lambda\) of a semisimple Lie algebra by Verma modules:

where \(w\) runs over the Weyl group, \(N\) is the number of positive roots and \(w\cdot\lambda=w(\lambda+\rho)-\rho\). Taking characters gives the Weyl character formula, and the same authors introduced the category \(\mathcal O\) in which these modules live. In the late 1980s, with Mikhail Kapranov and Andrei Zelevinsky, he defined A-hypergeometric (GKZ) systems: for an integer matrix \(A\) and parameter \(\beta\), the system \(\big(\sum_ja_{ij}x_j\partial_j-\beta_i\big)\Phi=0\) together with the toric equations \(\partial^u\Phi=\partial^v\Phi\) for \(Au=Av\). Classical hypergeometric functions are special cases, and the theory led to their 1994 book on discriminants and resultants.

The seminar, biology and schools, 1943–1989

In 1943 Gelfand began a weekly seminar at Moscow State University. It ran for more than forty-five years and became the most famous institution in Soviet mathematics. Sessions began late, ran for hours, and followed no fixed program. Gelfand interrupted speakers, demanded that they explain their results to the weakest listener in the room, called up students without warning, and switched topics at will. Many found it intimidating; many more found it the best mathematical education available anywhere. His favourite teaching maxim, as reported by the New York Times, was that you have to be fast only to catch fleas.

The seminar and his collaborations formed a school of extraordinary breadth. His doctoral students included Georgiy Shilov, Felix Berezin, Georgy Adelson-Velsky, Alexandre Kirillov, Endre Szemerédi, Joseph Bernstein, Andrei Zelevinsky, Victor Ginzburg, Alexander Goncharov and his son Sergei Gelfand; David Kazhdan, Edward Frenkel and many others came through the seminar. Long collaborations with Mark Naimark, Mark Graev and Ilya Piatetski-Shapiro produced the theory of automorphic representations in its modern form; their 1966 book Representation Theory and Automorphic Functions was a bridge to the Langlands program. In 1960 he proposed the index problem for elliptic operators, solved by Michael Atiyah and Isadore Singer. He gave invited addresses at the International Congresses of 1954, 1962 and 1970. In 1967 he founded the journal Functional Analysis and Its Applications, and from 1968 to 1970 he was president of the Moscow Mathematical Society.

Recognition at home was limited by anti-Jewish policy. He received State Prizes in 1951 and 1953 and the Lenin Prize in 1961, and became a corresponding member of the Academy of Sciences in 1953, but full membership came only in 1984. He worked for decades at the Institute of Applied Mathematics of the Academy, where he led computational work.

From around 1958 Gelfand built a second career in biology. After his son Aleksandr fell ill with leukaemia he organized a seminar on cell biology; he co-founded the Institute of Biological Physics of the Academy in 1960 with Sergei Fomin, and published on cell motility, the organization of movement, and medical diagnosis. Aleksandr died of the disease. In 1964 Gelfand founded the All-Union Mathematical Correspondence School, which sent problem booklets and personal feedback to schoolchildren across the Soviet Union, far from the elite schools of the capitals. He wrote short books for it with collaborators, on algebra, functions and graphs, the coordinate method and trigonometry.

Rutgers and legacy, 1989–2009

In 1978 Gelfand shared the first Wolf Prize in Mathematics with Carl Ludwig Siegel, cited for his work in functional analysis and group representations and for seminal contributions to many areas of mathematics and its applications. He received the Kyoto Prize in 1989. That year, aged 76, he left the Soviet Union, spent a year at Harvard and MIT, and in 1990 joined Rutgers University in New Jersey as a Distinguished Visiting Professor. He restarted the seminar there on a smaller scale and in 1992 founded the Gelfand Outreach Program, a correspondence program in mathematics for American schoolchildren. He continued to publish in mathematics and biology into his nineties. He received a MacArthur Fellowship in 1994 and the Steele Prize for Lifetime Achievement of the American Mathematical Society in 2005.

Gelfand died on 5 October 2009 at Robert Wood Johnson University Hospital in New Brunswick, New Jersey, near his home in Highland Park, aged 96. He was survived by his second wife Tatiana, his sons Sergei and Vladimir from his first marriage, to the mathematician Zorya Shapiro, and his daughter Tatiana.

Honours

• State Prizes of the USSR (1951, 1953), Lenin Prize (1961), Wolf Prize (1978), Wigner Medal, Kyoto Prize (1989), MacArthur Fellowship (1994), Steele Prize (2005). Three Orders of Lenin.
• Corresponding member (1953) and full member (1984) of the USSR Academy of Sciences.
• Foreign membership of the American Academy of Arts and Sciences (1964), the US National Academy of Sciences (1970), the Royal Irish Academy (1970), the Royal Swedish Academy (1974), the French Academy of Sciences (1976), the Royal Society (1977) and the Accademia dei Lincei (1988), among others; honorary doctorates from Oxford, Harvard, Paris and elsewhere.

Legacy

• Operator algebras: C*-algebras, the GNS construction and the Gelfand transform are the language of quantum theory and of noncommutative geometry.
• Representation theory: the unitary representations of non-compact groups, Gelfand pairs and the automorphic representations of Gelfand, Graev and Piatetski-Shapiro lead directly into the Langlands program.
• Analysis and physics: the Gelfand–Levitan equation in inverse scattering and solitons; rigged Hilbert spaces in quantum mechanics; integral geometry in tomography.
• Algebra and geometry: Gelfand–Fuchs cohomology, BGG, Gelfand–Kirillov dimension and GKZ systems are standard tools in Lie theory, algebraic geometry and mirror symmetry.
• Teaching: the seminar trained several generations of Moscow mathematicians, and the correspondence schools reached children far from the elite schools of Moscow.

Connections

People linked to Israel Gelfand across this collection. A name without a link has no page here yet. See Israel Gelfand in the network of all 337 lives →

Teachers and advisors

• Andrey Kolmogorov 1932-1935

Students

• Alexander Goncharov
• Alexandre Kirillov
• Andrei Zelevinsky
• David Kazhdan
• Endre Szemerédi
• Felix Berezin
• Georgiy Shilov
• Georgy Adelson-Velsky
• Joseph Bernstein
• Victor Ginzburg

Influenced by

• Norbert Wiener
• Stefan Banach

Influenced

• Maxim Kontsevich
• Michael Atiyah

Colleagues and collaborators

• Carl Ludwig Siegel
• Boris Levitan 1951
• Dmitrii Raikov 1943
• Dmitry Fuchs 1968-1970
• Ilya Piatetski-Shapiro
• Leonid Dikii
• Mark Graev
• Mark Naimark 1943-1950s
• Mikhail Kapranov
• Mikhail Tsetlin 1950
• Mstislav Keldysh
• Naum Vilenkin
• Sergei Fomin

Family

• Sergei Gelfand
• Zorya Shapiro

Also mentioned in this life

• Paul Dirac
• Peter Gustav Lejeune Dirichlet
• Joseph Fourier
• Ivar Fredholm
• Carl Friedrich Gauss
• Harish-Chandra
• Felix Hausdorff
• David Hilbert
• Robert Langlands
• Adrien-Marie Legendre
• Sophus Lie
• Joseph Liouville
• Hendrik Antoon Lorentz
• Frigyes Riesz
• Erwin Schrödinger
• Vito Volterra
• John von Neumann
• Hermann Weyl

Fields

• functional analysis 11 lives
• representation theory 8 lives
• mathematical physics 45 lives
• mathematical biology 3 lives

On CoursesHub

• Functional Analysis · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor, E. F. Robertson, Israel Moiseevich Gelfand, MacTutor History of Mathematics.
2. Israel Gelfand, Wikipedia.
3. A. A. Kirillov, Israel M. Gelfand 1913–2009, Biographical Memoirs, US National Academy of Sciences, 2023.
4. Obituary, New York Times, October 2009 (MacTutor copy).
5. Awards and honors, israelmgelfand.com.
6. Wolf Prize in Mathematics, list of laureates and citations.
7. V. Retakh (ed.), Israel Moiseevich Gelfand, Parts I and II, Notices of the AMS 60 (2013).

Mathematics

1. I. Gelfand, Normierte Ringe, Mat. Sbornik 9 (1941).
2. I. Gelfand, M. Naimark, On the imbedding of normed rings into the ring of operators in Hilbert space, Mat. Sbornik 12 (1943).
3. I. M. Gelfand, D. A. Raikov, G. E. Shilov, Commutative Normed Rings, 1960; English, Chelsea, 1964.
4. I. M. Gelfand, M. L. Tsetlin, Finite-dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk SSSR 71 (1950).
5. I. M. Gelfand, B. M. Levitan, On the determination of a differential equation from its spectral function, Izv. Akad. Nauk SSSR 15 (1951).
6. I. M. Gelfand, G. E. Shilov, N. Ya. Vilenkin, M. I. Graev, Generalized Functions, vols. 1–5, 1958–1966; AMS Chelsea reprint.
7. I. M. Gelfand, M. I. Graev, I. I. Piatetski-Shapiro, Representation Theory and Automorphic Functions, 1966; English, Saunders, 1969.
8. I. M. Gelfand, A. A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Publ. Math. IHÉS 31 (1966).
9. D. B. Fuchs, Cohomology of Infinite-Dimensional Lie Algebras, Consultants Bureau, 1986.
10. I. M. Gelfand, M. M. Kapranov, A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, 1994.
11. S. Helgason, Integral Geometry and Radon Transforms, Springer, 2011.