Andrius Kulikauskas

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See: Math, Clifford algebra, Geometry, Classical Lie Groups

Reach out to Steven Lehar and Math Future.

Clifford algebra, geometric algebra, and applications, D. Lundholm and L. Svensson, 2009.

Study and learn

  • How quadratic forms relate to Lie algebras and Clifford algebras.
  • How quadratic forms express geometric invariants.
  • How to interpret the Cayley-Dickson construction combinatorially and in terms of Clifford algebras.
  • Complex Clifford algebras as relevant for Bott periodicity of order 2.
  • Real Clifford algebras as relevant for Bott periodicity of order 8.

More questions

  • Relate the basis of the geometric algebra to the symmetric functions of eigenvalues.
  • Think of angles as bivectors - the basis for conformal geometry? And vectors for projective geometry? Trivectors for symplectic geometry?
  • Interpret the Clifford product as an action on simplexes.
  • Compare the definition of complex numbers, quaternions in terms of full Clifford algebras where q=-1, and as even subalgebras where q=1.
  • Relate the octonions to Clifford algebras. Learn how the complex, quaternions are defined in the Clifford algebras and how that would proceed further. And then compare that with the Cayley-Dickson construction. Give combinatorial interpretations of both and see how they differ.
  • In geometric algebra, why does e1e1=1 if anticommutativity is parallel?
  • How to identify knots with paths on simplexes?
  • Define an analogue of symmetric functions on Clifford algebra bases and the products of such symmetric functions.





Steven Lehar


  • Geometric algebra is a vector space with the field appended (like a zero?)
  • Quadratic form is one matrix multiplied by another matrix. Or better yet it is summing over all the entries of the matrix where they are written xi xj instead of Aij.
  • Whereas linear form, linear equations are described by a single matrix. Compare the importance of linear equations in geometry.
  • Sameness + difference. (Dvejybės atvaizdas) (Same means "combine like units" and different means "list separate units")
  • Real Clifford algebras are like negative numbers in that they have positive or negative or mixed signature. Whereas complex numbers do not respect any such distinctions. The reals expand the complexes by breaking symmetry.
  • Karoubi generalizes... with hermitian... see his video


Basis elements

  • The basis elements of geometric algebras are simplexes.
  • That is why the dimension is {$2^{n}$}.


  • The square equals +1, 0 or -1.
  • The square is the path from a vertex to itself.
  • This loop removes the vertex and at the same time can A) keep the rest of the simplex (adjusting the edge) - letting go of the slack, B) demolish the entire simplex - by cutting it, C) or it flips the perspective on the simplex - reorienting the slack.


  • Each terms is a simplex with the vertices ordered in a certain way.
  • Thus a term may be identified with a path on a simplex that visits all of its vertices.
  • A positive or negative term may be interpreted as a path for which the slack is at the end (in the positive case, infinitely increasing slack) or at the beginning (in the negative case, finite, decreasing slack).
  • The slack determines the direction we associate to the path. Reversing the order of a segment switches this direction, the parity, the sign, the location and nature of the slack.
  • Parentheses define slack - which is directed. And the negative signs indicate the direction.
  • Slack is always directed because one end is the "center" and the other is the "void". The two directions distinguish unbounded, increasing slack, and bounded, decreasing slack.


  • Thus the elements are simplicial sets.
  • Positive and negative numbers refer to the creating or dismantling of the simplex; the inside or the outside; the center or the totality/complement?
  • Think of a Clifford algebra as a simplex where the center is the origin from which the vectors go, and which is related to the self-standing scalars.
  • Each basis element is a "square root" of 1 or 0 or -1 - symbols of everything. And thus is a division into two. And what about a typical element? Is it a division? And/or an assembly?

Even subalgebra of Clifford algebra.

  • The even subalgebra is generated by the edges.
  • The complex numbers are generated by {$i=e_{1}e_{2}$}.
  • Identify complexes with a line-segment 1-simplex; and with a point 0-simplex.
  • The quaternions are generated by {$i=e_{1}e_{2}, j=e_{3}e_{1}, k=e_{2}e_{3}$}.
  • Identify quaternions with a triangle 2-simplex; and with a line segment 1-simplex.



  • Quadratic form with terms {$a_{ij}x_{i}x_{j}$} is expressed by an upper triangular matrix and hence, when the coefficients are real, as a symmetric matrix {$x_{T}Ax$}.
  • In the real case, the quadratic form q can be put into diagonal form so that it has only square terms with coefficients 1,–1 or 0.
  • If all of the coefficients are 0 then we have an external algebra or Grassmann algebra. Its product is the wedge product.
  • In the complex case, there is only one nondegenerate quadratic form, for which all of the coefficients are 1.
  • Classification of Clifford algebras Periodicity 2 in the complex case, and 8 in the real case.
  • Geometric algebra has all square terms 1 and so it is the same as the complex case.
  • Multiplication by i is rotation by 1/4.
  • Regarding root systems: Two rotations of 135 degrees in the plane give perpendicularity. But two rotations of 120 degrees can also give perpendicularity if we leave the plane and come back to it.


  • In the complex Clifford algebras, Bott periodicity works like a counting process, extending the (dual) chain, interpersed by having two distinct chains (forward and backwards).

Totality = Pseudoscalar

Totality (pseudoscalar) {$e_{1}e_{2}\dots e_{n}$} is an important concept for me and also for Clifford algebras.

The totality commutes with every element precisely when n is odd. That is because to move the element from the right hand side to the left hand side we need to swap with n-1 different elements (switching sign) and 1 element which is the same (thus not switching sign).

This is the basis for a twofold periodicity.

{$e_{1}e_{2}\dots e_{n}\cdot e_{i}=(-1)^{n-1}e_{i}\cdot e_{1}e_{2}\dots e_{n}$}

Squaring the totality yields a fourfold periodicity.

  • {$1\cdot 1=1$}
  • {$e_{1}\cdot e_{1}=Q(e_{1})$}
  • {$e_{1}e_{2}\cdot e_{1}e_{2}=-Q(e_{1})Q(e_{2})$}
  • {$e_{1}e_{2}e_{3}\cdot e_{1}e_{2}e_{3}=-Q(e_{1})Q(e_{2})Q(e_{3})$}
  • {$e_{1}e_{2}e_{3}e_{4}\cdot e_{1}e_{2}e_{3}e_{4}=Q(e_{1})Q(e_{2})Q(e_{3})Q(e_{4})$}
  • {$e_{1}e_{2}e_{3}e_{4}e_{5}\cdot e_{1}e_{2}e_{3}e_{4}e_{5}=Q(e_{1})Q(e_{2})Q(e_{3})Q(e_{4})Q(e_{5})$}

This is because in the sum 0+1+2+3+4... we start with even and then have

  • even + odd = odd
  • odd + even = odd
  • odd + odd = even
  • even + even = even

Where even means "stay the same" and odd means "switch". Note that here the two are dual!

Classifying Clifford algebras

For complex numbers there is essentially only one quadratic form {$\sum_{i=1}^{n}u_{i}^2$} because squaring a complex number can generate -1 as needed. Whereas the reals discriminate real and imaginary square roots and so they have a signature {$\sum_{i=1}^{p}u_{i}^2 - \sum_{i=1}^{q}u_{i}^2$}.


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Puslapis paskutinį kartą pakeistas 2018 kovo 23 d., 12:19