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In the mathematics of moduli theory, given an algebraic, reductive, Lie group and a finitely generated group , the -*character variety of* is a space of equivalence classes of group homomorphisms

- Formulation
- Examples
- Variants
- Connection to geometry
- Connection to skein modules
- See also
- References

More precisely, acts on by conjugation, and two homomorphisms are defined to be equivalent (denoted ) if and only if their orbit closures intersect. This is the weakest equivalence relation on the set of conjugation orbits that yields a Hausdorff space.

Formally, and when the algebraic group is defined over the complex numbers , the -character variety is the spectrum of prime ideals of the ring of invariants (i.e., the GIT quotient).

Here more generally one can consider algebraically closed fields of prime characteristic. In this generality, character varieties are only algebraic sets and are not actual varieties. To avoid technical issues, one often considers the associated reduced space by dividing by the radical of 0 (eliminating nilpotents). However, this does not necessarily yield an irreducible space either. Moreover, if we replace the complex group by a real group we may not even get an algebraic set. In particular, a maximal compact subgroup generally gives a semi-algebraic set. On the other hand, whenever is free we always get an honest variety; it is singular however.

For example, if and is free of rank two, then the character variety is , since by a theorem of Robert Fricke, Felix Klein, and Henri G. Vogt, its coordinate ring is isomorphic to the complex polynomial ring in 3 variables, . Restricting to gives a closed real three-dimensional ball (semi-algebraic, but not algebraic).

Another example, also studied by Vogt and Fricke–Klein is the case with and is free of rank three. Then the character variety is isomorphic to the hypersurface in given by the equation .

This construction of the character variety is not necessarily the same as that of Marc Culler and Peter Shalen (generated by evaluations of traces), although when they do agree, since Claudio Procesi has shown that in this case the ring of invariants is in fact generated by only traces. Since trace functions are invariant by all inner automorphisms, the Culler–Shalen construction essentially assumes that we are acting by on even if .^{[ clarification needed ]}

For instance, when is a free group of rank 2 and , the conjugation action is trivial and the -character variety is the torus

But the trace algebra is a strictly small subalgebra (there are less invariants). This provides an involutive action on the torus that needs to be accounted for to yield the Culler–Shalen character variety. The involution on this torus yields a 2-sphere. The point is that up to -conjugation all points are distinct, but the trace identifies elements with differing anti-diagonal elements (the involution).

There is an interplay between these moduli spaces and the moduli spaces of principal bundles, vector bundles, Higgs bundles, and geometric structures on topological spaces, given generally by the observation that, at least locally, equivalent objects in these categories are parameterized by conjugacy classes of holonomy homomorphisms. In other words, with respect to a base space for the bundles or a fixed topological space for the geometric structures, the holonomy homomorphism is a group homomorphism from to the structure group of the bundle.

The coordinate ring of the character variety has been related to skein modules in knot theory.^{ [1] }^{ [2] } The skein module is roughly a deformation (or quantization) of the character variety. It is closely related to topological quantum field theory in dimension 2+1.

In mathematics, a **Lie algebra** is a vector space together with an operation called the **Lie bracket**, an alternating bilinear map , that satisfies the Jacobi identity. The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.

In mathematics and theoretical physics, a **representation of a Lie group** is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the **Chern classes** are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc.

In mathematics, the **adjoint representation** of a Lie group *G* is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if *G* is , the Lie group of real *n*-by-*n* invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible *n*-by-*n* matrix to an endomorphism of the vector space of all linear transformations of defined by: .

In the mathematical field of representation theory, a **Lie algebra representation** or **representation of a Lie algebra** is a way of writing a Lie algebra as a set of matrices in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.

In mathematics, the **Chern–Weil homomorphism** is a basic construction in **Chern–Weil theory** that computes topological invariants of vector bundles and principal bundles on a smooth manifold *M* in terms of connections and curvature representing classes in the de Rham cohomology rings of *M*. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes.

In mathematics, specifically in algebraic geometry, the **Grothendieck–Riemann–Roch theorem** is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces.

The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly established, and the conjunction of these two theories is the study of the infinite-dimensional unitary representations of the Lorentz group. These have both historical importance in mainstream physics, as well as connections to more speculative present-day theories.

In mathematics, a **super vector space** is a -graded vector space, that is, a vector space over a field with a given decomposition of subspaces of grade and grade . The study of super vector spaces and their generalizations is sometimes called **super linear algebra**. These objects find their principal application in theoretical physics where they are used to describe the various algebraic aspects of supersymmetry.

In mathematics, a **zonal spherical function** or often just **spherical function** is a function on a locally compact group *G* with compact subgroup *K* that arises as the matrix coefficient of a *K*-invariant vector in an irreducible representation of *G*. The key examples are the matrix coefficients of the *spherical principal series*, the irreducible representations appearing in the decomposition of the unitary representation of *G* on *L*^{2}(*G*/*K*). In this case the commutant of *G* is generated by the algebra of biinvariant functions on *G* with respect to *K* acting by right convolution. It is commutative if in addition *G*/*K* is a symmetric space, for example when *G* is a connected semisimple Lie group with finite centre and *K* is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant *L*^{1} functions is larger; when *G* is a semisimple Lie group with maximal compact subgroup *K*, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.

This is a **glossary of algebraic geometry**.

In algebraic geometry, an affine **GIT quotient**, or affine **geometric invariant theory quotient**, of an affine scheme with an action by a group scheme *G* is the affine scheme , the prime spectrum of the ring of invariants of *A*, and is denoted by . A GIT quotient is a categorical quotient: any invariant morphism uniquely factors through it.

In mathematics, **Lie group–Lie algebra correspondence** allows one to study Lie groups, which are geometric objects, in terms of Lie algebras, which are linear objects. In this article, a Lie group refers to a real Lie group. For the complex and *p*-adic cases, see complex Lie group and *p*-adic Lie group.

In algebraic geometry and differential geometry, the **Nonabelian Hodge correspondence** or **Corlette–Simpson correspondence** is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.

In algebraic geometry, a **level structure** on a space *X* is an extra structure attached to *X* that shrinks or eliminates the automorphism group of *X*, by demanding automorphisms to preserve the level structure; attaching a level structure is often phrased as **rigidifying** the geometry of *X*.

In algebra, the **fixed-point subring** of an automorphism *f* of a ring *R* is the subring of the fixed points of *f*:

This is a glossary of properties and concepts in algebraic topology in mathematics.

In mathematics, the **moduli stack of elliptic curves**, denoted as or , is an algebraic stack over classifying elliptic curves. Note that it is a special case of the Moduli stack of algebraic curves . In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme to it correspond to elliptic curves over . The construction of this space spans over a century because of the various generalizations of elliptic curves as the field has developed. All of these generalizations are contained in .

This is a **glossary of representation theory** in mathematics.

In mathematics, the **representation theory of semisimple Lie algebras** is one of crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the **Cartan–Weyl theory**. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra ; in particular, it gives a way to parametrize irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight.

- ↑ Doug Bullock,
*Rings of -characters and the Kauffman bracket skein module*, Commentarii Mathematici Helvetici**72**(1997), no. 4, 521–542. MR 1600138 - ↑ Józef H. Przytycki, Adam S. Sikora,
*On skein algebras and -character varieties*, Topology**39**(2000), no. 1, 115–148. MR 1710996

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