Vladimir Fock - Steinberg symbol and tau function @IhesFr

Uploaded November 2025 | Updated September 2026, 3 weeks ago

Steinberg symbol is a multiplicative bicharacter of a ring satisfying additional (Steinberg) relation. Beilinson in 1982 suggested an explicit expression for a canonical symbol for the ring functions on a circle. This symbol can be interpreted as a multiplicative analogue of a residue and also as a cocycle defining the Heisenberg group. We will show how one can define the boson-fermion correspondence using the symbol and show that the finite-gap tau-function can be interpreted as an automorphic form.

Vladimir Fock (IRMA Strasbourg)

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Find this and many more scientific videos on carmin.tv - a French video platform for mathematics and their interactions with other sciences offering extra functionalities tailored to meet the needs of the research community.

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Vladimir Fock - Steinberg symbol and tau function

$Arthemy Kiselev - The Wronskians over Multidimension and Homotopy Lie Algebras The Wronskian determinant of $N$ functions in one variable $x$ helps us verify that they are linearly independent on an interval $(a, b)$ in ${mathbb R}$. Can we have an equally convenient procedure over multidimensional spaces ${mathbb R}^d$ with Cartesian coordinates $x, y, z, . . .$ ? Yes we can; let us study the definition of Wronskians for functions in many variables, and let us explore which differential-algebraic identities these structures satisfy. To see why the Wronskian determinants actually do satisfy a set of quadratic, Jacobi-type identities, we observe first that the Wronskian of size $2 times 2$ results from commutation of vector fields on the real line ${mathbb R}$. From differential geometry we know that vector fields are differential operators of strict order $p = 1$. By taking the alternated composition of $N = 2p$ differential operators of strict order $p gt 0$ on the affine line, we obtain the operator of same order $p$ with the Wronskian determinant for coefficient. As we had the Jacobi identity for the Lie algebra of vector fields, so we establish the (table of) quadratic, Jacobi-type identities for higher-order Wronskians of $N gt 1$ arguments. (In string theory, these identities govern homotopy deformations of Lie algebras, here of vector fields.) We prove that the new Wronskians over multidimensional base with $d$ coordinates $x, y, z,. . .$ do satisfy the (table of) identities for strongly homotopy Lie algebras. The problem is to understand how fast the dimension grows under iterated $N$-ary brackets. We spot a countable chain of finite-dimensional homotopy Lie algebras that generalize the vector field realization of $sl(2)$ on ${mathbb R}$; we explicitly calculate all the structure constants. Yet the four-dimensional analogue of $sl(2)$ over the plane ${mathbb R}^2$ with Cartesian coordinates $(x, y)$, now with ternary bracket from the Wronskian, is so far the only known finite-dimensional homotopy Lie algebra of this type over base dimension $gt 1$. The hunt is on; we conclude that Lie algebra $sl(2)$ is the prototype not only for semisimple complex Lie algebras encoded by root systems but also for countably many $N$-ary homotopy Lie algebras. Arthemy Kiselev (University of Groningen) Find this and many more scientific videos on https://www.carmin.tv/ - a French video platform for mathematics and their interactions with other sciences offering extra functionalities tailored to meet the needs of the research community.$

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